Optimal. Leaf size=413 \[ -\frac{15 \sqrt{\pi } b^{5/2} n^{5/2} e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{8 e^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} g n^{5/2} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{64 e^2}+\frac{15 b^2 n^2 (d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac{15 b^2 g n^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{32 e^2}+\frac{(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}-\frac{5 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2} \]
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Rubi [A] time = 0.508177, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2401, 2389, 2296, 2300, 2180, 2204, 2390, 2305, 2310} \[ -\frac{15 \sqrt{\pi } b^{5/2} n^{5/2} e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{8 e^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} g n^{5/2} e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{64 e^2}+\frac{15 b^2 n^2 (d+e x) (e f-d g) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac{15 b^2 g n^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{32 e^2}+\frac{(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}-\frac{5 b n (d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2} \]
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
Rubi steps
\begin{align*} \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx &=\int \left (\frac{(e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}+\frac{g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}\right ) \, dx\\ &=\frac{g \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx}{e}+\frac{(e f-d g) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx}{e}\\ &=\frac{g \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^{5/2} \, dx,x,d+e x\right )}{e^2}+\frac{(e f-d g) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{5/2} \, dx,x,d+e x\right )}{e^2}\\ &=\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac{(5 b g n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{4 e^2}-\frac{(5 b (e f-d g) n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{2 e^2}\\ &=-\frac{5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}+\frac{\left (15 b^2 g n^2\right ) \operatorname{Subst}\left (\int x \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{16 e^2}+\frac{\left (15 b^2 (e f-d g) n^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{4 e^2}\\ &=\frac{15 b^2 (e f-d g) n^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac{15 b^2 g n^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac{5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac{\left (15 b^3 g n^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{64 e^2}-\frac{\left (15 b^3 (e f-d g) n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{8 e^2}\\ &=\frac{15 b^2 (e f-d g) n^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac{15 b^2 g n^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac{5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac{\left (15 b^3 g n^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{64 e^2}-\frac{\left (15 b^3 (e f-d g) n^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{8 e^2}\\ &=\frac{15 b^2 (e f-d g) n^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac{15 b^2 g n^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac{5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}-\frac{\left (15 b^2 g n^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b n}+\frac{2 x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{32 e^2}-\frac{\left (15 b^2 (e f-d g) n^2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b n}+\frac{x^2}{b n}} \, dx,x,\sqrt{a+b \log \left (c (d+e x)^n\right )}\right )}{4 e^2}\\ &=-\frac{15 b^{5/2} e^{-\frac{a}{b n}} (e f-d g) n^{5/2} \sqrt{\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{8 e^2}-\frac{15 b^{5/2} e^{-\frac{2 a}{b n}} g n^{5/2} \sqrt{\frac{\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )}{64 e^2}+\frac{15 b^2 (e f-d g) n^2 (d+e x) \sqrt{a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac{15 b^2 g n^2 (d+e x)^2 \sqrt{a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac{5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac{5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac{(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac{g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2}\\ \end{align*}
Mathematica [A] time = 0.692662, size = 326, normalized size = 0.79 \[ \frac{(d+e x) \left (-80 b n (e f-d g) \left (3 \sqrt{\pi } b^{3/2} n^{3/2} e^{-\frac{a}{b n}} \left (c (d+e x)^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+2 \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (2 a+2 b \log \left (c (d+e x)^n\right )-3 b n\right )\right )-5 b g n (d+e x) \left (3 \sqrt{2 \pi } b^{3/2} n^{3/2} e^{-\frac{2 a}{b n}} \left (c (d+e x)^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \log \left (c (d+e x)^n\right )}}{\sqrt{b} \sqrt{n}}\right )+4 \sqrt{a+b \log \left (c (d+e x)^n\right )} \left (4 a+4 b \log \left (c (d+e x)^n\right )-3 b n\right )\right )+128 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}+64 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}\right )}{128 e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.487, size = 0, normalized size = 0. \begin{align*} \int \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{{\frac{5}{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{\left (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{5}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (g x + f\right )}{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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