Optimal. Leaf size=72 \[ \frac {a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac {3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac {3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac {1}{12 b^4 \left (a+b x^2\right )^6} \]
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Rubi [A] time = 0.05, 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} \[ \frac {a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac {3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac {3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac {1}{12 b^4 \left (a+b x^2\right )^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{10}}+\frac {3 a^2}{b^3 (a+b x)^9}-\frac {3 a}{b^3 (a+b x)^8}+\frac {1}{b^3 (a+b x)^7}\right ) \, dx,x,x^2\right )\\ &=\frac {a^3}{18 b^4 \left (a+b x^2\right )^9}-\frac {3 a^2}{16 b^4 \left (a+b x^2\right )^8}+\frac {3 a}{14 b^4 \left (a+b x^2\right )^7}-\frac {1}{12 b^4 \left (a+b x^2\right )^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.64 \[ -\frac {a^3+9 a^2 b x^2+36 a b^2 x^4+84 b^3 x^6}{1008 b^4 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 135, normalized size = 1.88 \[ -\frac {84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \, {\left (b^{13} x^{18} + 9 \, a b^{12} x^{16} + 36 \, a^{2} b^{11} x^{14} + 84 \, a^{3} b^{10} x^{12} + 126 \, a^{4} b^{9} x^{10} + 126 \, a^{5} b^{8} x^{8} + 84 \, a^{6} b^{7} x^{6} + 36 \, a^{7} b^{6} x^{4} + 9 \, a^{8} b^{5} x^{2} + a^{9} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 44, normalized size = 0.61 \[ -\frac {84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \, {\left (b x^{2} + a\right )}^{9} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.90 \[ \frac {a^{3}}{18 \left (b \,x^{2}+a \right )^{9} b^{4}}-\frac {3 a^{2}}{16 \left (b \,x^{2}+a \right )^{8} b^{4}}+\frac {3 a}{14 \left (b \,x^{2}+a \right )^{7} b^{4}}-\frac {1}{12 \left (b \,x^{2}+a \right )^{6} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 135, normalized size = 1.88 \[ -\frac {84 \, b^{3} x^{6} + 36 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + a^{3}}{1008 \, {\left (b^{13} x^{18} + 9 \, a b^{12} x^{16} + 36 \, a^{2} b^{11} x^{14} + 84 \, a^{3} b^{10} x^{12} + 126 \, a^{4} b^{9} x^{10} + 126 \, a^{5} b^{8} x^{8} + 84 \, a^{6} b^{7} x^{6} + 36 \, a^{7} b^{6} x^{4} + 9 \, a^{8} b^{5} x^{2} + a^{9} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 136, normalized size = 1.89 \[ -\frac {\frac {a^3}{1008\,b^4}+\frac {x^6}{12\,b}+\frac {a\,x^4}{28\,b^2}+\frac {a^2\,x^2}{112\,b^3}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.01, size = 143, normalized size = 1.99 \[ \frac {- a^{3} - 9 a^{2} b x^{2} - 36 a b^{2} x^{4} - 84 b^{3} x^{6}}{1008 a^{9} b^{4} + 9072 a^{8} b^{5} x^{2} + 36288 a^{7} b^{6} x^{4} + 84672 a^{6} b^{7} x^{6} + 127008 a^{5} b^{8} x^{8} + 127008 a^{4} b^{9} x^{10} + 84672 a^{3} b^{10} x^{12} + 36288 a^{2} b^{11} x^{14} + 9072 a b^{12} x^{16} + 1008 b^{13} x^{18}} \]
Verification of antiderivative is not currently implemented for this CAS.
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