Optimal. Leaf size=423 \[ \frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}} \]
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Rubi [A]
time = 0.80, antiderivative size = 423, normalized size of antiderivative = 1.00, number of
steps used = 14, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used
= {2445, 2458, 2389, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356}
\begin {gather*} -\frac {8 b^2 e^{3/2} n^2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g \sqrt {f+g x} (e f-d g)}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2389
Rule 2390
Rule 2445
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{5/2}} \, dx &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {(4 b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{3/2}} \, dx}{3 g}\\ &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {(4 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{3 g}\\ &=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {(4 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (\frac {e f-d g}{e}+\frac {g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{3 (e f-d g)}+\frac {(4 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 g (e f-d g)}\\ &=\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {\left (4 b^2 e n^2\right ) \text {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{3 g (e f-d g)}-\frac {\left (8 b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{3 g (e f-d g)}\\ &=\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {\left (8 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{3 g (e f-d g)^{3/2}}-\frac {\left (16 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 g^2 (e f-d g)}\\ &=\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {\left (16 b^2 e^{5/2} n^2\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^{3/2}}\\ &=\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}+\frac {\left (16 b^2 e^{5/2} n^2\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^{3/2}}\\ &=\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {\left (16 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^2}\\ &=\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}+\frac {\left (16 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{3 g (e f-d g)^2}\\ &=\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {\left (16 b^2 e^{3/2} n^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}\\ &=\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 g (e f-d g)^{3/2}}+\frac {8 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{3 g (e f-d g)^{3/2}}+\frac {8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g) \sqrt {f+g x}}-\frac {8 b e^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g (e f-d g)^{3/2}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{3 g (f+g x)^{3/2}}-\frac {16 b^2 e^{3/2} n^2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}-\frac {8 b^2 e^{3/2} n^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{3 g (e f-d g)^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.89, size = 419, normalized size = 0.99 \begin {gather*} \frac {2 \left (-2 b \sqrt {e f-d g} n \left (2 e^{3/2} (f+g x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )-\sqrt {e f-d g} (2 e (f+g x)+(-e f+d g) \log (d+e x))\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )-(e f-d g)^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+b^2 n^2 \left (3 e g (d+e x) (f+g x) \sqrt {\frac {e (f+g x)}{e f-d g}} \, _4F_3\left (1,1,1,\frac {5}{2};2,2,2;\frac {g (d+e x)}{-e f+d g}\right )+(e f-d g) \log (d+e x) \left (\left (d g+e g x \sqrt {\frac {e (f+g x)}{e f-d g}}+e f \left (-1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right ) \log (d+e x)-4 e (f+g x) \left (-1+\sqrt {\frac {e (f+g x)}{e f-d g}}+\sqrt {\frac {e (f+g x)}{e f-d g}} \log \left (\frac {1}{2} \left (1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )\right )\right )\right )\right )}{3 g (e f-d g)^2 (f+g x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{\left (g x +f \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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 \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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