Optimal. Leaf size=59 \[ -\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )+\frac {a p x^2}{4 b}-\frac {p x^4}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2454, 2395, 43} \[ -\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )+\frac {a p x^2}{4 b}-\frac {p x^4}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int x^3 \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \log \left (c (a+b x)^p\right ) \, dx,x,x^2\right )\\ &=\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )-\frac {1}{4} (b p) \operatorname {Subst}\left (\int \frac {x^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )-\frac {1}{4} (b p) \operatorname {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a p x^2}{4 b}-\frac {p x^4}{8}-\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 59, normalized size = 1.00 \[ -\frac {a^2 p \log \left (a+b x^2\right )}{4 b^2}+\frac {1}{4} x^4 \log \left (c \left (a+b x^2\right )^p\right )+\frac {a p x^2}{4 b}-\frac {p x^4}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 57, normalized size = 0.97 \[ -\frac {b^{2} p x^{4} - 2 \, b^{2} x^{4} \log \relax (c) - 2 \, a b p x^{2} - 2 \, {\left (b^{2} p x^{4} - a^{2} p\right )} \log \left (b x^{2} + a\right )}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 97, normalized size = 1.64 \[ \frac {\frac {{\left (2 \, {\left (b x^{2} + a\right )}^{2} \log \left (b x^{2} + a\right ) - 4 \, {\left (b x^{2} + a\right )} a \log \left (b x^{2} + a\right ) - {\left (b x^{2} + a\right )}^{2} + 4 \, {\left (b x^{2} + a\right )} a\right )} p}{b} + \frac {2 \, {\left ({\left (b x^{2} + a\right )}^{2} - 2 \, {\left (b x^{2} + a\right )} a\right )} \log \relax (c)}{b}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.32, size = 183, normalized size = 3.10 \[ -\frac {i \pi \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}{8}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{8}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{8}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{8}-\frac {p \,x^{4}}{8}+\frac {x^{4} \ln \relax (c )}{4}+\frac {x^{4} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{4}+\frac {a p \,x^{2}}{4 b}-\frac {a^{2} p \ln \left (b \,x^{2}+a \right )}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 55, normalized size = 0.93 \[ \frac {1}{4} \, x^{4} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {1}{8} \, b p {\left (\frac {2 \, a^{2} \log \left (b x^{2} + a\right )}{b^{3}} + \frac {b x^{4} - 2 \, a x^{2}}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.22, size = 51, normalized size = 0.86 \[ \frac {x^4\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{4}-\frac {p\,x^4}{8}-\frac {a^2\,p\,\ln \left (b\,x^2+a\right )}{4\,b^2}+\frac {a\,p\,x^2}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.46, size = 70, normalized size = 1.19 \[ \begin {cases} - \frac {a^{2} p \log {\left (a + b x^{2} \right )}}{4 b^{2}} + \frac {a p x^{2}}{4 b} + \frac {p x^{4} \log {\left (a + b x^{2} \right )}}{4} - \frac {p x^{4}}{8} + \frac {x^{4} \log {\relax (c )}}{4} & \text {for}\: b \neq 0 \\\frac {x^{4} \log {\left (a^{p} c \right )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________