Optimal. Leaf size=421 \[ \frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}-\frac {2 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4} \]
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Rubi [A]
time = 0.39, antiderivative size = 421, normalized size of antiderivative = 1.00, number of
steps used = 25, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used
= {2516, 2498, 269, 211, 2505, 266, 199, 327, 2512, 2463, 2441, 2352, 2440, 2438}
\begin {gather*} \frac {d^3 p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}-\frac {2 d^3 p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^4}-\frac {2 b^{3/2} p \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {2 \sqrt {b} d^2 p \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}+\frac {d^3 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}-\frac {b d p \log \left (a x^2+b\right )}{2 a e^2}+\frac {2 b p x}{3 a e}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 211
Rule 266
Rule 269
Rule 327
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2498
Rule 2505
Rule 2512
Rule 2516
Rubi steps
\begin {align*} \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx &=\int \left (\frac {d^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {d^2 \int \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx}{e^3}-\frac {d^3 \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx}{e^3}-\frac {d \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx}{e^2}+\frac {\int x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx}{e}\\ &=\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (2 b d^3 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{e^4}+\frac {\left (2 b d^2 p\right ) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^2} \, dx}{e^3}-\frac {(b d p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x} \, dx}{e^2}+\frac {(2 b p) \int \frac {1}{a+\frac {b}{x^2}} \, dx}{3 e}\\ &=\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (2 b d^3 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}{e^4}+\frac {\left (2 b d^2 p\right ) \int \frac {1}{b+a x^2} \, dx}{e^3}-\frac {(b d p) \int \frac {x}{b+a x^2} \, dx}{e^2}+\frac {(2 b p) \int \frac {x^2}{b+a x^2} \, dx}{3 e}\\ &=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}-\frac {\left (2 d^3 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{e^4}+\frac {\left (2 a d^3 p\right ) \int \frac {x \log (d+e x)}{b+a x^2} \, dx}{e^4}-\frac {\left (2 b^2 p\right ) \int \frac {1}{b+a x^2} \, dx}{3 a e}\\ &=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}+\frac {\left (2 a d^3 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}{e^4}+\frac {\left (2 d^3 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e^3}\\ &=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}-\frac {2 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}-\frac {\left (\sqrt {-a} d^3 p\right ) \int \frac {\log (d+e x)}{\sqrt {b}-\sqrt {-a} x} \, dx}{e^4}+\frac {\left (\sqrt {-a} d^3 p\right ) \int \frac {\log (d+e x)}{\sqrt {b}+\sqrt {-a} x} \, dx}{e^4}\\ &=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}-\frac {2 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}-\frac {\left (d^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}{e^3}-\frac {\left (d^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}{e^3}\\ &=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}-\frac {2 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}-\frac {\left (d^3 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 )}{e^4}-\frac {\left (d^3 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 )}{e^4}\\ &=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}-\frac {2 d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.31, size = 375, normalized size = 0.89 \begin {gather*} -\frac {\frac {12 \sqrt {b} d^2 e p \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {a}}-\frac {4 b e^3 p x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {b}{a x^2}\right )}{a}-6 d^2 e x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+3 d e^2 x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-2 e^3 x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {3 b d e^2 p \left (\log \left (a+\frac {b}{x^2}\right )+2 \log (x)\right )}{a}+6 d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)+6 d^3 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 )}{6 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{e x +d}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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