Optimal. Leaf size=42 \[ \frac{\left (a+b x^2\right )^6 (A b-a B)}{12 b^2}+\frac{B \left (a+b x^2\right )^7}{14 b^2} \]
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Rubi [A] time = 0.0661224, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 43} \[ \frac{\left (a+b x^2\right )^6 (A b-a B)}{12 b^2}+\frac{B \left (a+b x^2\right )^7}{14 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^5 (A+B x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(A b-a B) (a+b x)^5}{b}+\frac{B (a+b x)^6}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{(A b-a B) \left (a+b x^2\right )^6}{12 b^2}+\frac{B \left (a+b x^2\right )^7}{14 b^2}\\ \end{align*}
Mathematica [B] time = 0.0229003, size = 107, normalized size = 2.55 \[ \frac{1}{84} x^2 \left (105 a^2 b^2 x^6 (a B+A b)+70 a^3 b x^4 (a B+2 A b)+21 a^4 x^2 (a B+5 A b)+42 a^5 A+7 b^4 x^{10} (5 a B+A b)+42 a b^3 x^8 (2 a B+A b)+6 b^5 B x^{12}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 124, normalized size = 3. \begin{align*}{\frac{{b}^{5}B{x}^{14}}{14}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{12}}{12}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{8}}{8}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{6}}{6}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{4}}{4}}+{\frac{{a}^{5}A{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.980255, size = 161, normalized size = 3.83 \begin{align*} \frac{1}{14} \, B b^{5} x^{14} + \frac{1}{12} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{12} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{5}{4} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac{1}{2} \, A a^{5} x^{2} + \frac{5}{6} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac{1}{4} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.26913, size = 296, normalized size = 7.05 \begin{align*} \frac{1}{14} x^{14} b^{5} B + \frac{5}{12} x^{12} b^{4} a B + \frac{1}{12} x^{12} b^{5} A + x^{10} b^{3} a^{2} B + \frac{1}{2} x^{10} b^{4} a A + \frac{5}{4} x^{8} b^{2} a^{3} B + \frac{5}{4} x^{8} b^{3} a^{2} A + \frac{5}{6} x^{6} b a^{4} B + \frac{5}{3} x^{6} b^{2} a^{3} A + \frac{1}{4} x^{4} a^{5} B + \frac{5}{4} x^{4} b a^{4} A + \frac{1}{2} x^{2} a^{5} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.081339, size = 133, normalized size = 3.17 \begin{align*} \frac{A a^{5} x^{2}}{2} + \frac{B b^{5} x^{14}}{14} + x^{12} \left (\frac{A b^{5}}{12} + \frac{5 B a b^{4}}{12}\right ) + x^{10} \left (\frac{A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{8} \left (\frac{5 A a^{2} b^{3}}{4} + \frac{5 B a^{3} b^{2}}{4}\right ) + x^{6} \left (\frac{5 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{6}\right ) + x^{4} \left (\frac{5 A a^{4} b}{4} + \frac{B a^{5}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19393, size = 167, normalized size = 3.98 \begin{align*} \frac{1}{14} \, B b^{5} x^{14} + \frac{5}{12} \, B a b^{4} x^{12} + \frac{1}{12} \, A b^{5} x^{12} + B a^{2} b^{3} x^{10} + \frac{1}{2} \, A a b^{4} x^{10} + \frac{5}{4} \, B a^{3} b^{2} x^{8} + \frac{5}{4} \, A a^{2} b^{3} x^{8} + \frac{5}{6} \, B a^{4} b x^{6} + \frac{5}{3} \, A a^{3} b^{2} x^{6} + \frac{1}{4} \, B a^{5} x^{4} + \frac{5}{4} \, A a^{4} b x^{4} + \frac{1}{2} \, A a^{5} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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