Optimal. Leaf size=48 \[ -\frac {a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac {b^4 x^6}{6} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \begin {gather*} 3 a^2 b^2 x^2+4 a^3 b \log (x)-\frac {a^4}{2 x^2}+a b^3 x^4+\frac {b^4 x^6}{6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^3} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^3} \, dx}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^2} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (6 a^2 b^6+\frac {a^4 b^4}{x^2}+\frac {4 a^3 b^5}{x}+4 a b^7 x+b^8 x^2\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a^4}{2 x^2}+3 a^2 b^2 x^2+a b^3 x^4+\frac {b^4 x^6}{6}+4 a^3 b \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 48, normalized size = 1.00 \begin {gather*} -\frac {a^4}{2 x^2}+4 a^3 b \log (x)+3 a^2 b^2 x^2+a b^3 x^4+\frac {b^4 x^6}{6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 49, normalized size = 1.02 \begin {gather*} \frac {b^{4} x^{8} + 6 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} \log \relax (x) - 3 \, a^{4}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 56, normalized size = 1.17 \begin {gather*} \frac {1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac {4 \, a^{3} b x^{2} + a^{4}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 45, normalized size = 0.94 \begin {gather*} \frac {b^{4} x^{6}}{6}+a \,b^{3} x^{4}+3 a^{2} b^{2} x^{2}+4 a^{3} b \ln \relax (x )-\frac {a^{4}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.34, size = 46, normalized size = 0.96 \begin {gather*} \frac {1}{6} \, b^{4} x^{6} + a b^{3} x^{4} + 3 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b \log \left (x^{2}\right ) - \frac {a^{4}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 44, normalized size = 0.92 \begin {gather*} \frac {b^4\,x^6}{6}-\frac {a^4}{2\,x^2}+a\,b^3\,x^4+4\,a^3\,b\,\ln \relax (x)+3\,a^2\,b^2\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.17, size = 46, normalized size = 0.96 \begin {gather*} - \frac {a^{4}}{2 x^{2}} + 4 a^{3} b \log {\relax (x )} + 3 a^{2} b^{2} x^{2} + a b^{3} x^{4} + \frac {b^{4} x^{6}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________