Optimal. Leaf size=275 \[ \frac{2 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}+\frac{2 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
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Rubi [A] time = 1.01186, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {2400, 2401, 2389, 2300, 2180, 2204, 2390, 2310, 2445} \[ \frac{2 \sqrt{\pi } (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}+\frac{2 \sqrt{2 \pi } h (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
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}{\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}{\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)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{4 \int \frac{g+h x}{\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{(2 (f g-e h)) \int \frac{1}{\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)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{4 \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 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(2 (f g-e h)) \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^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)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{(4 h) \int \frac{e+f 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 )+\operatorname{Subst}\left (\frac{(4 (f g-e h)) \int \frac{1}{\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 )-\operatorname{Subst}\left (\frac{\left (2 (f g-e h) (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^2 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)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{(4 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^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 (f g-e h)) \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^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) (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^2 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) \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^2 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (4 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^2 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) (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^2 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) \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^2 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\operatorname{Subst}\left (\frac{\left (8 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^2 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) (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^2 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) \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^2 p^{3/2} q^{3/2}}+\frac{2 e^{-\frac{2 a}{b p q}} 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^2 p^{3/2} q^{3/2}}-\frac{2 (e+f x) (g+h x)}{b f p q \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\\ \end{align*}
Mathematica [A] time = 1.34486, size = 435, normalized size = 1.58 \[ \frac{2 (e+f x) e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \left (\sqrt{b} \sqrt{p} \sqrt{q} e^{\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \left ((e h+f g) \sqrt{-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )-f (g+h x) e^{\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}}\right )-2 \sqrt{\pi } e h e^{\frac{a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )+\sqrt{2 \pi } h (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )\right )}{b^{3/2} f^2 p^{3/2} q^{3/2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.263, size = 0, normalized size = 0. \begin{align*} \int{(hx+g) \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{h x + g}{{\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{g + h x}{\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{h x + g}{{\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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