3.7.8 \(\int \frac {(f+g x) (a+b \log (c (d+e x^2)^p))}{(h x)^{5/2}} \, dx\) [608]

Optimal. Leaf size=588 \[ -\frac {2 \sqrt {2} b e^{3/4} f p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{5/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g p \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {2 \sqrt {2} b e^{3/4} f p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{3 d^{3/4} h^{5/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g p \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}-\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h (h x)^{3/2}}-\frac {2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^2 \sqrt {h x}}-\frac {\sqrt {2} b e^{3/4} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{5/2}}+\frac {\sqrt {2} b \sqrt [4]{e} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {\sqrt {2} b e^{3/4} f p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{3 d^{3/4} h^{5/2}}-\frac {\sqrt {2} b \sqrt [4]{e} g p \log \left (\sqrt {d} \sqrt {h}+\sqrt {e} \sqrt {h} x+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x}\right )}{\sqrt [4]{d} h^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.48, antiderivative size = 588, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2517, 2526, 2505, 217, 1179, 642, 1176, 631, 210, 303} \begin {gather*} -\frac {2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h (h x)^{3/2}}-\frac {2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{h^2 \sqrt {h x}}-\frac {2 \sqrt {2} b e^{3/4} f 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^{5/2}}+\frac {2 \sqrt {2} b e^{3/4} f 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^{5/2}}-\frac {2 \sqrt {2} b \sqrt [4]{e} g p \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{d} h^{5/2}}+\frac {2 \sqrt {2} b \sqrt [4]{e} g p \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{d} h^{5/2}}-\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{5/2}}+\frac {\sqrt {2} b e^{3/4} 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 )}{3 d^{3/4} h^{5/2}}+\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{5/2}}-\frac {\sqrt {2} b \sqrt [4]{e} 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 )}{\sqrt [4]{d} h^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(5/2),x]

[Out]

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

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

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Mathematica [A]
time = 0.28, size = 271, normalized size = 0.46 \begin {gather*} \frac {2 x^{5/2} \left (\frac {2 b \sqrt [4]{e} g p \left (\tan ^{-1}\left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{-d}}\right )+\tanh ^{-1}\left (\frac {d \sqrt [4]{e} \sqrt {x}}{(-d)^{5/4}}\right )\right )}{\sqrt [4]{-d}}-\frac {b e^{3/4} f p \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )-2 \tan ^{-1}\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 \sqrt {2} d^{3/4}}-\frac {f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac {g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{\sqrt {x}}\right )}{(h x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)*(a + b*Log[c*(d + e*x^2)^p]))/(h*x)^(5/2),x]

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right ) \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{\left (h x \right )^{\frac {5}{2}}}\, dx\]

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)^(5/2),x)

[Out]

int((g*x+f)*(a+b*ln(c*(e*x^2+d)^p))/(h*x)^(5/2),x)

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Maxima [A]
time = 0.52, size = 495, normalized size = 0.84 \begin {gather*} -\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 )} b g p e}{h^{2}} - \frac {2 \, b g x^{2} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{\left (h x\right )^{\frac {5}{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 f p e}{3 \, h^{3}} - \frac {2 \, a g x^{2}}{\left (h x\right )^{\frac {5}{2}}} - \frac {2 \, b f \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{3 \, \left (h x\right )^{\frac {3}{2}} h} - \frac {2 \, a f}{3 \, \left (h x\right )^{\frac {3}{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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1339 vs. \(2 (388) = 776\).
time = 0.39, size = 1339, normalized size = 2.28 \begin {gather*} -\frac {2 \, {\left (h^{3} x^{2} \sqrt {-\frac {6 \, b^{2} f g p^{2} e + d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}} \log \left (-32 \, {\left (81 \, b^{3} d^{2} g^{4} p^{3} e - b^{3} f^{4} p^{3} e^{3}\right )} \sqrt {h x} + 32 \, {\left (9 \, b^{2} d^{2} f g^{2} h^{3} p^{2} e - 3 \, d^{3} g h^{8} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}} - b^{2} d f^{3} h^{3} p^{2} e^{2}\right )} \sqrt {-\frac {6 \, b^{2} f g p^{2} e + d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}}\right ) - h^{3} x^{2} \sqrt {-\frac {6 \, b^{2} f g p^{2} e + d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}} \log \left (-32 \, {\left (81 \, b^{3} d^{2} g^{4} p^{3} e - b^{3} f^{4} p^{3} e^{3}\right )} \sqrt {h x} - 32 \, {\left (9 \, b^{2} d^{2} f g^{2} h^{3} p^{2} e - 3 \, d^{3} g h^{8} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}} - b^{2} d f^{3} h^{3} p^{2} e^{2}\right )} \sqrt {-\frac {6 \, b^{2} f g p^{2} e + d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}}\right ) + h^{3} x^{2} \sqrt {-\frac {6 \, b^{2} f g p^{2} e - d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}} \log \left (-32 \, {\left (81 \, b^{3} d^{2} g^{4} p^{3} e - b^{3} f^{4} p^{3} e^{3}\right )} \sqrt {h x} + 32 \, {\left (9 \, b^{2} d^{2} f g^{2} h^{3} p^{2} e + 3 \, d^{3} g h^{8} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}} - b^{2} d f^{3} h^{3} p^{2} e^{2}\right )} \sqrt {-\frac {6 \, b^{2} f g p^{2} e - d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}}\right ) - h^{3} x^{2} \sqrt {-\frac {6 \, b^{2} f g p^{2} e - d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}} \log \left (-32 \, {\left (81 \, b^{3} d^{2} g^{4} p^{3} e - b^{3} f^{4} p^{3} e^{3}\right )} \sqrt {h x} - 32 \, {\left (9 \, b^{2} d^{2} f g^{2} h^{3} p^{2} e + 3 \, d^{3} g h^{8} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}} - b^{2} d f^{3} h^{3} p^{2} e^{2}\right )} \sqrt {-\frac {6 \, b^{2} f g p^{2} e - d h^{5} \sqrt {-\frac {81 \, b^{4} d^{2} g^{4} p^{4} e - 18 \, b^{4} d f^{2} g^{2} p^{4} e^{2} + b^{4} f^{4} p^{4} e^{3}}{d^{3} h^{10}}}}{d h^{5}}}\right ) + {\left (3 \, a g x + a f + {\left (3 \, b g p x + b f p\right )} \log \left (x^{2} e + d\right ) + {\left (3 \, b g x + b f\right )} \log \left (c\right )\right )} \sqrt {h x}\right )}}{3 \, h^{3} x^{2}} \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)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(h^3*x^2*sqrt(-(6*b^2*f*g*p^2*e + d*h^5*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*
p^4*e^3)/(d^3*h^10)))/(d*h^5))*log(-32*(81*b^3*d^2*g^4*p^3*e - b^3*f^4*p^3*e^3)*sqrt(h*x) + 32*(9*b^2*d^2*f*g^
2*h^3*p^2*e - 3*d^3*g*h^8*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)
) - b^2*d*f^3*h^3*p^2*e^2)*sqrt(-(6*b^2*f*g*p^2*e + d*h^5*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e
^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)))/(d*h^5))) - h^3*x^2*sqrt(-(6*b^2*f*g*p^2*e + d*h^5*sqrt(-(81*b^4*d^2*g^4*p^
4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)))/(d*h^5))*log(-32*(81*b^3*d^2*g^4*p^3*e - b^3*f^
4*p^3*e^3)*sqrt(h*x) - 32*(9*b^2*d^2*f*g^2*h^3*p^2*e - 3*d^3*g*h^8*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*
g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)) - b^2*d*f^3*h^3*p^2*e^2)*sqrt(-(6*b^2*f*g*p^2*e + d*h^5*sqrt(-(81*b
^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)))/(d*h^5))) + h^3*x^2*sqrt(-(6*b^2*f
*g*p^2*e - d*h^5*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)))/(d*h^5
))*log(-32*(81*b^3*d^2*g^4*p^3*e - b^3*f^4*p^3*e^3)*sqrt(h*x) + 32*(9*b^2*d^2*f*g^2*h^3*p^2*e + 3*d^3*g*h^8*sq
rt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)) - b^2*d*f^3*h^3*p^2*e^2)*s
qrt(-(6*b^2*f*g*p^2*e - d*h^5*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h
^10)))/(d*h^5))) - h^3*x^2*sqrt(-(6*b^2*f*g*p^2*e - d*h^5*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e
^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)))/(d*h^5))*log(-32*(81*b^3*d^2*g^4*p^3*e - b^3*f^4*p^3*e^3)*sqrt(h*x) - 32*(9
*b^2*d^2*f*g^2*h^3*p^2*e + 3*d^3*g*h^8*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^
3)/(d^3*h^10)) - b^2*d*f^3*h^3*p^2*e^2)*sqrt(-(6*b^2*f*g*p^2*e - d*h^5*sqrt(-(81*b^4*d^2*g^4*p^4*e - 18*b^4*d*
f^2*g^2*p^4*e^2 + b^4*f^4*p^4*e^3)/(d^3*h^10)))/(d*h^5))) + (3*a*g*x + a*f + (3*b*g*p*x + b*f*p)*log(x^2*e + d
) + (3*b*g*x + b*f)*log(c))*sqrt(h*x))/(h^3*x^2)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)**(5/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

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Giac [A]
time = 4.40, size = 443, normalized size = 0.75 \begin {gather*} \frac {\frac {2 \, {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b f h p e^{\frac {11}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b g 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 (-2\right )}}{d h} + \frac {2 \, {\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b f h p e^{\frac {11}{4}} + 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b g 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 (-2\right )}}{d h} + \frac {{\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b f h p e^{\frac {11}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b g p e^{\frac {9}{4}}\right )} e^{\left (-2\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 h} - \frac {{\left (\sqrt {2} \left (d h^{2}\right )^{\frac {1}{4}} b f h p e^{\frac {11}{4}} - 3 \, \sqrt {2} \left (d h^{2}\right )^{\frac {3}{4}} b g p e^{\frac {9}{4}}\right )} e^{\left (-2\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 h} - \frac {2 \, {\left (3 \, b g h^{2} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b g h^{2} p x \log \left (h^{2}\right ) + b f h^{2} p \log \left (h^{2} x^{2} e + d h^{2}\right ) - b f h^{2} p \log \left (h^{2}\right ) + 3 \, b g h^{2} x \log \left (c\right ) + 3 \, a g h^{2} x + b f h^{2} \log \left (c\right ) + a f h^{2}\right )}}{\sqrt {h x} h x}}{3 \, h^{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)^(5/2),x, algorithm="giac")

[Out]

1/3*(2*(sqrt(2)*(d*h^2)^(1/4)*b*f*h*p*e^(11/4) + 3*sqrt(2)*(d*h^2)^(3/4)*b*g*p*e^(9/4))*arctan(1/2*sqrt(2)*(sq
rt(2)*(d*h^2)^(1/4)*e^(-1/4) + 2*sqrt(h*x))*e^(1/4)/(d*h^2)^(1/4))*e^(-2)/(d*h) + 2*(sqrt(2)*(d*h^2)^(1/4)*b*f
*h*p*e^(11/4) + 3*sqrt(2)*(d*h^2)^(3/4)*b*g*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^(-2)/(d*h) + (sqrt(2)*(d*h^2)^(1/4)*b*f*h*p*e^(11/4) - 3*sqrt(2)*(d*h^2)^
(3/4)*b*g*p*e^(9/4))*e^(-2)*log(sqrt(2)*(d*h^2)^(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d*h) -
(sqrt(2)*(d*h^2)^(1/4)*b*f*h*p*e^(11/4) - 3*sqrt(2)*(d*h^2)^(3/4)*b*g*p*e^(9/4))*e^(-2)*log(-sqrt(2)*(d*h^2)^
(1/4)*sqrt(h*x)*e^(-1/4) + h*x + sqrt(d*h^2)*e^(-1/2))/(d*h) - 2*(3*b*g*h^2*p*x*log(h^2*x^2*e + d*h^2) - 3*b*g
*h^2*p*x*log(h^2) + b*f*h^2*p*log(h^2*x^2*e + d*h^2) - b*f*h^2*p*log(h^2) + 3*b*g*h^2*x*log(c) + 3*a*g*h^2*x +
b*f*h^2*log(c) + a*f*h^2)/(sqrt(h*x)*h*x))/h^3

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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 )}^{5/2}} \,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)^(5/2),x)

[Out]

int(((f + g*x)*(a + b*log(c*(d + e*x^2)^p)))/(h*x)^(5/2), x)

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