Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^7}{14 b^2}-\frac{a \left (a+b x^2\right )^6}{12 b^2} \]
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Rubi [A] time = 0.036031, antiderivative size = 34, normalized size of antiderivative = 1., 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} \[ \frac{\left (a+b x^2\right )^7}{14 b^2}-\frac{a \left (a+b x^2\right )^6}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (a+b x^2\right )^5 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^5}{b}+\frac{(a+b x)^6}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac{a \left (a+b x^2\right )^6}{12 b^2}+\frac{\left (a+b x^2\right )^7}{14 b^2}\\ \end{align*}
Mathematica [A] time = 0.0020954, size = 66, normalized size = 1.94 \[ a^2 b^3 x^{10}+\frac{5}{4} a^3 b^2 x^8+\frac{5}{6} a^4 b x^6+\frac{a^5 x^4}{4}+\frac{5}{12} a b^4 x^{12}+\frac{b^5 x^{14}}{14} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 57, normalized size = 1.7 \begin{align*}{\frac{{b}^{5}{x}^{14}}{14}}+{\frac{5\,a{b}^{4}{x}^{12}}{12}}+{a}^{2}{b}^{3}{x}^{10}+{\frac{5\,{a}^{3}{b}^{2}{x}^{8}}{4}}+{\frac{5\,{a}^{4}b{x}^{6}}{6}}+{\frac{{a}^{5}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.03753, size = 76, normalized size = 2.24 \begin{align*} \frac{1}{14} \, b^{5} x^{14} + \frac{5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{4} \, a^{5} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1988, size = 130, normalized size = 3.82 \begin{align*} \frac{1}{14} x^{14} b^{5} + \frac{5}{12} x^{12} b^{4} a + x^{10} b^{3} a^{2} + \frac{5}{4} x^{8} b^{2} a^{3} + \frac{5}{6} x^{6} b a^{4} + \frac{1}{4} x^{4} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.072762, size = 63, normalized size = 1.85 \begin{align*} \frac{a^{5} x^{4}}{4} + \frac{5 a^{4} b x^{6}}{6} + \frac{5 a^{3} b^{2} x^{8}}{4} + a^{2} b^{3} x^{10} + \frac{5 a b^{4} x^{12}}{12} + \frac{b^{5} x^{14}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31139, size = 76, normalized size = 2.24 \begin{align*} \frac{1}{14} \, b^{5} x^{14} + \frac{5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{4} \, a^{5} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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