3.466 \(\int (g+h x) (a+b \log (c (d (e+f x)^p)^q))^{3/2} \, dx\)

Optimal. Leaf size=396 \[ \frac{3 \sqrt{\pi } b^{3/2} p^{3/2} q^{3/2} (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{4 f^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} h p^{3/2} q^{3/2} (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{16 f^2}+\frac{(e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}-\frac{3 b p q (e+f x) (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}-\frac{3 b h p q (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2} \]

[Out]

(3*b^(3/2)*(f*g - e*h)*p^(3/2)*Sqrt[Pi]*q^(3/2)*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*S
qrt[p]*Sqrt[q])])/(4*E^(a/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (3*b^(3/2)*h*p^(3/2)*Sqrt[Pi/2]*q^(3
/2)*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(16*E^((2*a)/(
b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (3*b*(f*g - e*h)*p*q*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)
^q]])/(2*f^2) - (3*b*h*p*q*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(8*f^2) + ((f*g - e*h)*(e + f*x)*
(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f^2 + (h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/(2*f^2)

________________________________________________________________________________________

Rubi [A]  time = 1.01785, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310, 2445} \[ \frac{3 \sqrt{\pi } b^{3/2} p^{3/2} q^{3/2} (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{4 f^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} h p^{3/2} q^{3/2} (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{16 f^2}+\frac{(e+f x) (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}-\frac{3 b p q (e+f x) (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}-\frac{3 b h p q (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2),x]

[Out]

(3*b^(3/2)*(f*g - e*h)*p^(3/2)*Sqrt[Pi]*q^(3/2)*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*S
qrt[p]*Sqrt[q])])/(4*E^(a/(b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (3*b^(3/2)*h*p^(3/2)*Sqrt[Pi/2]*q^(3
/2)*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(16*E^((2*a)/(
b*p*q))*f^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (3*b*(f*g - e*h)*p*q*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)
^q]])/(2*f^2) - (3*b*h*p*q*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(8*f^2) + ((f*g - e*h)*(e + f*x)*
(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f^2 + (h*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/(2*f^2)

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2445

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin{align*} \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int (g+h x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{(f g-e h) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f}+\frac{h (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(f g-e h) \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h \operatorname{Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2} \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(f g-e h) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2} \, dx,x,e+f x\right )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}-\operatorname{Subst}\left (\frac{(3 b h p q) \operatorname{Subst}\left (\int x \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{4 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(3 b (f g-e h) p q) \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 b (f g-e h) p q (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\frac{3 b h p q (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2}+\frac{(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}+\operatorname{Subst}\left (\frac{\left (3 b^2 h p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{16 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (3 b^2 (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{4 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 b (f g-e h) p q (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\frac{3 b h p q (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2}+\frac{(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}+\operatorname{Subst}\left (\frac{\left (3 b^2 h p q (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac{2}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{p q}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{16 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (3 b^2 (f g-e h) p q (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p q}}}{\sqrt{a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{4 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 b (f g-e h) p q (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\frac{3 b h p q (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2}+\frac{(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}+\operatorname{Subst}\left (\frac{\left (3 b h p q (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac{2}{p q}}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b p q}+\frac{2 x^2}{b p q}} \, dx,x,\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{8 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (3 b (f g-e h) p q (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b p q}+\frac{x^2}{b p q}} \, dx,x,\sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{2 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{3 b^{3/2} e^{-\frac{a}{b p q}} (f g-e h) p^{3/2} \sqrt{\pi } q^{3/2} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{4 f^2}+\frac{3 b^{3/2} e^{-\frac{2 a}{b p q}} h p^{3/2} \sqrt{\frac{\pi }{2}} q^{3/2} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{16 f^2}-\frac{3 b (f g-e h) p q (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\frac{3 b h p q (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2}+\frac{(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}+\frac{h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2}\\ \end{align*}

Mathematica [A]  time = 0.566567, size = 348, normalized size = 0.88 \[ \frac{(e+f x) \left (24 b p q (f g-e h) \left (\sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} e^{-\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )-2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )+3 b h p q (e+f x) \left (\sqrt{2 \pi } \sqrt{b} \sqrt{p} \sqrt{q} e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )-4 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )+32 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+16 h (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}\right )}{32 f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2),x]

[Out]

((e + f*x)*(32*(f*g - e*h)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2) + 16*h*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p
)^q])^(3/2) + 3*b*h*p*q*(e + f*x)*((Sqrt[b]*Sqrt[p]*Sqrt[2*Pi]*Sqrt[q]*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e +
f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - 4*Sqrt[a + b*Lo
g[c*(d*(e + f*x)^p)^q]]) + 24*b*(f*g - e*h)*p*q*((Sqrt[b]*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*Erfi[Sqrt[a + b*Log[c*(d*(e
+ f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) - 2*Sqrt[a + b*Log[
c*(d*(e + f*x)^p)^q]])))/(32*f^2)

________________________________________________________________________________________

Maple [F]  time = 0.272, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(3/2),x)

[Out]

int((h*x+g)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (h x + g\right )}{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(3/2),x, algorithm="maxima")

[Out]

integrate((h*x + g)*(b*log(((f*x + e)^p*d)^q*c) + a)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*ln(c*(d*(f*x+e)**p)**q))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (h x + g\right )}{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)*(a+b*log(c*(d*(f*x+e)^p)^q))^(3/2),x, algorithm="giac")

[Out]

integrate((h*x + g)*(b*log(((f*x + e)^p*d)^q*c) + a)^(3/2), x)