\(\int \frac {(f+g x)^2 (a+b \log (c (d+e x^2)^p))}{(h x)^{7/2}} \, dx\) [614]

Optimal result

Integrand size = 31, antiderivative size = 935 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=-\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{5/4} f^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}-\frac {4 \sqrt {2} b e^{3/4} f g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} f^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} f g p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}-\frac {\sqrt {2} b e^{5/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}} \]

[Out]

-2/5*f^2*(a+b*ln(c*(e*x^2+d)^p))/h/(h*x)^(5/2)-4/3*f*g*(a+b*ln(c*(e*x^2+d)^p))/h^2/(h*x)^(3/2)+2/5*b*e^(5/4)*f
^2*p*arctan(1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(5/4)/h^(7/2)-4/3*b*e^(3/4)*f*g*p*arctan(
1-e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(3/4)/h^(7/2)-2*b*e^(1/4)*g^2*p*arctan(1-e^(1/4)*2^(1
/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(1/4)/h^(7/2)-2/5*b*e^(5/4)*f^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1
/2)/d^(1/4)/h^(1/2))*2^(1/2)/d^(5/4)/h^(7/2)+4/3*b*e^(3/4)*f*g*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/
h^(1/2))*2^(1/2)/d^(3/4)/h^(7/2)+2*b*e^(1/4)*g^2*p*arctan(1+e^(1/4)*2^(1/2)*(h*x)^(1/2)/d^(1/4)/h^(1/2))*2^(1/
2)/d^(1/4)/h^(7/2)-1/5*b*e^(5/4)*f^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2
))*2^(1/2)/d^(5/4)/h^(7/2)-2/3*b*e^(3/4)*f*g*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2)*(h
*x)^(1/2))*2^(1/2)/d^(3/4)/h^(7/2)+b*e^(1/4)*g^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)-d^(1/4)*e^(1/4)*2^(1/2
)*(h*x)^(1/2))*2^(1/2)/d^(1/4)/h^(7/2)+1/5*b*e^(5/4)*f^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4)*e^(1/4
)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/d^(5/4)/h^(7/2)+2/3*b*e^(3/4)*f*g*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^(1/4
)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/d^(3/4)/h^(7/2)-b*e^(1/4)*g^2*p*ln(d^(1/2)*h^(1/2)+x*e^(1/2)*h^(1/2)+d^
(1/4)*e^(1/4)*2^(1/2)*(h*x)^(1/2))*2^(1/2)/d^(1/4)/h^(7/2)-8/5*b*e*f^2*p/d/h^3/(h*x)^(1/2)-2*g^2*(a+b*ln(c*(e*
x^2+d)^p))/h^3/(h*x)^(1/2)

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 935, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2517, 2526, 2505, 331, 303, 1176, 631, 210, 1179, 642, 217} \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\frac {2 \sqrt {2} b e^{5/4} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f^2}{5 h (h x)^{5/2}}-\frac {\sqrt {2} b e^{5/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{5 d^{5/4} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f^2}{5 d^{5/4} h^{7/2}}-\frac {8 b e p f^2}{5 d h^3 \sqrt {h x}}-\frac {4 \sqrt {2} b e^{3/4} g p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right ) f}{3 d^{3/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} g p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right ) f}{3 d^{3/4} h^{7/2}}-\frac {4 g \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right ) f}{3 h^2 (h x)^{3/2}}-\frac {2 \sqrt {2} b e^{3/4} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} g p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right ) f}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{h^3 \sqrt {h x}}+\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {e} \sqrt {h} x+\sqrt {d} \sqrt {h}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}} \]

[In]

Int[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(7/2),x]

[Out]

(-8*b*e*f^2*p)/(5*d*h^3*Sqrt[h*x]) + (2*Sqrt[2]*b*e^(5/4)*f^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4
)*Sqrt[h])])/(5*d^(5/4)*h^(7/2)) - (4*Sqrt[2]*b*e^(3/4)*f*g*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*
Sqrt[h])])/(3*d^(3/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(1/4)*g^2*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sq
rt[h])])/(d^(1/4)*h^(7/2)) - (2*Sqrt[2]*b*e^(5/4)*f^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h
])])/(5*d^(5/4)*h^(7/2)) + (4*Sqrt[2]*b*e^(3/4)*f*g*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])
])/(3*d^(3/4)*h^(7/2)) + (2*Sqrt[2]*b*e^(1/4)*g^2*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])
/(d^(1/4)*h^(7/2)) - (2*f^2*(a + b*Log[c*(d + e*x^2)^p]))/(5*h*(h*x)^(5/2)) - (4*f*g*(a + b*Log[c*(d + e*x^2)^
p]))/(3*h^2*(h*x)^(3/2)) - (2*g^2*(a + b*Log[c*(d + e*x^2)^p]))/(h^3*Sqrt[h*x]) - (Sqrt[2]*b*e^(5/4)*f^2*p*Log
[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(5*d^(5/4)*h^(7/2)) - (2*Sqrt[2]*b*
e^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(7/2)
) + (Sqrt[2]*b*e^(1/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^
(1/4)*h^(7/2)) + (Sqrt[2]*b*e^(5/4)*f^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sq
rt[h*x]])/(5*d^(5/4)*h^(7/2)) + (2*Sqrt[2]*b*e^(3/4)*f*g*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h]*x + Sqrt[2]*d
^(1/4)*e^(1/4)*Sqrt[h*x]])/(3*d^(3/4)*h^(7/2)) - (Sqrt[2]*b*e^(1/4)*g^2*p*Log[Sqrt[d]*Sqrt[h] + Sqrt[e]*Sqrt[h
]*x + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[h*x]])/(d^(1/4)*h^(7/2))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2517

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(r
_.), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/h, Subst[Int[x^(k*(m + 1) - 1)*(f + g*(x^k/h))^r*(a + b*Lo
g[c*(d + e*(x^(k*n)/h^n))^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, g, h, p, r}, x] && Fract
ionQ[m] && IntegerQ[n] && IntegerQ[r]

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (f+\frac {g x^2}{h}\right )^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^6} \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {f^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{x^6}+\frac {2 f g \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h x^4}+\frac {g^2 \left (a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )\right )}{h^2 x^2}\right ) \, dx,x,\sqrt {h x}\right )}{h} \\ & = \frac {\left (2 g^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^2} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {(4 f g) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^4} \, dx,x,\sqrt {h x}\right )}{h^2}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e x^4}{h^2}\right )^p\right )}{x^6} \, dx,x,\sqrt {h x}\right )}{h} \\ & = -\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}+\frac {\left (8 b e g^2 p\right ) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^5}+\frac {(16 b e f g p) \text {Subst}\left (\int \frac {1}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 h^4}+\frac {\left (8 b e f^2 p\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (d+\frac {e x^4}{h^2}\right )} \, dx,x,\sqrt {h x}\right )}{5 h^3} \\ & = -\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}-\frac {\left (8 b e^2 f^2 p\right ) \text {Subst}\left (\int \frac {x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{5 d h^5}+\frac {(8 b e f g p) \text {Subst}\left (\int \frac {\sqrt {d} h-\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 \sqrt {d} h^5}+\frac {(8 b e f g p) \text {Subst}\left (\int \frac {\sqrt {d} h+\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{3 \sqrt {d} h^5}-\frac {\left (4 b \sqrt {e} g^2 p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h-\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^5}+\frac {\left (4 b \sqrt {e} g^2 p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h+\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{h^5} \\ & = -\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}+\frac {\left (4 b e^{3/2} f^2 p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h-\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{5 d h^5}-\frac {\left (4 b e^{3/2} f^2 p\right ) \text {Subst}\left (\int \frac {\sqrt {d} h+\sqrt {e} x^2}{d+\frac {e x^4}{h^2}} \, dx,x,\sqrt {h x}\right )}{5 d h^5}-\frac {\left (2 \sqrt {2} b e^{3/4} f g p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}-\frac {\left (2 \sqrt {2} b e^{3/4} f g p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}+\frac {\left (\sqrt {2} b \sqrt [4]{e} g^2 p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {\left (\sqrt {2} b \sqrt [4]{e} g^2 p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {\left (4 b \sqrt {e} f g p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{3 \sqrt {d} h^3}+\frac {\left (4 b \sqrt {e} f g p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{3 \sqrt {d} h^3}+\frac {\left (2 b g^2 p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{h^3}+\frac {\left (2 b g^2 p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{h^3} \\ & = -\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}-\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {\left (\sqrt {2} b e^{5/4} f^2 p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}+2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}-\frac {\left (\sqrt {2} b e^{5/4} f^2 p\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h}}{\sqrt [4]{e}}-2 x}{-\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}+\frac {\left (4 \sqrt {2} b e^{3/4} f g p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {\left (4 \sqrt {2} b e^{3/4} f g p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}+\frac {\left (2 \sqrt {2} b \sqrt [4]{e} g^2 p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {\left (2 \sqrt {2} b \sqrt [4]{e} g^2 p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {\left (2 b e f^2 p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{5 d h^3}-\frac {\left (2 b e f^2 p\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {d} h}{\sqrt {e}}+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt {h x}\right )}{5 d h^3} \\ & = -\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}-\frac {4 \sqrt {2} b e^{3/4} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}-\frac {\sqrt {2} b e^{5/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {\left (2 \sqrt {2} b e^{5/4} f^2 p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}+\frac {\left (2 \sqrt {2} b e^{5/4} f^2 p\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}} \\ & = -\frac {8 b e f^2 p}{5 d h^3 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{5/4} f^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}-\frac {4 \sqrt {2} b e^{3/4} f g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} f^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}+\frac {4 \sqrt {2} b e^{3/4} f g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g^2 p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{7/2}}-\frac {2 f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {4 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {2 g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^3 \sqrt {h x}}-\frac {\sqrt {2} b e^{5/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} f^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} f g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}-\frac {\sqrt {2} b \sqrt [4]{e} g^2 p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{7/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.64 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.36 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\frac {2 x^{7/2} \left (\frac {2 b \sqrt [4]{e} g^2 p \left (\arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\text {arctanh}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {4 b e f^2 p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\frac {e x^2}{d}\right )}{5 d \sqrt {x}}-\frac {\sqrt {2} b e^{3/4} f g p \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )\right )}{3 d^{3/4}}-\frac {f^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 x^{5/2}}-\frac {2 f g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {g^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{7/2}} \]

[In]

Integrate[((f + g*x)^2*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(7/2),x]

[Out]

(2*x^(7/2)*((2*b*e^(1/4)*g^2*p*(ArcTan[(e^(1/4)*Sqrt[x])/(-d)^(1/4)] + ArcTanh[(d*e^(1/4)*Sqrt[x])/(-d)^(5/4)]
))/(-d)^(1/4) - (4*b*e*f^2*p*Hypergeometric2F1[-1/4, 1, 3/4, -((e*x^2)/d)])/(5*d*Sqrt[x]) - (Sqrt[2]*b*e^(3/4)
*f*g*p*(2*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] + Lo
g[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x] - Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqr
t[e]*x]))/(3*d^(3/4)) - (f^2*(a + b*Log[c*(d + e*x^2)^p]))/(5*x^(5/2)) - (2*f*g*(a + b*Log[c*(d + e*x^2)^p]))/
(3*x^(3/2)) - (g^2*(a + b*Log[c*(d + e*x^2)^p]))/Sqrt[x]))/(h*x)^(7/2)

Maple [F]

\[\int \frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {7}{2}}}d x\]

[In]

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(7/2),x)

[Out]

int((g*x+f)^2*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(7/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2205 vs. \(2 (653) = 1306\).

Time = 0.46 (sec) , antiderivative size = 2205, normalized size of antiderivative = 2.36 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(d*h^4*x^3*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85
500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^
2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6
+ 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 + 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*
e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 1
0*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt((d^2*h^7*sqrt(-(81*b^4*e
^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)
*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))) - d*h^4*x^3*sqrt((d^2*h^7*sqrt(-(81*b
^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*
g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*
d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 - 32
*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 -
85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 10*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f
^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt((d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^
4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 60*(b^2*e^2*f^3*g - 5*b
^2*d*e*f*g^3)*p^2)/(d^2*h^7))) - d*h^4*x^3*sqrt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 401
50*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 60*(b^2*e^2*f^3*g
- 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(32*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4
- 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*d^4*e*g^8)*sqrt(h*x)*p^3 + 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(8
1*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4
*e*g^8)*p^4/(d^5*h^14)) + 10*(9*b^2*d^2*e^3*f^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sq
rt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^
2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))) + d*h^4*x
^3*sqrt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e
^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) - 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))*log(3
2*(81*b^3*e^5*f^8 - 1620*b^3*d*e^4*f^6*g^2 + 2150*b^3*d^2*e^3*f^4*g^4 - 40500*b^3*d^3*e^2*f^2*g^6 + 50625*b^3*
d^4*e*g^8)*sqrt(h*x)*p^3 - 32*(3*(d^4*e*f^2 - 5*d^5*g^2)*h^11*sqrt(-(81*b^4*e^5*f^8 - 3420*b^4*d*e^4*f^6*g^2 +
40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^14)) + 10*(9*b^2*d^2*
e^3*f^5*g - 190*b^2*d^3*e^2*f^3*g^3 + 225*b^2*d^4*e*f*g^5)*h^4*p^2)*sqrt(-(d^2*h^7*sqrt(-(81*b^4*e^5*f^8 - 342
0*b^4*d*e^4*f^6*g^2 + 40150*b^4*d^2*e^3*f^4*g^4 - 85500*b^4*d^3*e^2*f^2*g^6 + 50625*b^4*d^4*e*g^8)*p^4/(d^5*h^
14)) - 60*(b^2*e^2*f^3*g - 5*b^2*d*e*f*g^3)*p^2)/(d^2*h^7))) + (10*a*d*f*g*x + 3*a*d*f^2 + 3*(4*b*e*f^2*p + 5*
a*d*g^2)*x^2 + (15*b*d*g^2*p*x^2 + 10*b*d*f*g*p*x + 3*b*d*f^2*p)*log(e*x^2 + d) + (15*b*d*g^2*x^2 + 10*b*d*f*g
*x + 3*b*d*f^2)*log(c))*sqrt(h*x))/(d*h^4*x^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**2*(a+b*ln(c*(e*x**2+d)**p))/(h*x)**(7/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 1088, normalized size of antiderivative = 1.16 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x, algorithm="maxima")

[Out]

-2*b*g^2*x^3*log((e*x^2 + d)^p*c)/(h*x)^(7/2) + 1/5*b*e*f^2*p*(e*(sqrt(2)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1
/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*s
qrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(
2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1
/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)) - sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h
) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(
1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)))/d - 8/(sqrt(h*x)*d))/h^3 - b*e*g^2*p*
(sqrt(2)*log(sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt
(2)*log(sqrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(1/4)*e^(3/4)) - sqrt(2)*l
og(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sq
rt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(e)) -
sqrt(2)*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)
*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sq
rt(e)))/h^3 - 2*a*g^2*x^3/(h*x)^(7/2) - 4/3*b*f*g*x^2*log((e*x^2 + d)^p*c)/(h*x)^(7/2) + 2/3*(sqrt(2)*h^2*log(
sqrt(e)*h*x + sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) - sqrt(2)*h^2*log(s
qrt(e)*h*x - sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(1/4) + sqrt(d)*h)/((d*h^2)^(3/4)*e^(1/4)) + sqrt(2)*h*log(-(sq
rt(2)*sqrt(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt(-sqrt(d)*s
qrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(d)) + sqrt(2)
*h*log(-(sqrt(2)*sqrt(-sqrt(d)*sqrt(e)*h) - sqrt(2)*(d*h^2)^(1/4)*e^(1/4) - 2*sqrt(h*x)*sqrt(e))/(sqrt(2)*sqrt
(-sqrt(d)*sqrt(e)*h) + sqrt(2)*(d*h^2)^(1/4)*e^(1/4) + 2*sqrt(h*x)*sqrt(e)))/(sqrt(-sqrt(d)*sqrt(e)*h)*sqrt(d)
))*b*e*f*g*p/h^4 - 4/3*a*f*g*x^2/(h*x)^(7/2) - 2/5*b*f^2*log((e*x^2 + d)^p*c)/((h*x)^(5/2)*h) - 2/5*a*f^2/((h*
x)^(5/2)*h)

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 675, normalized size of antiderivative = 0.72 \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=-\frac {\frac {2 \, {\left (15 \, b g^{2} h^{3} p x^{2} + 10 \, b f g h^{3} p x + 3 \, b f^{2} h^{3} p\right )} \log \left (e h^{2} x^{2} + d h^{2}\right )}{\sqrt {h x} h^{2} x^{2}} - \frac {2 \, {\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p - 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p + 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} p\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} + 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e^{2} h} - \frac {2 \, {\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p - 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p + 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} p\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} - 2 \, \sqrt {h x}\right )}}{2 \, \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}}}\right )}{d^{2} e^{2} h} - \frac {{\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p + 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p - 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} p\right )} \log \left (h x + \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e^{2} h} + \frac {{\left (10 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {1}{4}} b d e^{2} f g h p + 3 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b e f^{2} p - 15 \, \sqrt {2} \left (d e^{3} h^{2}\right )^{\frac {3}{4}} b d g^{2} p\right )} \log \left (h x - \sqrt {2} \left (\frac {d h^{2}}{e}\right )^{\frac {1}{4}} \sqrt {h x} + \sqrt {\frac {d h^{2}}{e}}\right )}{d^{2} e^{2} h} - \frac {2 \, {\left (15 \, b d g^{2} h^{3} p x^{2} \log \left (h^{2}\right ) - 12 \, b e f^{2} h^{3} p x^{2} + 10 \, b d f g h^{3} p x \log \left (h^{2}\right ) - 15 \, b d g^{2} h^{3} x^{2} \log \left (c\right ) - 15 \, a d g^{2} h^{3} x^{2} + 3 \, b d f^{2} h^{3} p \log \left (h^{2}\right ) - 10 \, b d f g h^{3} x \log \left (c\right ) - 10 \, a d f g h^{3} x - 3 \, b d f^{2} h^{3} \log \left (c\right ) - 3 \, a d f^{2} h^{3}\right )}}{\sqrt {h x} d h^{2} x^{2}}}{15 \, h^{4}} \]

[In]

integrate((g*x+f)^2*(a+b*log(c*(e*x^2+d)^p))/(h*x)^(7/2),x, algorithm="giac")

[Out]

-1/15*(2*(15*b*g^2*h^3*p*x^2 + 10*b*f*g*h^3*p*x + 3*b*f^2*h^3*p)*log(e*h^2*x^2 + d*h^2)/(sqrt(h*x)*h^2*x^2) -
2*(10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^2*f*g*h*p - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p + 15*sqrt(2)*(d*e^3*h^
2)^(3/4)*b*d*g^2*p)*arctan(1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) + 2*sqrt(h*x))/(d*h^2/e)^(1/4))/(d^2*e^2*h) -
2*(10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^2*f*g*h*p - 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p + 15*sqrt(2)*(d*e^3*h^
2)^(3/4)*b*d*g^2*p)*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2/e)^(1/4) - 2*sqrt(h*x))/(d*h^2/e)^(1/4))/(d^2*e^2*h) -
(10*sqrt(2)*(d*e^3*h^2)^(1/4)*b*d*e^2*f*g*h*p + 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p - 15*sqrt(2)*(d*e^3*h^2
)^(3/4)*b*d*g^2*p)*log(h*x + sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d^2*e^2*h) + (10*sqrt(2)*(d*e
^3*h^2)^(1/4)*b*d*e^2*f*g*h*p + 3*sqrt(2)*(d*e^3*h^2)^(3/4)*b*e*f^2*p - 15*sqrt(2)*(d*e^3*h^2)^(3/4)*b*d*g^2*p
)*log(h*x - sqrt(2)*(d*h^2/e)^(1/4)*sqrt(h*x) + sqrt(d*h^2/e))/(d^2*e^2*h) - 2*(15*b*d*g^2*h^3*p*x^2*log(h^2)
- 12*b*e*f^2*h^3*p*x^2 + 10*b*d*f*g*h^3*p*x*log(h^2) - 15*b*d*g^2*h^3*x^2*log(c) - 15*a*d*g^2*h^3*x^2 + 3*b*d*
f^2*h^3*p*log(h^2) - 10*b*d*f*g*h^3*x*log(c) - 10*a*d*f*g*h^3*x - 3*b*d*f^2*h^3*log(c) - 3*a*d*f^2*h^3)/(sqrt(
h*x)*d*h^2*x^2))/h^4

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{7/2}} \,d x \]

[In]

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(7/2),x)

[Out]

int(((f + g*x)^2*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(7/2), x)