Optimal. Leaf size=75 \[ \frac {a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}-\frac {a x^2 (A b-a B)}{2 b^3}+\frac {x^4 (A b-a B)}{4 b^2}+\frac {B x^6}{6 b} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} \frac {a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}+\frac {x^4 (A b-a B)}{4 b^2}-\frac {a x^2 (A b-a B)}{2 b^3}+\frac {B x^6}{6 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (A+B x)}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a (-A b+a B)}{b^3}+\frac {(A b-a B) x}{b^2}+\frac {B x^2}{b}-\frac {a^2 (-A b+a B)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a (A b-a B) x^2}{2 b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^6}{6 b}+\frac {a^2 (A b-a B) \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 71, normalized size = 0.95 \begin {gather*} \frac {b x^2 \left (6 a^2 B-3 a b \left (2 A+B x^2\right )+b^2 x^2 \left (3 A+2 B x^2\right )\right )+6 a^2 (A b-a B) \log \left (a+b x^2\right )}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (A+B x^2\right )}{a+b x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 75, normalized size = 1.00 \begin {gather*} \frac {2 \, B b^{3} x^{6} - 3 \, {\left (B a b^{2} - A b^{3}\right )} x^{4} + 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} - 6 \, {\left (B a^{3} - A a^{2} b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.31, size = 77, normalized size = 1.03 \begin {gather*} \frac {2 \, B b^{2} x^{6} - 3 \, B a b x^{4} + 3 \, A b^{2} x^{4} + 6 \, B a^{2} x^{2} - 6 \, A a b x^{2}}{12 \, b^{3}} - \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 86, normalized size = 1.15 \begin {gather*} \frac {B \,x^{6}}{6 b}+\frac {A \,x^{4}}{4 b}-\frac {B a \,x^{4}}{4 b^{2}}-\frac {A a \,x^{2}}{2 b^{2}}+\frac {B \,a^{2} x^{2}}{2 b^{3}}+\frac {A \,a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}}-\frac {B \,a^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.03, size = 74, normalized size = 0.99 \begin {gather*} \frac {2 \, B b^{2} x^{6} - 3 \, {\left (B a b - A b^{2}\right )} x^{4} + 6 \, {\left (B a^{2} - A a b\right )} x^{2}}{12 \, b^{3}} - \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 76, normalized size = 1.01 \begin {gather*} x^4\,\left (\frac {A}{4\,b}-\frac {B\,a}{4\,b^2}\right )+\frac {B\,x^6}{6\,b}-\frac {\ln \left (b\,x^2+a\right )\,\left (B\,a^3-A\,a^2\,b\right )}{2\,b^4}-\frac {a\,x^2\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.32, size = 70, normalized size = 0.93 \begin {gather*} \frac {B x^{6}}{6 b} - \frac {a^{2} \left (- A b + B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{4}} + x^{4} \left (\frac {A}{4 b} - \frac {B a}{4 b^{2}}\right ) + x^{2} \left (- \frac {A a}{2 b^{2}} + \frac {B a^{2}}{2 b^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________