Optimal. Leaf size=112 \[ -\frac {a^5 A}{8 x^8}-\frac {a^4 (a B+5 A b)}{6 x^6}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {5 a^2 b^2 (a B+A b)}{x^2}+\frac {1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac {1}{4} b^5 B x^4 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 112, 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, 76} \[ -\frac {5 a^2 b^2 (a B+A b)}{x^2}-\frac {a^4 (a B+5 A b)}{6 x^6}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {a^5 A}{8 x^8}+\frac {1}{2} b^4 x^2 (5 a B+A b)+5 a b^3 \log (x) (2 a B+A b)+\frac {1}{4} b^5 B x^4 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 76
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^4 (A b+5 a B)+\frac {a^5 A}{x^5}+\frac {a^4 (5 A b+a B)}{x^4}+\frac {5 a^3 b (2 A b+a B)}{x^3}+\frac {10 a^2 b^2 (A b+a B)}{x^2}+\frac {5 a b^3 (A b+2 a B)}{x}+b^5 B x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5 A}{8 x^8}-\frac {a^4 (5 A b+a B)}{6 x^6}-\frac {5 a^3 b (2 A b+a B)}{4 x^4}-\frac {5 a^2 b^2 (A b+a B)}{x^2}+\frac {1}{2} b^4 (A b+5 a B) x^2+\frac {1}{4} b^5 B x^4+5 a b^3 (A b+2 a B) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 116, normalized size = 1.04 \[ 5 a b^3 \log (x) (2 a B+A b)-\frac {a^5 \left (3 A+4 B x^2\right )+10 a^4 b x^2 \left (2 A+3 B x^2\right )+60 a^3 b^2 x^4 \left (A+2 B x^2\right )+120 a^2 A b^3 x^6-60 a b^4 B x^{10}-6 b^5 x^{10} \left (2 A+B x^2\right )}{24 x^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 123, normalized size = 1.10 \[ \frac {6 \, B b^{5} x^{12} + 12 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 120 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} \log \relax (x) - 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 3 \, A a^{5} - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.37, size = 150, normalized size = 1.34 \[ \frac {1}{4} \, B b^{5} x^{4} + \frac {5}{2} \, B a b^{4} x^{2} + \frac {1}{2} \, A b^{5} x^{2} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left (x^{2}\right ) - \frac {250 \, B a^{2} b^{3} x^{8} + 125 \, A a b^{4} x^{8} + 120 \, B a^{3} b^{2} x^{6} + 120 \, A a^{2} b^{3} x^{6} + 30 \, B a^{4} b x^{4} + 60 \, A a^{3} b^{2} x^{4} + 4 \, B a^{5} x^{2} + 20 \, A a^{4} b x^{2} + 3 \, A a^{5}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 124, normalized size = 1.11 \[ \frac {B \,b^{5} x^{4}}{4}+\frac {A \,b^{5} x^{2}}{2}+\frac {5 B a \,b^{4} x^{2}}{2}+5 A a \,b^{4} \ln \relax (x )+10 B \,a^{2} b^{3} \ln \relax (x )-\frac {5 A \,a^{2} b^{3}}{x^{2}}-\frac {5 B \,a^{3} b^{2}}{x^{2}}-\frac {5 A \,a^{3} b^{2}}{2 x^{4}}-\frac {5 B \,a^{4} b}{4 x^{4}}-\frac {5 A \,a^{4} b}{6 x^{6}}-\frac {B \,a^{5}}{6 x^{6}}-\frac {A \,a^{5}}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 123, normalized size = 1.10 \[ \frac {1}{4} \, B b^{5} x^{4} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} \log \left (x^{2}\right ) - \frac {120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 3 \, A a^{5} + 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 122, normalized size = 1.09 \[ \ln \relax (x)\,\left (10\,B\,a^2\,b^3+5\,A\,a\,b^4\right )-\frac {\frac {A\,a^5}{8}+x^4\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^2\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+x^6\,\left (5\,B\,a^3\,b^2+5\,A\,a^2\,b^3\right )}{x^8}+x^2\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+\frac {B\,b^5\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.73, size = 129, normalized size = 1.15 \[ \frac {B b^{5} x^{4}}{4} + 5 a b^{3} \left (A b + 2 B a\right ) \log {\relax (x )} + x^{2} \left (\frac {A b^{5}}{2} + \frac {5 B a b^{4}}{2}\right ) + \frac {- 3 A a^{5} + x^{6} \left (- 120 A a^{2} b^{3} - 120 B a^{3} b^{2}\right ) + x^{4} \left (- 60 A a^{3} b^{2} - 30 B a^{4} b\right ) + x^{2} \left (- 20 A a^{4} b - 4 B a^{5}\right )}{24 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________