Optimal. Leaf size=311 \[ -\frac{\sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} h \sqrt{p} \sqrt{q} (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{4 f^2}+\frac{(e+f x) (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac{h (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2} \]
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Rubi [A] time = 0.811227, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{\sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} (e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{2 f^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} h \sqrt{p} \sqrt{q} (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )}{4 f^2}+\frac{(e+f x) (f g-e h) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac{h (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2} \]
Antiderivative was successfully verified.
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Rule 2401
Rule 2389
Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rule 2390
Rule 2305
Rule 2310
Rule 2445
Rubi steps
\begin{align*} \int (g+h x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx &=\operatorname{Subst}\left (\int (g+h x) \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, 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) \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}{f}+\frac{h (e+f x) \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )}}{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) \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, 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 \sqrt{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, 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 \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, 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 \sqrt{a+b \log \left (c d^q x^{p q}\right )} \, 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) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac{h (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\operatorname{Subst}\left (\frac{(b h p q) \operatorname{Subst}\left (\int \frac{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{(b (f g-e h) p q) \operatorname{Subst}\left (\int \frac{1}{\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{(f g-e h) (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac{h (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\operatorname{Subst}\left (\frac{\left (b 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 )}{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{\left (b (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 )}{2 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) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac{h (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\operatorname{Subst}\left (\frac{\left (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 )}{2 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 ((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 )}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{\sqrt{b} e^{-\frac{a}{b p q}} (f g-e h) \sqrt{p} \sqrt{\pi } \sqrt{q} (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 )}{2 f^2}-\frac{\sqrt{b} e^{-\frac{2 a}{b p q}} h \sqrt{p} \sqrt{\frac{\pi }{2}} \sqrt{q} (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 )}{4 f^2}+\frac{(f g-e h) (e+f x) \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{f^2}+\frac{h (e+f x)^2 \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}\\ \end{align*}
Mathematica [A] time = 0.389771, size = 298, normalized size = 0.96 \[ -\frac{(e+f x) e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \left (4 \sqrt{\pi } \sqrt{b} \sqrt{p} \sqrt{q} e^{\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )+\sqrt{2 \pi } \sqrt{b} h \sqrt{p} \sqrt{q} (e+f x) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt{b} \sqrt{p} \sqrt{q}}\right )-4 e^{\frac{2 a}{b p q}} (-e h+2 f g+f h x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{2}{p q}} \sqrt{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )}{8 f^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.27, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) \sqrt{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }\, 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{\left (h x + g\right )} \sqrt{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}\,{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 \sqrt{a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}} \left (g + h x\right )\, 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{\left (h x + g\right )} \sqrt{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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