3.7.9 \(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{7/2}} \, dx\) [609]

Optimal. Leaf size=620 \[ -\frac {8 b e f p}{5 d h^3 \sqrt {h x}}+\frac {2 \sqrt {2} b e^{5/4} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{3/4} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{5/4} f p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{5 d^{5/4} h^{7/2}}+\frac {2 \sqrt {2} b e^{3/4} g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{7/2}}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac {\sqrt {2} b e^{5/4} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}-\frac {\sqrt {2} b e^{3/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{5 d^{5/4} h^{7/2}}+\frac {\sqrt {2} b e^{3/4} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{7/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 620, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2517, 2526, 2505, 331, 303, 1176, 631, 210, 1179, 642, 217} \begin {gather*} -\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac {2 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 \sqrt {2} b e^{5/4} f p \text {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 {2 \sqrt {2} b e^{5/4} f p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{5 d^{5/4} h^{7/2}}-\frac {2 \sqrt {2} b e^{3/4} g p \text {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 e^{3/4} g p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{3 d^{3/4} h^{7/2}}-\frac {\sqrt {2} b e^{5/4} f p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{5 d^{5/4} h^{7/2}}+\frac {\sqrt {2} b e^{5/4} f p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{5 d^{5/4} h^{7/2}}-\frac {\sqrt {2} b e^{3/4} g p \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 d^{3/4} h^{7/2}}+\frac {\sqrt {2} b e^{3/4} g p \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}+\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x\right )}{3 d^{3/4} h^{7/2}}-\frac {8 b e f p}{5 d h^3 \sqrt {h x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.12, size = 309, normalized size = 0.50 \begin {gather*} \frac {2 x^{7/2} \left (-\frac {4 b e f p \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\frac {e x^2}{d}\right )}{5 d \sqrt {x}}-\frac {1}{6} b g p \left (\frac {2 \left (\frac {\sqrt {2} e^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{d}}-\frac {\sqrt {2} e^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{d}}\right )}{\sqrt {d}}+\frac {\frac {\sqrt {2} e^{3/4} \log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt [4]{d}}-\frac {\sqrt {2} e^{3/4} \log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {x}+\sqrt {e} x\right )}{\sqrt [4]{d}}}{\sqrt {d}}\right )-\frac {f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 x^{5/2}}-\frac {g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}\right )}{(h x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(7/2)*((-4*b*e*f*p*Hypergeometric2F1[-1/4, 1, 3/4, -((e*x^2)/d)])/(5*d*Sqrt[x]) - (b*g*p*((2*((Sqrt[2]*e^
(3/4)*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/d^(1/4) - (Sqrt[2]*e^(3/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sq
rt[x])/d^(1/4)])/d^(1/4)))/Sqrt[d] + ((Sqrt[2]*e^(3/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]
*x])/d^(1/4) - (Sqrt[2]*e^(3/4)*Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/d^(1/4))/Sqrt[d]))
/6 - (f*(a + b*Log[c*(d + e*x^2)^p]))/(5*x^(5/2)) - (g*(a + b*Log[c*(d + e*x^2)^p]))/(3*x^(3/2))))/(h*x)^(7/2)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {7}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 512, normalized size = 0.83 \begin {gather*} \frac {b f p {\left (\frac {{\left (\frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}}} - \frac {\sqrt {2} e^{\left (-\frac {3}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {1}{4}}} - \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}} - \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {3}{4}\right )}}{\sqrt {\sqrt {d} h}}\right )} e}{d} - \frac {8}{\sqrt {h x} d}\right )} e}{5 \, h^{3}} - \frac {2 \, b g x^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{3 \, \left (h x\right )^{\frac {7}{2}}} + \frac {{\left (\frac {\sqrt {2} h^{2} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} + \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} h^{2} e^{\left (-\frac {1}{4}\right )} \log \left (h x e^{\frac {1}{2}} - \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\frac {1}{4}} + \sqrt {d} h\right )}{\left (d h^{2}\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} h \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} + 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}} + \frac {2 \, \sqrt {2} h \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\frac {1}{4}} - 2 \, \sqrt {h x} e^{\frac {1}{2}}\right )} e^{\left (-\frac {1}{4}\right )}}{2 \, \sqrt {\sqrt {d} h}}\right ) e^{\left (-\frac {1}{4}\right )}}{\sqrt {\sqrt {d} h} \sqrt {d}}\right )} b g p e}{3 \, h^{4}} - \frac {2 \, a g x^{2}}{3 \, \left (h x\right )^{\frac {7}{2}}} - \frac {2 \, b f \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{5 \, \left (h x\right )^{\frac {5}{2}} h} - \frac {2 \, a f}{5 \, \left (h x\right )^{\frac {5}{2}} h} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1393 vs. \(2 (407) = 814\).
time = 0.45, size = 1393, normalized size = 2.25 \begin {gather*} -\frac {2 \, {\left (d h^{4} x^{3} \sqrt {\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} + 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}} \log \left (-32 \, {\left (625 \, b^{3} d^{2} g^{4} p^{3} e^{2} - 81 \, b^{3} f^{4} p^{3} e^{4}\right )} \sqrt {h x} + 32 \, {\left (3 \, d^{4} f h^{11} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} + 125 \, b^{2} d^{3} g^{3} h^{4} p^{2} e - 45 \, b^{2} d^{2} f^{2} g h^{4} p^{2} e^{2}\right )} \sqrt {\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} + 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}}\right ) - d h^{4} x^{3} \sqrt {\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} + 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}} \log \left (-32 \, {\left (625 \, b^{3} d^{2} g^{4} p^{3} e^{2} - 81 \, b^{3} f^{4} p^{3} e^{4}\right )} \sqrt {h x} - 32 \, {\left (3 \, d^{4} f h^{11} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} + 125 \, b^{2} d^{3} g^{3} h^{4} p^{2} e - 45 \, b^{2} d^{2} f^{2} g h^{4} p^{2} e^{2}\right )} \sqrt {\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} + 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}}\right ) - d h^{4} x^{3} \sqrt {-\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} - 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}} \log \left (-32 \, {\left (625 \, b^{3} d^{2} g^{4} p^{3} e^{2} - 81 \, b^{3} f^{4} p^{3} e^{4}\right )} \sqrt {h x} + 32 \, {\left (3 \, d^{4} f h^{11} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} - 125 \, b^{2} d^{3} g^{3} h^{4} p^{2} e + 45 \, b^{2} d^{2} f^{2} g h^{4} p^{2} e^{2}\right )} \sqrt {-\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} - 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}}\right ) + d h^{4} x^{3} \sqrt {-\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} - 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}} \log \left (-32 \, {\left (625 \, b^{3} d^{2} g^{4} p^{3} e^{2} - 81 \, b^{3} f^{4} p^{3} e^{4}\right )} \sqrt {h x} - 32 \, {\left (3 \, d^{4} f h^{11} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} - 125 \, b^{2} d^{3} g^{3} h^{4} p^{2} e + 45 \, b^{2} d^{2} f^{2} g h^{4} p^{2} e^{2}\right )} \sqrt {-\frac {d^{2} h^{7} \sqrt {-\frac {625 \, b^{4} d^{2} g^{4} p^{4} e^{3} - 450 \, b^{4} d f^{2} g^{2} p^{4} e^{4} + 81 \, b^{4} f^{4} p^{4} e^{5}}{d^{5} h^{14}}} - 30 \, b^{2} f g p^{2} e^{2}}{d^{2} h^{7}}}\right ) + {\left (12 \, b f p x^{2} e + 5 \, a d g x + 3 \, a d f + {\left (5 \, b d g p x + 3 \, b d f p\right )} \log \left (x^{2} e + d\right ) + {\left (5 \, b d g x + 3 \, b d f\right )} \log \left (c\right )\right )} \sqrt {h x}\right )}}{15 \, d h^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(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(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5
)/(d^5*h^14)) + 30*b^2*f*g*p^2*e^2)/(d^2*h^7))*log(-32*(625*b^3*d^2*g^4*p^3*e^2 - 81*b^3*f^4*p^3*e^4)*sqrt(h*x
) + 32*(3*d^4*f*h^11*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^1
4)) + 125*b^2*d^3*g^3*h^4*p^2*e - 45*b^2*d^2*f^2*g*h^4*p^2*e^2)*sqrt((d^2*h^7*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 -
450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) + 30*b^2*f*g*p^2*e^2)/(d^2*h^7))) - d*h^4*x^3*sqr
t((d^2*h^7*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) + 30*b
^2*f*g*p^2*e^2)/(d^2*h^7))*log(-32*(625*b^3*d^2*g^4*p^3*e^2 - 81*b^3*f^4*p^3*e^4)*sqrt(h*x) - 32*(3*d^4*f*h^11
*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) + 125*b^2*d^3*g^
3*h^4*p^2*e - 45*b^2*d^2*f^2*g*h^4*p^2*e^2)*sqrt((d^2*h^7*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p
^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) + 30*b^2*f*g*p^2*e^2)/(d^2*h^7))) - d*h^4*x^3*sqrt(-(d^2*h^7*sqrt(-(6
25*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) - 30*b^2*f*g*p^2*e^2)/(d^
2*h^7))*log(-32*(625*b^3*d^2*g^4*p^3*e^2 - 81*b^3*f^4*p^3*e^4)*sqrt(h*x) + 32*(3*d^4*f*h^11*sqrt(-(625*b^4*d^2
*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) - 125*b^2*d^3*g^3*h^4*p^2*e + 45*b^
2*d^2*f^2*g*h^4*p^2*e^2)*sqrt(-(d^2*h^7*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^
4*p^4*e^5)/(d^5*h^14)) - 30*b^2*f*g*p^2*e^2)/(d^2*h^7))) + d*h^4*x^3*sqrt(-(d^2*h^7*sqrt(-(625*b^4*d^2*g^4*p^4
*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) - 30*b^2*f*g*p^2*e^2)/(d^2*h^7))*log(-32*(6
25*b^3*d^2*g^4*p^3*e^2 - 81*b^3*f^4*p^3*e^4)*sqrt(h*x) - 32*(3*d^4*f*h^11*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450
*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^14)) - 125*b^2*d^3*g^3*h^4*p^2*e + 45*b^2*d^2*f^2*g*h^4*p^
2*e^2)*sqrt(-(d^2*h^7*sqrt(-(625*b^4*d^2*g^4*p^4*e^3 - 450*b^4*d*f^2*g^2*p^4*e^4 + 81*b^4*f^4*p^4*e^5)/(d^5*h^
14)) - 30*b^2*f*g*p^2*e^2)/(d^2*h^7))) + (12*b*f*p*x^2*e + 5*a*d*g*x + 3*a*d*f + (5*b*d*g*p*x + 3*b*d*f*p)*log
(x^2*e + d) + (5*b*d*g*x + 3*b*d*f)*log(c))*sqrt(h*x))/(d*h^4*x^3)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3064 deep

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Giac [A]
time = 3.09, size = 478, normalized size = 0.77 \begin {gather*} \frac {\frac {2 \, {\left (5 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d g h p e^{\frac {7}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f p e^{\frac {9}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{4}\right )} + 2 \, \sqrt {h x}\right )} e^{\frac {1}{4}}}{2 \, \left (d h^{2}\right )^{\frac {1}{4}}}\right ) e^{\left (-1\right )}}{d^{2} h} + \frac {2 \, {\left (5 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d g h p e^{\frac {7}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f p e^{\frac {9}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} e^{\left (-\frac {1}{4}\right )} - 2 \, \sqrt {h x}\right )} e^{\frac {1}{4}}}{2 \, \left (d h^{2}\right )^{\frac {1}{4}}}\right ) e^{\left (-1\right )}}{d^{2} h} + \frac {{\left (5 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d g h p e^{\frac {7}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f p e^{\frac {9}{4}}\right )} e^{\left (-1\right )} \log \left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\left (-\frac {1}{4}\right )} + h x + \sqrt {d h^{2}} e^{\left (-\frac {1}{2}\right )}\right )}{d^{2} h} - \frac {{\left (5 \, \sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b d g h p e^{\frac {7}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b f p e^{\frac {9}{4}}\right )} e^{\left (-1\right )} \log \left (-\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} \sqrt {h x} e^{\left (-\frac {1}{4}\right )} + h x + \sqrt {d h^{2}} e^{\left (-\frac {1}{2}\right )}\right )}{d^{2} h} - \frac {2 \, {\left (12 \, b f h^{3} p x^{2} e + 5 \, b d g h^{3} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 5 \, b d g h^{3} p x \log \left (h^{2}\right ) + 3 \, b d f h^{3} p \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b d f h^{3} p \log \left (h^{2}\right ) + 5 \, b d g h^{3} x \log \left (c\right ) + 5 \, a d g h^{3} x + 3 \, b d f h^{3} \log \left (c\right ) + 3 \, a d f h^{3}\right )}}{\sqrt {h x} d h^{2} x^{2}}}{15 \, h^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(2*(5*sqrt(2)*(d*h^2)^(1/4)*b*d*g*h*p*e^(7/4) - 3*sqrt(2)*(d*h^2)^(3/4)*b*f*p*e^(9/4))*arctan(1/2*sqrt(2)
*(sqrt(2)*(d*h^2)^(1/4)*e^(-1/4) + 2*sqrt(h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-1)/(d^2*h) + 2*(5*sqrt(2)*(d*h^2)^(
1/4)*b*d*g*h*p*e^(7/4) - 3*sqrt(2)*(d*h^2)^(3/4)*b*f*p*e^(9/4))*arctan(-1/2*sqrt(2)*(sqrt(2)*(d*h^2)^(1/4)*e^(
-1/4) - 2*sqrt(h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-1)/(d^2*h) + (5*sqrt(2)*(d*h^2)^(1/4)*b*d*g*h*p*e^(7/4) + 3*sq
rt(2)*(d*h^2)^(3/4)*b*f*p*e^(9/4))*e^(-1)*log(sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(
-1/2))/(d^2*h) - (5*sqrt(2)*(d*h^2)^(1/4)*b*d*g*h*p*e^(7/4) + 3*sqrt(2)*(d*h^2)^(3/4)*b*f*p*e^(9/4))*e^(-1)*lo
g(-sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d^2*h) - 2*(12*b*f*h^3*p*x^2*e + 5*
b*d*g*h^3*p*x*log(h^2*x^2*e + d*h^2) - 5*b*d*g*h^3*p*x*log(h^2) + 3*b*d*f*h^3*p*log(h^2*x^2*e + d*h^2) - 3*b*d
*f*h^3*p*log(h^2) + 5*b*d*g*h^3*x*log(c) + 5*a*d*g*h^3*x + 3*b*d*f*h^3*log(c) + 3*a*d*f*h^3)/(sqrt(h*x)*d*h^2*
x^2))/h^4

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{{\left (h\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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