3.5.74 \(\int \frac {(g+h x)^2}{(a+b \log (c (d (e+f x)^p)^q))^{3/2}} \, dx\) [474]

Optimal. Leaf size=404 \[ \frac {2 e^{-\frac {a}{b p q}} (f g-e h)^2 \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 e^{-\frac {3 a}{b p q}} h^2 \sqrt {3 \pi } (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]

[Out]

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

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Rubi [A]
time = 1.64, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2447, 2448, 2436, 2337, 2211, 2235, 2437, 2347, 2495} \begin {gather*} \frac {4 \sqrt {2 \pi } h (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 \sqrt {\pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 \sqrt {3 \pi } h^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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \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]

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

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 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 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 2447

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(d
+ e*x)*(f + g*x)^q*((a + b*Log[c*(d + e*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[q*((e*f - d*g)/(b*e*n*(p + 1))), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 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 \frac {(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 \frac {(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 )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(4 (f g-e h)) \int \frac {g+h x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {6 \int \left (\frac {(f g-e h)^2}{f^2 \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {2 h (f g-e h) (e+f x)}{f^2 \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {h^2 (e+f x)^2}{f^2 \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}\right ) \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(4 (f g-e h)) \int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}+\frac {h (e+f x)}{f \sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}\right ) \, dx}{b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {\left (6 h^2\right ) \int \frac {(e+f x)^2}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(4 h (f g-e h)) \int \frac {e+f x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(12 h (f g-e h)) \int \frac {e+f x}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (4 (f g-e h)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (6 (f g-e h)^2\right ) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}} \, dx}{b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {\left (6 h^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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(4 h (f g-e h)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(12 h (f g-e h)) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c d^q x^{p q}\right )}} \, dx,x,e+f x\right )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (4 (f g-e h)^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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (6 (f g-e h)^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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {\left (6 h^2 (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (4 h (f g-e h) (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (12 h (f g-e h) (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (4 (f g-e h)^2 (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (6 (f g-e h)^2 (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 )}{b f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\text {Subst}\left (\frac {\left (12 h^2 (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (8 h (f g-e h) (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (24 h (f g-e h) (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\left (8 (f g-e h)^2 (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {\left (12 (f g-e h)^2 (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 )}{b^2 f^3 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {2 e^{-\frac {a}{b p q}} (f g-e h)^2 \sqrt {\pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) \sqrt {2 \pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 e^{-\frac {3 a}{b p q}} h^2 \sqrt {3 \pi } (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 )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1821\) vs. \(2(404)=808\).
time = 2.76, size = 1821, normalized size = 4.51 \begin {gather*} -\frac {2 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (\sqrt {b} e e^{\frac {3 a}{b p q}} f^2 g^2 \sqrt {p} \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+\sqrt {b} e^{\frac {3 a}{b p q}} f^3 g^2 \sqrt {p} \sqrt {q} x \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+2 \sqrt {b} e e^{\frac {3 a}{b p q}} f^2 g h \sqrt {p} \sqrt {q} x \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+2 \sqrt {b} e^{\frac {3 a}{b p q}} f^3 g h \sqrt {p} \sqrt {q} x^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+\sqrt {b} e e^{\frac {3 a}{b p q}} f^2 h^2 \sqrt {p} \sqrt {q} x^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+\sqrt {b} e^{\frac {3 a}{b p q}} f^3 h^2 \sqrt {p} \sqrt {q} x^3 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}}+4 e e^{\frac {2 a}{b p q}} f g h \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}+2 e^2 e^{\frac {2 a}{b p q}} h^2 \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}-2 e^{\frac {a}{b p q}} f g h \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}-e e^{\frac {a}{b p q}} h^2 \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{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 ) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}-\sqrt {b} h^2 \sqrt {p} \sqrt {3 \pi } \sqrt {q} (e+f x)^3 \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}+3 \sqrt {b} e e^{\frac {a}{b p q}} h^2 \sqrt {p} \sqrt {2 \pi } \sqrt {q} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}-3 \sqrt {b} e^2 e^{\frac {2 a}{b p q}} h^2 \sqrt {p} \sqrt {\pi } \sqrt {q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}+3 \sqrt {b} e^2 e^{\frac {2 a}{b p q}} h^2 \sqrt {p} \sqrt {\pi } \sqrt {q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \text {erf}\left (\sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}-3 \sqrt {b} e e^{\frac {a}{b p q}} h^2 \sqrt {p} \sqrt {2 \pi } \sqrt {q} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \text {erf}\left (\sqrt {2} \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}+\sqrt {b} h^2 \sqrt {p} \sqrt {3 \pi } \sqrt {q} (e+f x)^3 \text {erf}\left (\sqrt {3} \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}-\sqrt {b} e^{\frac {2 a}{b p q}} f^2 g^2 \sqrt {p} \sqrt {q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}-2 \sqrt {b} e e^{\frac {2 a}{b p q}} f g h \sqrt {p} \sqrt {q} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \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]

(-2*(Sqrt[b]*e*E^((3*a)/(b*p*q))*f^2*g^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(3/(p*q)) + Sqrt[b]*E^((3*a)/(b
*p*q))*f^3*g^2*Sqrt[p]*Sqrt[q]*x*(c*(d*(e + f*x)^p)^q)^(3/(p*q)) + 2*Sqrt[b]*e*E^((3*a)/(b*p*q))*f^2*g*h*Sqrt[
p]*Sqrt[q]*x*(c*(d*(e + f*x)^p)^q)^(3/(p*q)) + 2*Sqrt[b]*E^((3*a)/(b*p*q))*f^3*g*h*Sqrt[p]*Sqrt[q]*x^2*(c*(d*(
e + f*x)^p)^q)^(3/(p*q)) + Sqrt[b]*e*E^((3*a)/(b*p*q))*f^2*h^2*Sqrt[p]*Sqrt[q]*x^2*(c*(d*(e + f*x)^p)^q)^(3/(p
*q)) + Sqrt[b]*E^((3*a)/(b*p*q))*f^3*h^2*Sqrt[p]*Sqrt[q]*x^3*(c*(d*(e + f*x)^p)^q)^(3/(p*q)) + 4*e*E^((2*a)/(b
*p*q))*f*g*h*Sqrt[Pi]*(e + f*x)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqr
t[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]] + 2*e^2*E^((2*a)/(b*p*q))*h^2*Sqrt[Pi]*(e + f*x)*(
c*(d*(e + f*x)^p)^q)^(2/(p*q))*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b
*Log[c*(d*(e + f*x)^p)^q]] - 2*E^(a/(b*p*q))*f*g*h*Sqrt[2*Pi]*(e + f*x)^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*Erfi
[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]
] - e*E^(a/(b*p*q))*h^2*Sqrt[2*Pi]*(e + f*x)^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*
(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]] - Sqrt[b]*h^2*Sqrt[p]*Sqr
t[3*Pi]*Sqrt[q]*(e + f*x)^3*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] + 3*Sqrt[b]*e*E^(a/(b*p*q))*h^2*
Sqrt[p]*Sqrt[2*Pi]*Sqrt[q]*(e + f*x)^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])
/(b*p*q))] - 3*Sqrt[b]*e^2*E^((2*a)/(b*p*q))*h^2*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*(c*(d*(e + f*x)^p)^q)^(2/(
p*q))*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] + 3*Sqrt[b]*e^2*E^((2*a)/(b*p*q))*h^2*Sqrt[p]*Sqrt[Pi]
*Sqrt[q]*(e + f*x)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*Erf[Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]]*Sqrt
[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] - 3*Sqrt[b]*e*E^(a/(b*p*q))*h^2*Sqrt[p]*Sqrt[2*Pi]*Sqrt[q]*(e +
f*x)^2*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*Erf[Sqrt[2]*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]]*Sqrt[-((
a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] + Sqrt[b]*h^2*Sqrt[p]*Sqrt[3*Pi]*Sqrt[q]*(e + f*x)^3*Erf[Sqrt[3]*Sqr
t[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]]*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] - Sqrt[b]*E^
((2*a)/(b*p*q))*f^2*g^2*Sqrt[p]*Sqrt[q]*(e + f*x)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*Gamma[1/2, -((a + b*Log[c*(d
*(e + f*x)^p)^q])/(b*p*q))]*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))] - 2*Sqrt[b]*e*E^((2*a)/(b*p*q))*
f*g*h*Sqrt[p]*Sqrt[q]*(e + f*x)*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^p)^q])/
(b*p*q))]*Sqrt[-((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]))/(b^(3/2)*E^((3*a)/(b*p*q))*f^3*p^(3/2)*q^(3/2)*(
c*(d*(e + f*x)^p)^q)^(3/(p*q))*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (h x +g \right )^{2}}{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{\frac {3}{2}}}\, dx\]

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 \frac {\left (g + h x\right )^{2}}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {3}{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((g + h*x)**2/(a + b*log(c*(d*(e + f*x)**p)**q))**(3/2), 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 \frac {{\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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