Optimal. Leaf size=107 \[ \frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{2 b^2 p}-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{2 b^2 p} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2454, 2399, 2389, 2300, 2178, 2390, 2310} \[ \frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{2 b^2 p}-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{2 b^2 p} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2178
Rule 2300
Rule 2310
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 )^p\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a}{b \log \left (c (a+b x)^p\right )}+\frac {a+b x}{b \log \left (c (a+b x)^p\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b x}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{2 b}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b^2}-\frac {a \operatorname {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b^2}\\ &=\frac {\left (\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b^2 p}-\frac {\left (a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b^2 p}\\ &=-\frac {a \left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b^2 p}+\frac {\left (a+b x^2\right )^2 \left (c \left (a+b x^2\right )^p\right )^{-2/p} \text {Ei}\left (\frac {2 \log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b^2 p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 96, normalized size = 0.90 \[ -\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-2/p} \left (a \left (c \left (a+b x^2\right )^p\right )^{\frac {1}{p}} \text {Ei}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )-\left (a+b x^2\right ) \text {Ei}\left (\frac {2 \log \left (c \left (b x^2+a\right )^p\right )}{p}\right )\right )}{2 b^2 p} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 68, normalized size = 0.64 \[ -\frac {a c^{\left (\frac {1}{p}\right )} \operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right ) - \operatorname {log\_integral}\left ({\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} c^{\frac {2}{p}}\right )}{2 \, b^{2} c^{\frac {2}{p}} p} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 73, normalized size = 0.68 \[ -\frac {\frac {a {\rm Ei}\left (\frac {\log \relax (c)}{p} + \log \left (b x^{2} + a\right )\right )}{b c^{\left (\frac {1}{p}\right )} p} - \frac {{\rm Ei}\left (\frac {2 \, \log \relax (c)}{p} + 2 \, \log \left (b x^{2} + a\right )\right )}{b c^{\frac {2}{p}} p}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________