Optimal. Leaf size=404 \[ \frac{4 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{3 \pi } h^2 (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
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Rubi [A] time = 2.24988, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310, 2445} \[ \frac{4 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac{2 \sqrt{3 \pi } h^2 (e+f x)^3 e^{-\frac{3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)^2}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
Antiderivative was successfully verified.
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Rule 2400
Rule 2401
Rule 2389
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2310
Rule 2445
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 &=\operatorname{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 )}}+\operatorname{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 )-\operatorname{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 )}}+\operatorname{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 )-\operatorname{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 )}}+\operatorname{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 )-\operatorname{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 )+\operatorname{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 )-\operatorname{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 )+\operatorname{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 )}}+\operatorname{Subst}\left (\frac{\left (6 h^2\right ) \operatorname{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 )-\operatorname{Subst}\left (\frac{(4 h (f g-e h)) \operatorname{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 )+\operatorname{Subst}\left (\frac{(12 h (f g-e h)) \operatorname{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 )-\operatorname{Subst}\left (\frac{\left (4 (f g-e h)^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 )}{b f^3 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 (f g-e h)^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 )}{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 )}}+\operatorname{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 ) \operatorname{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 )-\operatorname{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 ) \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 )}{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 )+\operatorname{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 ) \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 )}{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 )-\operatorname{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 ) \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 )}{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 )+\operatorname{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 ) \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 )}{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 )}}+\operatorname{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 ) \operatorname{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 )-\operatorname{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 ) \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 )}{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 )+\operatorname{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 ) \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 )}{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 )-\operatorname{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 ) \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 )}{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 )+\operatorname{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 ) \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 )}{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*}
Mathematica [B] time = 2.48928, size = 1040, normalized size = 2.57 \[ \frac{2 \left (e^{-\frac{3 a}{b p q}} h^2 \sqrt{3 \pi } (e+f x)^3 \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 ) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{3}{p q}}-2 e e^{-\frac{2 a}{b p q}} h^2 \sqrt{2 \pi } (e+f x)^2 \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 )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}}+2 e^{-\frac{2 a}{b p q}} f g h \sqrt{2 \pi } (e+f x)^2 \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 )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}}+e^2 e^{-\frac{a}{b p q}} h^2 \sqrt{\pi } (e+f x) \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 )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}-4 e e^{-\frac{a}{b p q}} f g h \sqrt{\pi } (e+f x) \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 )} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}+\sqrt{b} e^{-\frac{a}{b p q}} f^2 g^2 \sqrt{p} \sqrt{q} (e+f x) \text{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}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}+2 \sqrt{b} e e^{-\frac{a}{b p q}} f g h \sqrt{p} \sqrt{q} (e+f x) \text{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}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}}-\sqrt{b} f^3 h^2 \sqrt{p} \sqrt{q} x^3-\sqrt{b} e f^2 h^2 \sqrt{p} \sqrt{q} x^2-2 \sqrt{b} f^3 g h \sqrt{p} \sqrt{q} x^2-\sqrt{b} f^3 g^2 \sqrt{p} \sqrt{q} x-2 \sqrt{b} e f^2 g h \sqrt{p} \sqrt{q} x-\sqrt{b} e f^2 g^2 \sqrt{p} \sqrt{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 )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.887, size = 0, normalized size = 0. \begin{align*} \int{ \left ( hx+g \right ) ^{2} \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.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (h x + g\right )}^{2}}{{\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.
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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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (h x + g\right )}^{2}}{{\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.
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