Optimal. Leaf size=71 \[ \frac {\text {Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{b^2 c^2}-\frac {a \text {li}\left (c \left (b x^2+a\right )\right )}{2 b^2 c}-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2454, 2400, 2399, 2389, 2298, 2390, 2309, 2178} \[ \frac {\text {Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{b^2 c^2}-\frac {a \text {li}\left (c \left (b x^2+a\right )\right )}{2 b^2 c}-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2298
Rule 2309
Rule 2389
Rule 2390
Rule 2399
Rule 2400
Rule 2454
Rubi steps
\begin {align*} \int \frac {x^3}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\log ^2(c (a+b x))} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}+\operatorname {Subst}\left (\int \frac {x}{\log (c (a+b x))} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}+\operatorname {Subst}\left (\int \left (-\frac {a}{b \log (c (a+b x))}+\frac {a+b x}{b \log (c (a+b x))}\right ) \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac {\operatorname {Subst}\left (\int \frac {a+b x}{\log (c (a+b x))} \, dx,x,x^2\right )}{b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\log (c (a+b x))} \, dx,x,x^2\right )}{b}\\ &=-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac {\operatorname {Subst}\left (\int \frac {x}{\log (c x)} \, dx,x,a+b x^2\right )}{b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{b^2}\\ &=-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}-\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}+\frac {\operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )\right )\right )}{b^2 c^2}\\ &=\frac {\text {Ei}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{b^2 c^2}-\frac {x^2 \left (a+b x^2\right )}{2 b \log \left (c \left (a+b x^2\right )\right )}-\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 66, normalized size = 0.93 \[ -\frac {-\frac {2 \text {Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{c^2}+\frac {a \text {Ei}\left (\log \left (c \left (b x^2+a\right )\right )\right )}{c}+\frac {b x^2 \left (a+b x^2\right )}{\log \left (c \left (a+b x^2\right )\right )}}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 99, normalized size = 1.39 \[ -\frac {b^{2} c^{2} x^{4} + a b c^{2} x^{2} + {\left (a c \operatorname {log\_integral}\left (b c x^{2} + a c\right ) - 2 \, \operatorname {log\_integral}\left (b^{2} c^{2} x^{4} + 2 \, a b c^{2} x^{2} + a^{2} c^{2}\right )\right )} \log \left (b c x^{2} + a c\right )}{2 \, b^{2} c^{2} \log \left (b c x^{2} + a c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 89, normalized size = 1.25 \[ -\frac {a c {\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right ) - \frac {{\left (b c x^{2} + a c\right )} a c}{\log \left ({\left (b x^{2} + a\right )} c\right )} + \frac {{\left (b c x^{2} + a c\right )}^{2}}{\log \left ({\left (b x^{2} + a\right )} c\right )} - 2 \, {\rm Ei}\left (2 \, \log \left ({\left (b x^{2} + a\right )} c\right )\right )}{2 \, b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\ln \left (\left (b \,x^{2}+a \right ) c \right )^{2}}\, dx \]
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 \[ -\frac {b x^{4} + a x^{2}}{2 \, {\left (b \log \left (b x^{2} + a\right ) + b \log \relax (c)\right )}} + \int \frac {2 \, b x^{3} + a x}{b \log \left (b x^{2} + a\right ) + b \log \relax (c)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{{\ln \left (c\,\left (b\,x^2+a\right )\right )}^2} \,d x \]
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 \[ \frac {- a x^{2} - b x^{4}}{2 b \log {\left (c \left (a + b x^{2}\right ) \right )}} + \frac {\int \frac {a x}{\log {\left (a c + b c x^{2} \right )}}\, dx + \int \frac {2 b x^{3}}{\log {\left (a c + b c x^{2} \right )}}\, dx}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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