Optimal. Leaf size=88 \[ a^5 A \log (x)+\frac {5}{2} a^4 A b x^2+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{3} a^2 A b^3 x^6+\frac {5}{8} a A b^4 x^8+\frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{10} A b^5 x^{10} \]
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Rubi [A] time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 80, 43} \[ \frac {5}{3} a^2 A b^3 x^6+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{2} a^4 A b x^2+a^5 A \log (x)+\frac {5}{8} a A b^4 x^8+\frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{10} A b^5 x^{10} \]
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{2} A \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (a+b x^2\right )^6}{12 b}+\frac {1}{2} A \operatorname {Subst}\left (\int \left (5 a^4 b+\frac {a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {5}{2} a^4 A b x^2+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{3} a^2 A b^3 x^6+\frac {5}{8} a A b^4 x^8+\frac {1}{10} A b^5 x^{10}+\frac {B \left (a+b x^2\right )^6}{12 b}+a^5 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 113, normalized size = 1.28 \[ a^5 A \log (x)+\frac {1}{2} a^4 x^2 (a B+5 A b)+\frac {5}{4} a^3 b x^4 (a B+2 A b)+\frac {5}{3} a^2 b^2 x^6 (a B+A b)+\frac {1}{10} b^4 x^{10} (5 a B+A b)+\frac {5}{8} a b^3 x^8 (2 a B+A b)+\frac {1}{12} b^5 B x^{12} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 117, normalized size = 1.33 \[ \frac {1}{12} \, B b^{5} x^{12} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + A a^{5} \log \relax (x) + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 126, normalized size = 1.43 \[ \frac {1}{12} \, B b^{5} x^{12} + \frac {1}{2} \, B a b^{4} x^{10} + \frac {1}{10} \, A b^{5} x^{10} + \frac {5}{4} \, B a^{2} b^{3} x^{8} + \frac {5}{8} \, A a b^{4} x^{8} + \frac {5}{3} \, B a^{3} b^{2} x^{6} + \frac {5}{3} \, A a^{2} b^{3} x^{6} + \frac {5}{4} \, B a^{4} b x^{4} + \frac {5}{2} \, A a^{3} b^{2} x^{4} + \frac {1}{2} \, B a^{5} x^{2} + \frac {5}{2} \, A a^{4} b x^{2} + \frac {1}{2} \, A a^{5} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 124, normalized size = 1.41 \[ \frac {B \,b^{5} x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 A a \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 A \,a^{2} b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 A \,a^{3} b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 A \,a^{4} b \,x^{2}}{2}+\frac {B \,a^{5} x^{2}}{2}+A \,a^{5} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 120, normalized size = 1.36 \[ \frac {1}{12} \, B b^{5} x^{12} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + \frac {1}{2} \, A a^{5} \log \left (x^{2}\right ) + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 105, normalized size = 1.19 \[ x^2\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )+x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )+\frac {B\,b^5\,x^{12}}{12}+A\,a^5\,\ln \relax (x)+\frac {5\,a^2\,b^2\,x^6\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^4\,\left (2\,A\,b+B\,a\right )}{4}+\frac {5\,a\,b^3\,x^8\,\left (A\,b+2\,B\,a\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 134, normalized size = 1.52 \[ A a^{5} \log {\relax (x )} + \frac {B b^{5} x^{12}}{12} + x^{10} \left (\frac {A b^{5}}{10} + \frac {B a b^{4}}{2}\right ) + x^{8} \left (\frac {5 A a b^{4}}{8} + \frac {5 B a^{2} b^{3}}{4}\right ) + x^{6} \left (\frac {5 A a^{2} b^{3}}{3} + \frac {5 B a^{3} b^{2}}{3}\right ) + x^{4} \left (\frac {5 A a^{3} b^{2}}{2} + \frac {5 B a^{4} b}{4}\right ) + x^{2} \left (\frac {5 A a^{4} b}{2} + \frac {B a^{5}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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