Optimal. Leaf size=72 \[ -\frac {a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac {3 a^2 \left (a+b x^2\right )^7}{14 b^4}+\frac {\left (a+b x^2\right )^9}{18 b^4}-\frac {3 a \left (a+b x^2\right )^8}{16 b^4} \]
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Rubi [A] time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, 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} \[ \frac {3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac {a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac {\left (a+b x^2\right )^9}{18 b^4}-\frac {3 a \left (a+b x^2\right )^8}{16 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2\right )^5 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3 \left (a+b x^2\right )^6}{12 b^4}+\frac {3 a^2 \left (a+b x^2\right )^7}{14 b^4}-\frac {3 a \left (a+b x^2\right )^8}{16 b^4}+\frac {\left (a+b x^2\right )^9}{18 b^4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 69, normalized size = 0.96 \[ \frac {a^5 x^8}{8}+\frac {1}{2} a^4 b x^{10}+\frac {5}{6} a^3 b^2 x^{12}+\frac {5}{7} a^2 b^3 x^{14}+\frac {5}{16} a b^4 x^{16}+\frac {b^5 x^{18}}{18} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 57, normalized size = 0.79 \[ \frac {1}{18} x^{18} b^{5} + \frac {5}{16} x^{16} b^{4} a + \frac {5}{7} x^{14} b^{3} a^{2} + \frac {5}{6} x^{12} b^{2} a^{3} + \frac {1}{2} x^{10} b a^{4} + \frac {1}{8} x^{8} a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 57, normalized size = 0.79 \[ \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 58, normalized size = 0.81 \[ \frac {1}{18} b^{5} x^{18}+\frac {5}{16} a \,b^{4} x^{16}+\frac {5}{7} a^{2} b^{3} x^{14}+\frac {5}{6} a^{3} b^{2} x^{12}+\frac {1}{2} a^{4} b \,x^{10}+\frac {1}{8} a^{5} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 57, normalized size = 0.79 \[ \frac {1}{18} \, b^{5} x^{18} + \frac {5}{16} \, a b^{4} x^{16} + \frac {5}{7} \, a^{2} b^{3} x^{14} + \frac {5}{6} \, a^{3} b^{2} x^{12} + \frac {1}{2} \, a^{4} b x^{10} + \frac {1}{8} \, a^{5} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 57, normalized size = 0.79 \[ \frac {a^5\,x^8}{8}+\frac {a^4\,b\,x^{10}}{2}+\frac {5\,a^3\,b^2\,x^{12}}{6}+\frac {5\,a^2\,b^3\,x^{14}}{7}+\frac {5\,a\,b^4\,x^{16}}{16}+\frac {b^5\,x^{18}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 65, normalized size = 0.90 \[ \frac {a^{5} x^{8}}{8} + \frac {a^{4} b x^{10}}{2} + \frac {5 a^{3} b^{2} x^{12}}{6} + \frac {5 a^{2} b^{3} x^{14}}{7} + \frac {5 a b^{4} x^{16}}{16} + \frac {b^{5} x^{18}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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