Optimal. Leaf size=45 \[ \frac {\text {Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{2 b^2 c^2}-\frac {a \text {li}\left (c \left (b x^2+a\right )\right )}{2 b^2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2454, 2399, 2389, 2298, 2390, 2309, 2178} \[ \frac {\text {Ei}\left (2 \log \left (c \left (b x^2+a\right )\right )\right )}{2 b^2 c^2}-\frac {a \text {li}\left (c \left (b x^2+a\right )\right )}{2 b^2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2298
Rule 2309
Rule 2389
Rule 2390
Rule 2399
Rule 2454
Rubi steps
\begin {align*} \int \frac {x^3}{\log \left (c \left (a+b x^2\right )\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\log (c (a+b x))} \, dx,x,x^2\right )\\ &=\frac {1}{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 {\operatorname {Subst}\left (\int \frac {a+b x}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\log (c (a+b x))} \, dx,x,x^2\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b^2}\\ &=-\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 )}{2 b^2 c^2}\\ &=\frac {\text {Ei}\left (2 \log \left (c \left (a+b x^2\right )\right )\right )}{2 b^2 c^2}-\frac {a \text {li}\left (c \left (a+b x^2\right )\right )}{2 b^2 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 41, normalized size = 0.91 \[ \frac {\text {Ei}\left (2 \log \left (b c x^2+a c\right )\right )-a c \text {Ei}\left (\log \left (b c x^2+a c\right )\right )}{2 b^2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 54, normalized size = 1.20 \[ -\frac {a c \operatorname {log\_integral}\left (b c x^{2} + a c\right ) - \operatorname {log\_integral}\left (b^{2} c^{2} x^{4} + 2 \, a b c^{2} x^{2} + a^{2} c^{2}\right )}{2 \, b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 38, normalized size = 0.84 \[ -\frac {a c {\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right ) - {\rm Ei}\left (2 \, \log \left ({\left (b x^{2} + a\right )} c\right )\right )}{2 \, b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\ln \left (\left (b \,x^{2}+a \right ) c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )} c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{\ln \left (c\,\left (b\,x^2+a\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\log {\left (a c + b c x^{2} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________