Optimal. Leaf size=413 \[ -\frac {15 b^{5/2} e^{-\frac {a}{b n}} (e f-d g) n^{5/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b n}} g n^{5/2} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{64 e^2}+\frac {15 b^2 (e f-d g) n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e^2}+\frac {15 b^2 g n^2 (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{32 e^2}-\frac {5 b (e f-d g) n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac {5 b g n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{8 e^2}+\frac {(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{2 e^2} \]
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Rubi [A]
time = 0.38, antiderivative size = 413, normalized size of antiderivative = 1.00, 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 = {2448, 2436,
2333, 2337, 2211, 2235, 2437, 2342, 2347} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2342
Rule 2347
Rule 2436
Rule 2437
Rule 2448
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 \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.45, size = 326, normalized size = 0.79 \begin {gather*} \frac {(d+e x) \left (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}-80 b (e f-d g) n \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \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-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )-5 b g n (d+e x) \left (3 b^{3/2} e^{-\frac {2 a}{b n}} n^{3/2} \sqrt {2 \pi } \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-3 b n+4 b \log \left (c (d+e x)^n\right )\right )\right )\right )}{128 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (g x +f \right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{\frac {5}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (f+g\,x\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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