Optimal. Leaf size=72 \[ -\frac {a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac {3 a^2 \left (a+b x^2\right )^8}{16 b^4}+\frac {\left (a+b x^2\right )^{10}}{20 b^4}-\frac {a \left (a+b x^2\right )^9}{6 b^4} \]
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Rubi [A] time = 0.12, antiderivative size = 72, 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} \[ \frac {3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac {a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac {\left (a+b x^2\right )^{10}}{20 b^4}-\frac {a \left (a+b x^2\right )^9}{6 b^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac {\int x^7 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int x^3 \left (a b+b^2 x\right )^6 \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^3 \left (a b+b^2 x\right )^6}{b^3}+\frac {3 a^2 \left (a b+b^2 x\right )^7}{b^4}-\frac {3 a \left (a b+b^2 x\right )^8}{b^5}+\frac {\left (a b+b^2 x\right )^9}{b^6}\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {a^3 \left (a+b x^2\right )^7}{14 b^4}+\frac {3 a^2 \left (a+b x^2\right )^8}{16 b^4}-\frac {a \left (a+b x^2\right )^9}{6 b^4}+\frac {\left (a+b x^2\right )^{10}}{20 b^4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 82, normalized size = 1.14 \[ \frac {a^6 x^8}{8}+\frac {3}{5} a^5 b x^{10}+\frac {5}{4} a^4 b^2 x^{12}+\frac {10}{7} a^3 b^3 x^{14}+\frac {15}{16} a^2 b^4 x^{16}+\frac {1}{3} a b^5 x^{18}+\frac {b^6 x^{20}}{20} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 68, normalized size = 0.94 \[ \frac {1}{20} x^{20} b^{6} + \frac {1}{3} x^{18} b^{5} a + \frac {15}{16} x^{16} b^{4} a^{2} + \frac {10}{7} x^{14} b^{3} a^{3} + \frac {5}{4} x^{12} b^{2} a^{4} + \frac {3}{5} x^{10} b a^{5} + \frac {1}{8} x^{8} a^{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 68, normalized size = 0.94 \[ \frac {1}{20} \, b^{6} x^{20} + \frac {1}{3} \, a b^{5} x^{18} + \frac {15}{16} \, a^{2} b^{4} x^{16} + \frac {10}{7} \, a^{3} b^{3} x^{14} + \frac {5}{4} \, a^{4} b^{2} x^{12} + \frac {3}{5} \, a^{5} b x^{10} + \frac {1}{8} \, a^{6} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 0.96 \[ \frac {1}{20} b^{6} x^{20}+\frac {1}{3} a \,b^{5} x^{18}+\frac {15}{16} a^{2} b^{4} x^{16}+\frac {10}{7} a^{3} b^{3} x^{14}+\frac {5}{4} a^{4} b^{2} x^{12}+\frac {3}{5} a^{5} b \,x^{10}+\frac {1}{8} a^{6} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 68, normalized size = 0.94 \[ \frac {1}{20} \, b^{6} x^{20} + \frac {1}{3} \, a b^{5} x^{18} + \frac {15}{16} \, a^{2} b^{4} x^{16} + \frac {10}{7} \, a^{3} b^{3} x^{14} + \frac {5}{4} \, a^{4} b^{2} x^{12} + \frac {3}{5} \, a^{5} b x^{10} + \frac {1}{8} \, a^{6} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 68, normalized size = 0.94 \[ \frac {a^6\,x^8}{8}+\frac {3\,a^5\,b\,x^{10}}{5}+\frac {5\,a^4\,b^2\,x^{12}}{4}+\frac {10\,a^3\,b^3\,x^{14}}{7}+\frac {15\,a^2\,b^4\,x^{16}}{16}+\frac {a\,b^5\,x^{18}}{3}+\frac {b^6\,x^{20}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 78, normalized size = 1.08 \[ \frac {a^{6} x^{8}}{8} + \frac {3 a^{5} b x^{10}}{5} + \frac {5 a^{4} b^{2} x^{12}}{4} + \frac {10 a^{3} b^{3} x^{14}}{7} + \frac {15 a^{2} b^{4} x^{16}}{16} + \frac {a b^{5} x^{18}}{3} + \frac {b^{6} x^{20}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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