Optimal. Leaf size=414 \[ \frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3} \]
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Rubi [A]
time = 0.37, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used =
{2516, 2504, 2436, 2332, 2505, 269, 331, 211, 2441, 2352, 2512, 266, 2463, 2440, 2438}
\begin {gather*} -\frac {e^2 p \text {PolyLog}\left (2,\frac {b}{a x^2}+1\right )}{2 d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^3}-\frac {2 \sqrt {a} e p \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {e^2 \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {2 e p}{d^2 x}+\frac {p}{2 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 269
Rule 331
Rule 2332
Rule 2352
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2504
Rule 2505
Rule 2512
Rule 2516
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3 (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d x^3}-\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x^2}+\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^3 x}-\frac {e^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx}{d^3}\\ &=\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {\text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x^2}\right )}{2 d}-\frac {e^2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x^2}\right )}{2 d^3}+\frac {(2 b e p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^4} \, dx}{d^2}-\frac {\left (2 b e^2 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{d^3}\\ &=\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+\frac {b}{x^2}\right )}{2 b d}+\frac {(2 b e p) \int \frac {1}{x^2 \left (b+a x^2\right )} \, dx}{d^2}+\frac {\left (b e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x^2}\right )}{2 d^3}-\frac {\left (2 b e^2 p\right ) \int \left (\frac {\log (d+e x)}{b x}-\frac {a x \log (d+e x)}{b \left (b+a x^2\right )}\right ) \, dx}{d^3}\\ &=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}-\frac {(2 a e p) \int \frac {1}{b+a x^2} \, dx}{d^2}-\frac {\left (2 e^2 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{d^3}+\frac {\left (2 a e^2 p\right ) \int \frac {x \log (d+e x)}{b+a x^2} \, dx}{d^3}\\ &=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}+\frac {\left (2 a e^2 p\right ) \int \left (-\frac {\sqrt {-a} \log (d+e x)}{2 a \left (\sqrt {b}-\sqrt {-a} x\right )}+\frac {\sqrt {-a} \log (d+e x)}{2 a \left (\sqrt {b}+\sqrt {-a} x\right )}\right ) \, dx}{d^3}+\frac {\left (2 e^3 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d^3}\\ &=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}-\frac {2 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}-\frac {\left (\sqrt {-a} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {b}-\sqrt {-a} x} \, dx}{d^3}+\frac {\left (\sqrt {-a} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {b}+\sqrt {-a} x} \, dx}{d^3}\\ &=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}-\frac {2 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{d+e x} \, dx}{d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right )}{d+e x} \, dx}{d^3}\\ &=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}-\frac {2 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-a} x}{-\sqrt {-a} d+\sqrt {b} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-a} x}{\sqrt {-a} d+\sqrt {b} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}\\ &=\frac {p}{2 d x^2}-\frac {2 e p}{d^2 x}-\frac {2 \sqrt {a} e p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b} d^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b d}+\frac {e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d^2 x}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )}{2 d^3}-\frac {e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{d^3}-\frac {2 e^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{d^3}-\frac {e^2 p \text {Li}_2\left (1+\frac {b}{a x^2}\right )}{2 d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{d^3}-\frac {2 e^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 364, normalized size = 0.88 \begin {gather*} \frac {4 d e p \left (-\frac {1}{x}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}\right )+\frac {2 d e \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+d^2 \left (\frac {p}{x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{b}\right )-2 e^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)-e^2 \left (\log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )+p \text {Li}_2\left (1+\frac {b}{a x^2}\right )\right )-2 e^2 p \left (2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)-\log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-\log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-\text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )-\text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )+2 \text {Li}_2\left (1+\frac {e x}{d}\right )\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{x^{3} \left (e x +d \right )}\, dx\]
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]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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