Optimal. Leaf size=314 \[ \frac {2}{15} \left (-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}+\frac {15}{d^3 r \sqrt {d+e x^r}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {3}{d r \left (d+e x^r\right )^{5/2}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^r+d}}\right )}{d^{7/2} r^2}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{d^{7/2} r^2}+\frac {92 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{15 d^{7/2} r^2}-\frac {4 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r^2}-\frac {32 b n}{15 d^3 r^2 \sqrt {d+e x^r}}-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}} \]
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Rubi [A] time = 0.47, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {266, 51, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac {2 b n \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{d^{7/2} r^2}+\frac {2}{15} \left (\frac {15}{d^3 r \sqrt {d+e x^r}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}+\frac {3}{d r \left (d+e x^r\right )^{5/2}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {32 b n}{15 d^3 r^2 \sqrt {d+e x^r}}-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{d^{7/2} r^2}+\frac {92 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{15 d^{7/2} r^2}-\frac {4 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 2315
Rule 2348
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^{7/2}} \, dx &=\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {2}{5 d r x \left (d+e x^r\right )^{5/2}}+\frac {2}{3 d^2 r x \left (d+e x^r\right )^{3/2}}+\frac {2}{d^3 r x \sqrt {d+e x^r}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r x}\right ) \, dx\\ &=\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {(2 b n) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{x} \, dx}{d^{7/2} r}-\frac {(2 b n) \int \frac {1}{x \sqrt {d+e x^r}} \, dx}{d^3 r}-\frac {(2 b n) \int \frac {1}{x \left (d+e x^r\right )^{3/2}} \, dx}{3 d^2 r}-\frac {(2 b n) \int \frac {1}{x \left (d+e x^r\right )^{5/2}} \, dx}{5 d r}\\ &=\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {(2 b n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx,x,x^r\right )}{d^{7/2} r^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{d^3 r^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^r\right )}{3 d^2 r^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^{5/2}} \, dx,x,x^r\right )}{5 d r^2}\\ &=-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}}-\frac {4 b n}{3 d^3 r^2 \sqrt {d+e x^r}}+\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {(4 b n) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x^r}\right )}{d^{7/2} r^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{3 d^3 r^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^r\right )}{5 d^2 r^2}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{d^3 e r^2}\\ &=-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}}-\frac {32 b n}{15 d^3 r^2 \sqrt {d+e x^r}}+\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r^2}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{d^{7/2} r^2}+\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x^r}\right )}{d^4 r^2}-\frac {(2 b n) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{5 d^3 r^2}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{3 d^3 e r^2}\\ &=-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}}-\frac {32 b n}{15 d^3 r^2 \sqrt {d+e x^r}}+\frac {16 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 d^{7/2} r^2}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{d^{7/2} r^2}+\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{d^{7/2} r^2}+\frac {(4 b n) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x^r}\right )}{d^4 r^2}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{5 d^3 e r^2}\\ &=-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}}-\frac {32 b n}{15 d^3 r^2 \sqrt {d+e x^r}}+\frac {92 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{15 d^{7/2} r^2}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{d^{7/2} r^2}+\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{d^{7/2} r^2}-\frac {(4 b n) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x^r}}{\sqrt {d}}}\right )}{d^{7/2} r^2}\\ &=-\frac {4 b n}{15 d^2 r^2 \left (d+e x^r\right )^{3/2}}-\frac {32 b n}{15 d^3 r^2 \sqrt {d+e x^r}}+\frac {92 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{15 d^{7/2} r^2}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{d^{7/2} r^2}+\frac {2}{15} \left (\frac {3}{d r \left (d+e x^r\right )^{5/2}}+\frac {5}{d^2 r \left (d+e x^r\right )^{3/2}}+\frac {15}{d^3 r \sqrt {d+e x^r}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{d^{7/2} r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{d^{7/2} r^2}-\frac {2 b n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x^r}}{\sqrt {d}}}\right )}{d^{7/2} r^2}\\ \end {align*}
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Mathematica [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^{7/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{\frac {7}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \,x^{r}+d \right )^{\frac {7}{2}} x}\, dx \]
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 \[ \frac {1}{15} \, a {\left (\frac {15 \, \log \left (\frac {\sqrt {e x^{r} + d} - \sqrt {d}}{\sqrt {e x^{r} + d} + \sqrt {d}}\right )}{d^{\frac {7}{2}} r} + \frac {2 \, {\left (15 \, {\left (e x^{r} + d\right )}^{2} + 5 \, {\left (e x^{r} + d\right )} d + 3 \, d^{2}\right )}}{{\left (e x^{r} + d\right )}^{\frac {5}{2}} d^{3} r}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{{\left (e^{3} x x^{3 \, r} + 3 \, d e^{2} x x^{2 \, r} + 3 \, d^{2} e x x^{r} + d^{3} x\right )} \sqrt {e x^{r} + d}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x^r\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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