Optimal. Leaf size=100 \[ -\frac {a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac {3 a^5}{b^7 \left (a+b x^2\right )}+\frac {15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac {5 a^3 x^2}{b^6}+\frac {3 a^2 x^4}{2 b^5}-\frac {a x^6}{2 b^4}+\frac {x^8}{8 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {3 a^2 x^4}{2 b^5}-\frac {5 a^3 x^2}{b^6}+\frac {3 a^5}{b^7 \left (a+b x^2\right )}-\frac {a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac {15 a^4 \log \left (a+b x^2\right )}{2 b^7}-\frac {a x^6}{2 b^4}+\frac {x^8}{8 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {10 a^3}{b^6}+\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{b^4}+\frac {x^3}{b^3}+\frac {a^6}{b^6 (a+b x)^3}-\frac {6 a^5}{b^6 (a+b x)^2}+\frac {15 a^4}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {5 a^3 x^2}{b^6}+\frac {3 a^2 x^4}{2 b^5}-\frac {a x^6}{2 b^4}+\frac {x^8}{8 b^3}-\frac {a^6}{4 b^7 \left (a+b x^2\right )^2}+\frac {3 a^5}{b^7 \left (a+b x^2\right )}+\frac {15 a^4 \log \left (a+b x^2\right )}{2 b^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 85, normalized size = 0.85 \[ \frac {-\frac {2 a^6}{\left (a+b x^2\right )^2}+\frac {24 a^5}{a+b x^2}+60 a^4 \log \left (a+b x^2\right )-40 a^3 b x^2+12 a^2 b^2 x^4-4 a b^3 x^6+b^4 x^8}{8 b^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 125, normalized size = 1.25 \[ \frac {b^{6} x^{12} - 2 \, a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} - 20 \, a^{3} b^{3} x^{6} - 68 \, a^{4} b^{2} x^{4} - 16 \, a^{5} b x^{2} + 22 \, a^{6} + 60 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{2} + a^{6}\right )} \log \left (b x^{2} + a\right )}{8 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.61, size = 102, normalized size = 1.02 \[ \frac {15 \, a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{7}} - \frac {45 \, a^{4} b^{2} x^{4} + 78 \, a^{5} b x^{2} + 34 \, a^{6}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{7}} + \frac {b^{9} x^{8} - 4 \, a b^{8} x^{6} + 12 \, a^{2} b^{7} x^{4} - 40 \, a^{3} b^{6} x^{2}}{8 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 91, normalized size = 0.91 \[ \frac {x^{8}}{8 b^{3}}-\frac {a \,x^{6}}{2 b^{4}}+\frac {3 a^{2} x^{4}}{2 b^{5}}-\frac {a^{6}}{4 \left (b \,x^{2}+a \right )^{2} b^{7}}-\frac {5 a^{3} x^{2}}{b^{6}}+\frac {3 a^{5}}{\left (b \,x^{2}+a \right ) b^{7}}+\frac {15 a^{4} \ln \left (b \,x^{2}+a \right )}{2 b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.29, size = 99, normalized size = 0.99 \[ \frac {12 \, a^{5} b x^{2} + 11 \, a^{6}}{4 \, {\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac {15 \, a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{7}} + \frac {b^{3} x^{8} - 4 \, a b^{2} x^{6} + 12 \, a^{2} b x^{4} - 40 \, a^{3} x^{2}}{8 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 100, normalized size = 1.00 \[ \frac {\frac {11\,a^6}{4\,b}+3\,a^5\,x^2}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}+\frac {x^8}{8\,b^3}-\frac {a\,x^6}{2\,b^4}+\frac {15\,a^4\,\ln \left (b\,x^2+a\right )}{2\,b^7}+\frac {3\,a^2\,x^4}{2\,b^5}-\frac {5\,a^3\,x^2}{b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.47, size = 104, normalized size = 1.04 \[ \frac {15 a^{4} \log {\left (a + b x^{2} \right )}}{2 b^{7}} - \frac {5 a^{3} x^{2}}{b^{6}} + \frac {3 a^{2} x^{4}}{2 b^{5}} - \frac {a x^{6}}{2 b^{4}} + \frac {11 a^{6} + 12 a^{5} b x^{2}}{4 a^{2} b^{7} + 8 a b^{8} x^{2} + 4 b^{9} x^{4}} + \frac {x^{8}}{8 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________