Optimal. Leaf size=371 \[ -\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3} \]
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Rubi [A]
time = 0.28, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 15, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used =
{2516, 2504, 2442, 36, 29, 31, 2505, 211, 2441, 2352, 2512, 266, 2463, 2440, 2438}
\begin {gather*} \frac {e^2 p \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^3}-\frac {2 \sqrt {b} e p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}+\frac {b p \log (x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2504
Rule 2505
Rule 2512
Rule 2516
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx &=\int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x^3}-\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x^2}+\frac {e^2 \log \left (c \left (a+b x^2\right )^p\right )}{d^3 x}-\frac {e^3 \log \left (c \left (a+b x^2\right )^p\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{d^3}\\ &=\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {\text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right )}{2 d}+\frac {e^2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {(2 b e p) \int \frac {1}{a+b x^2} \, dx}{d^2}+\frac {\left (2 b e^2 p\right ) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{d^3}\\ &=-\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {(b p) \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^2\right )}{2 d}-\frac {\left (b e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{2 d^3}+\frac {\left (2 b e^2 p\right ) \int \left (-\frac {\log (d+e x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\log (d+e x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{d^3}\\ &=-\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}+\frac {(b p) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 a d}-\frac {\left (b^2 p\right ) \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^2\right )}{2 a d}-\frac {\left (\sqrt {b} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{d^3}+\frac {\left (\sqrt {b} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{d^3}\\ &=-\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx}{d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx}{d^3}\\ &=-\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} d+\sqrt {-a} 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 {b} x}{\sqrt {b} d+\sqrt {-a} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}\\ &=-\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 320, normalized size = 0.86 \begin {gather*} \frac {-\frac {4 \sqrt {b} d e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {b d^2 p \left (2 \log (x)-\log \left (a+b x^2\right )\right )}{a}-\frac {d^2 \log \left (c \left (a+b x^2\right )^p\right )}{x^2}+\frac {2 d e \log \left (c \left (a+b x^2\right )^p\right )}{x}-2 e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 e^2 p \left (\left (\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right )+\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right )\right ) \log (d+e x)+\text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+\text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )\right )+e^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \text {Li}_2\left (1+\frac {b x^2}{a}\right )\right )}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.40, size = 1071, normalized size = 2.89
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1071\) |
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 (b\,x^2+a\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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