Optimal. Leaf size=64 \[ \frac {a^3}{9 b^4 (a+b x)^9}-\frac {3 a^2}{8 b^4 (a+b x)^8}+\frac {3 a}{7 b^4 (a+b x)^7}-\frac {1}{6 b^4 (a+b x)^6} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {a^3}{9 b^4 (a+b x)^9}-\frac {3 a^2}{8 b^4 (a+b x)^8}+\frac {3 a}{7 b^4 (a+b x)^7}-\frac {1}{6 b^4 (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^{10}} \, dx &=\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\\ &=\frac {a^3}{9 b^4 (a+b x)^9}-\frac {3 a^2}{8 b^4 (a+b x)^8}+\frac {3 a}{7 b^4 (a+b x)^7}-\frac {1}{6 b^4 (a+b x)^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.66 \begin {gather*} -\frac {a^3+9 a^2 b x+36 a b^2 x^2+84 b^3 x^3}{504 b^4 (a+b x)^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{(a+b x)^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.81, size = 131, normalized size = 2.05 \begin {gather*} -\frac {84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \, {\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 40, normalized size = 0.62 \begin {gather*} -\frac {84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \, {\left (b x + a\right )}^{9} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 57, normalized size = 0.89 \begin {gather*} \frac {a^{3}}{9 \left (b x +a \right )^{9} b^{4}}-\frac {3 a^{2}}{8 \left (b x +a \right )^{8} b^{4}}+\frac {3 a}{7 \left (b x +a \right )^{7} b^{4}}-\frac {1}{6 \left (b x +a \right )^{6} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 131, normalized size = 2.05 \begin {gather*} -\frac {84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \, {\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 48, normalized size = 0.75 \begin {gather*} \frac {\frac {3\,a}{7\,{\left (a+b\,x\right )}^7}-\frac {1}{6\,{\left (a+b\,x\right )}^6}-\frac {3\,a^2}{8\,{\left (a+b\,x\right )}^8}+\frac {a^3}{9\,{\left (a+b\,x\right )}^9}}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.70, size = 139, normalized size = 2.17 \begin {gather*} \frac {- a^{3} - 9 a^{2} b x - 36 a b^{2} x^{2} - 84 b^{3} x^{3}}{504 a^{9} b^{4} + 4536 a^{8} b^{5} x + 18144 a^{7} b^{6} x^{2} + 42336 a^{6} b^{7} x^{3} + 63504 a^{5} b^{8} x^{4} + 63504 a^{4} b^{9} x^{5} + 42336 a^{3} b^{10} x^{6} + 18144 a^{2} b^{11} x^{7} + 4536 a b^{12} x^{8} + 504 b^{13} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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