3.5.65 \(\int (g+h x)^2 (a+b \log (c (d (e+f x)^p)^q))^{3/2} \, dx\) [465]

Optimal. Leaf size=625 \[ \frac {3 b^{3/2} e^{-\frac {a}{b p q}} (f g-e h)^2 p^{3/2} \sqrt {\pi } q^{3/2} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^3}+\frac {3 b^{3/2} e^{-\frac {2 a}{b p q}} h (f g-e h) p^{3/2} \sqrt {\frac {\pi }{2}} q^{3/2} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{8 f^3}+\frac {b^{3/2} e^{-\frac {3 a}{b p q}} h^2 p^{3/2} \sqrt {\frac {\pi }{3}} q^{3/2} (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{12 f^3}-\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3} \]

[Out]

(-e*h+f*g)^2*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(3/2)/f^3+h*(-e*h+f*g)*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^
(3/2)/f^3+1/3*h^2*(f*x+e)^3*(a+b*ln(c*(d*(f*x+e)^p)^q))^(3/2)/f^3+1/36*b^(3/2)*h^2*p^(3/2)*q^(3/2)*(f*x+e)^3*e
rfi(3^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))*3^(1/2)*Pi^(1/2)/exp(3*a/b/p/q)/f^3/((c
*(d*(f*x+e)^p)^q)^(3/p/q))+3/16*b^(3/2)*h*(-e*h+f*g)*p^(3/2)*q^(3/2)*(f*x+e)^2*erfi(2^(1/2)*(a+b*ln(c*(d*(f*x+
e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))*2^(1/2)*Pi^(1/2)/exp(2*a/b/p/q)/f^3/((c*(d*(f*x+e)^p)^q)^(2/p/q))+3/4
*b^(3/2)*(-e*h+f*g)^2*p^(3/2)*q^(3/2)*(f*x+e)*erfi((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))*
Pi^(1/2)/exp(a/b/p/q)/f^3/((c*(d*(f*x+e)^p)^q)^(1/p/q))-3/2*b*(-e*h+f*g)^2*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)
^q))^(1/2)/f^3-3/4*b*h*(-e*h+f*g)*p*q*(f*x+e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/f^3-1/6*b*h^2*p*q*(f*x+e)^3*
(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/f^3

________________________________________________________________________________________

Rubi [A]
time = 1.36, antiderivative size = 625, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2448, 2436, 2333, 2337, 2211, 2235, 2437, 2342, 2347, 2495} \begin {gather*} \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}} (f g-e h) \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 )}{8 f^3}+\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)^2 \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^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} h^2 p^{3/2} q^{3/2} (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{12 f^3}+\frac {h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {(e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}-\frac {3 b h p q (e+f x)^2 (f g-e h) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {3 b p q (e+f x) (f g-e h)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^(3/2)*(f*g - e*h)^2*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]
*Sqrt[p]*Sqrt[q])])/(4*E^(a/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (3*b^(3/2)*h*(f*g - e*h)*p^(3/2)*S
qrt[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])])/
(8*E^((2*a)/(b*p*q))*f^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (b^(3/2)*h^2*p^(3/2)*Sqrt[Pi/3]*q^(3/2)*(e + f*x)^
3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(12*E^((3*a)/(b*p*q))*f^3*(c
*(d*(e + f*x)^p)^q)^(3/(p*q))) - (3*b*(f*g - e*h)^2*p*q*(e + f*x)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(2*f^3
) - (3*b*h*(f*g - e*h)*p*q*(e + f*x)^2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(4*f^3) - (b*h^2*p*q*(e + f*x)^3*
Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(6*f^3) + ((f*g - e*h)^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2
))/f^3 + (h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2))/f^3 + (h^2*(e + f*x)^3*(a + b*Log[
c*(d*(e + f*x)^p)^q])^(3/2))/(3*f^3)

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2333

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 2337

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 2342

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 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2436

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 2437

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 2448

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 2495

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)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx &=\text {Subst}\left (\int (g+h x)^2 \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 )\\ &=\text {Subst}\left (\int \left (\frac {(f g-e h)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f^2}+\frac {2 h (f g-e h) (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f^2}+\frac {h^2 (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}}{f^2}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {h^2 \int (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 h (f g-e h)) \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h)^2 \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2} \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\text {Subst}\left (\frac {h^2 \text {Subst}\left (\int x^2 \left (a+b \log \left (c d^q x^{p q}\right )\right )^{3/2} \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(2 h (f g-e h)) \text {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^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(f g-e h)^2 \text {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^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}-\text {Subst}\left (\frac {\left (b h^2 p q\right ) \text {Subst}\left (\int x^2 \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(3 b h (f g-e h) p q) \text {Subst}\left (\int x \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (3 b (f g-e h)^2 p q\right ) \text {Subst}\left (\int \sqrt {a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 f^3},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)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}+\text {Subst}\left (\frac {\left (b^2 h^2 p^2 q^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{12 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 h (f g-e h) p^2 q^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{8 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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^3},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)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}+\text {Subst}\left (\frac {\left (b^2 h^2 p q (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{p q}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{12 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 h (f g-e h) p q (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {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 )}{8 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b^2 (f g-e h)^2 p q (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {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^3},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)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}+\text {Subst}\left (\frac {\left (b h^2 p q (e+f x)^3 \left (c d^q (e+f x)^{p q}\right )^{-\frac {3}{p q}}\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b p q}+\frac {3 x^2}{b p q}} \, dx,x,\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}\right )}{6 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b h (f g-e h) p q (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac {2}{p q}}\right ) \text {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 )}{4 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (3 b (f g-e h)^2 p q (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}\right ) \text {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^3},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)^2 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^3}+\frac {3 b^{3/2} e^{-\frac {2 a}{b p q}} h (f g-e 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 )}{8 f^3}+\frac {b^{3/2} e^{-\frac {3 a}{b p q}} h^2 p^{3/2} \sqrt {\frac {\pi }{3}} q^{3/2} (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{12 f^3}-\frac {3 b (f g-e h)^2 p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{4 f^3}-\frac {b h^2 p q (e+f x)^3 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{6 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{3 f^3}\\ \end {align*}

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Mathematica [A]
time = 0.90, size = 545, normalized size = 0.87 \begin {gather*} \frac {(e+f x) \left (144 (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+144 h (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}+48 h^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+4 b h^2 p q (e+f x)^2 \left (\sqrt {b} e^{-\frac {3 a}{b p q}} \sqrt {p} \sqrt {3 \pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-6 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )+27 b h (f g-e h) p q (e+f x) \left (\sqrt {b} e^{-\frac {2 a}{b p q}} \sqrt {p} \sqrt {2 \pi } \sqrt {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 )+108 b (f g-e h)^2 p q \left (\sqrt {b} e^{-\frac {a}{b p q}} \sqrt {p} \sqrt {\pi } \sqrt {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 )\right )}{144 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)*(144*(f*g - e*h)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2) + 144*h*(f*g - e*h)*(e + f*x)*(a + b*Log[
c*(d*(e + f*x)^p)^q])^(3/2) + 48*h^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/2) + 4*b*h^2*p*q*(e + f*x
)^2*((Sqrt[b]*Sqrt[p]*Sqrt[3*Pi]*Sqrt[q]*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*
Sqrt[q])])/(E^((3*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(3/(p*q))) - 6*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]) + 27*
b*h*(f*g - e*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*Log[c*
(d*(e + f*x)^p)^q]]) + 108*b*(f*g - e*h)^2*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]])))/(144*f^3)

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{2} \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {3}{2}} \left (g + h x\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g+h\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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