Optimal. Leaf size=313 \[ \frac {2 d p x}{e^2}-\frac {p x^2}{2 e}-\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^3} \]
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Rubi [A]
time = 0.23, antiderivative size = 313, normalized size of antiderivative = 1.00, number
of steps used = 17, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules
used = {2516, 2498, 327, 211, 2504, 2436, 2332, 2512, 266, 2463, 2441, 2440, 2438}
\begin {gather*} -\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^3}-\frac {2 \sqrt {a} d p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}-\frac {d^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}+\frac {2 d p x}{e^2}-\frac {p x^2}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 327
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2498
Rule 2504
Rule 2512
Rule 2516
Rubi steps
\begin {align*} \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx &=\int \left (-\frac {d \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (a+b x^2\right )^p\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{e^2}+\frac {d^2 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{e^2}+\frac {\int x \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{e}\\ &=-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {\text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 e}-\frac {\left (2 b d^2 p\right ) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{e^3}+\frac {(2 b d p) \int \frac {x^2}{a+b x^2} \, dx}{e^2}\\ &=\frac {2 d p x}{e^2}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b e}-\frac {\left (2 b d^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}{e^3}-\frac {(2 a d p) \int \frac {1}{a+b x^2} \, dx}{e^2}\\ &=\frac {2 d p x}{e^2}-\frac {p x^2}{2 e}-\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {\left (\sqrt {b} d^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{e^3}-\frac {\left (\sqrt {b} d^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{e^3}\\ &=\frac {2 d p x}{e^2}-\frac {p x^2}{2 e}-\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {\left (d^2 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}{e^2}+\frac {\left (d^2 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}{e^2}\\ &=\frac {2 d p x}{e^2}-\frac {p x^2}{2 e}-\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {\left (d^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 )}{e^3}+\frac {\left (d^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 )}{e^3}\\ &=\frac {2 d p x}{e^2}-\frac {p x^2}{2 e}-\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 271, normalized size = 0.87 \begin {gather*} \frac {-e^2 p x^2+4 d e p \left (x-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}\right )-2 d e x \log \left (c \left (a+b x^2\right )^p\right )+\frac {e^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}+2 d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )-2 d^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 )}{2 e^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 = 0.48, size = 825, normalized size = 2.64
method | result | size |
risch | \(\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} d^{2} \ln \left (e x +d \right )}{2 e^{3}}-\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) d x}{2 e^{2}}+\frac {2 d p x}{e^{2}}-\frac {p \,d^{2} \dilog \left (\frac {e \sqrt {-b a}-\left (e x +d \right ) b +b d}{e \sqrt {-b a}+b d}\right )}{e^{3}}-\frac {p \,d^{2} \dilog \left (\frac {e \sqrt {-b a}+\left (e x +d \right ) b -b d}{e \sqrt {-b a}-b d}\right )}{e^{3}}-\frac {p \,d^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-b a}-\left (e x +d \right ) b +b d}{e \sqrt {-b a}+b d}\right )}{e^{3}}-\frac {p \,d^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-b a}+\left (e x +d \right ) b -b d}{e \sqrt {-b a}-b d}\right )}{e^{3}}-\frac {\ln \left (c \right ) d x}{e^{2}}+\frac {\ln \left (c \right ) d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {2 p a d \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {b a}}\right )}{e^{2} \sqrt {b a}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d x}{e^{2}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d^{2} \ln \left (e x +d \right )}{e^{3}}+\frac {\ln \left (c \right ) x^{2}}{2 e}+\frac {p a \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{2 b e}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) d x}{2 e^{2}}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) d^{2} \ln \left (e x +d \right )}{2 e^{3}}+\frac {5 p \,d^{2}}{2 e^{3}}-\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3} x^{2}}{4 e}-\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3} d^{2} \ln \left (e x +d \right )}{2 e^{3}}-\frac {p \,x^{2}}{2 e}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) x^{2}}{4 e}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) x^{2}}{2 e}+\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) x^{2}}{4 e}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} x^{2}}{4 e}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} d x}{2 e^{2}}+\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) d^{2} \ln \left (e x +d \right )}{2 e^{3}}+\frac {i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3} d x}{2 e^{2}}\) | \(825\) |
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^{2} \log {\left (c \left (a + 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]
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 {x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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