Optimal. Leaf size=53 \[ \frac {a^2 \left (a+b x^2\right )^7}{14 b^3}+\frac {\left (a+b x^2\right )^9}{18 b^3}-\frac {a \left (a+b x^2\right )^8}{8 b^3} \]
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Rubi [A] time = 0.09, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac {a^2 \left (a+b x^2\right )^7}{14 b^3}+\frac {\left (a+b x^2\right )^9}{18 b^3}-\frac {a \left (a+b x^2\right )^8}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x^5 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int x^2 \left (a b+b^2 x\right )^6 \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2 \left (a b+b^2 x\right )^6}{b^2}-\frac {2 a \left (a b+b^2 x\right )^7}{b^3}+\frac {\left (a b+b^2 x\right )^8}{b^4}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {a^2 \left (a+b x^2\right )^7}{14 b^3}-\frac {a \left (a+b x^2\right )^8}{8 b^3}+\frac {\left (a+b x^2\right )^9}{18 b^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 82, normalized size = 1.55 \[ \frac {a^6 x^6}{6}+\frac {3}{4} a^5 b x^8+\frac {3}{2} a^4 b^2 x^{10}+\frac {5}{3} a^3 b^3 x^{12}+\frac {15}{14} a^2 b^4 x^{14}+\frac {3}{8} a b^5 x^{16}+\frac {b^6 x^{18}}{18} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 68, normalized size = 1.28 \[ \frac {1}{18} x^{18} b^{6} + \frac {3}{8} x^{16} b^{5} a + \frac {15}{14} x^{14} b^{4} a^{2} + \frac {5}{3} x^{12} b^{3} a^{3} + \frac {3}{2} x^{10} b^{2} a^{4} + \frac {3}{4} x^{8} b a^{5} + \frac {1}{6} x^{6} a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 68, normalized size = 1.28 \[ \frac {1}{18} \, b^{6} x^{18} + \frac {3}{8} \, a b^{5} x^{16} + \frac {15}{14} \, a^{2} b^{4} x^{14} + \frac {5}{3} \, a^{3} b^{3} x^{12} + \frac {3}{2} \, a^{4} b^{2} x^{10} + \frac {3}{4} \, a^{5} b x^{8} + \frac {1}{6} \, a^{6} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 1.30 \[ \frac {1}{18} b^{6} x^{18}+\frac {3}{8} a \,b^{5} x^{16}+\frac {15}{14} a^{2} b^{4} x^{14}+\frac {5}{3} a^{3} b^{3} x^{12}+\frac {3}{2} a^{4} b^{2} x^{10}+\frac {3}{4} a^{5} b \,x^{8}+\frac {1}{6} a^{6} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 68, normalized size = 1.28 \[ \frac {1}{18} \, b^{6} x^{18} + \frac {3}{8} \, a b^{5} x^{16} + \frac {15}{14} \, a^{2} b^{4} x^{14} + \frac {5}{3} \, a^{3} b^{3} x^{12} + \frac {3}{2} \, a^{4} b^{2} x^{10} + \frac {3}{4} \, a^{5} b x^{8} + \frac {1}{6} \, a^{6} x^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 68, normalized size = 1.28 \[ \frac {a^6\,x^6}{6}+\frac {3\,a^5\,b\,x^8}{4}+\frac {3\,a^4\,b^2\,x^{10}}{2}+\frac {5\,a^3\,b^3\,x^{12}}{3}+\frac {15\,a^2\,b^4\,x^{14}}{14}+\frac {3\,a\,b^5\,x^{16}}{8}+\frac {b^6\,x^{18}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 80, normalized size = 1.51 \[ \frac {a^{6} x^{6}}{6} + \frac {3 a^{5} b x^{8}}{4} + \frac {3 a^{4} b^{2} x^{10}}{2} + \frac {5 a^{3} b^{3} x^{12}}{3} + \frac {15 a^{2} b^{4} x^{14}}{14} + \frac {3 a b^{5} x^{16}}{8} + \frac {b^{6} x^{18}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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