Optimal. Leaf size=179 \[ -\frac {15 b^{5/2} e^{-\frac {a}{b n}} n^{5/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{8 e}+\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e} \]
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Rubi [A]
time = 0.11, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2436, 2333,
2337, 2211, 2235} \begin {gather*} -\frac {15 \sqrt {\pi } b^{5/2} n^{5/2} e^{-\frac {a}{b n}} (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}+\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2436
Rubi steps
\begin {align*} \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2} \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{5/2} \, dx,x,d+e x\right )}{e}\\ &=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}-\frac {(5 b 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}\\ &=-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}+\frac {\left (15 b^2 n^2\right ) \text {Subst}\left (\int \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{4 e}\\ &=\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}-\frac {\left (15 b^3 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}\\ &=\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}-\frac {\left (15 b^3 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}\\ &=\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}-\frac {\left (15 b^2 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}\\ &=-\frac {15 b^{5/2} e^{-\frac {a}{b n}} 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}+\frac {15 b^2 n^2 (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{4 e}-\frac {5 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}{e}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 152, normalized size = 0.85 \begin {gather*} \frac {(d+e x) \left (8 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}-5 b 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 )\right )}{8 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{\frac {5}{2}}\, dx\]
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}}\, 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 = -0.01 \begin {gather*} \int {\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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