Optimal. Leaf size=52 \[ \frac {x^4}{60 a^3 (a+b x)^4}+\frac {x^4}{15 a^2 (a+b x)^5}+\frac {x^4}{6 a (a+b x)^6} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.23, 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}{6 b^4 (a+b x)^6}-\frac {3 a^2}{5 b^4 (a+b x)^5}+\frac {3 a}{4 b^4 (a+b x)^4}-\frac {1}{3 b^4 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x)^7} \, dx &=\int \left (-\frac {a^3}{b^3 (a+b x)^7}+\frac {3 a^2}{b^3 (a+b x)^6}-\frac {3 a}{b^3 (a+b x)^5}+\frac {1}{b^3 (a+b x)^4}\right ) \, dx\\ &=\frac {a^3}{6 b^4 (a+b x)^6}-\frac {3 a^2}{5 b^4 (a+b x)^5}+\frac {3 a}{4 b^4 (a+b x)^4}-\frac {1}{3 b^4 (a+b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.81 \begin {gather*} -\frac {a^3+6 a^2 b x+15 a b^2 x^2+20 b^3 x^3}{60 b^4 (a+b x)^6} \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)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.12, size = 98, normalized size = 1.88 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 40, normalized size = 0.77 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \, {\left (b x + a\right )}^{6} 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 = 1.10 \begin {gather*} \frac {a^{3}}{6 \left (b x +a \right )^{6} b^{4}}-\frac {3 a^{2}}{5 \left (b x +a \right )^{5} b^{4}}+\frac {3 a}{4 \left (b x +a \right )^{4} b^{4}}-\frac {1}{3 \left (b x +a \right )^{3} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 98, normalized size = 1.88 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 48, normalized size = 0.92 \begin {gather*} \frac {\frac {3\,a}{4\,{\left (a+b\,x\right )}^4}-\frac {1}{3\,{\left (a+b\,x\right )}^3}-\frac {3\,a^2}{5\,{\left (a+b\,x\right )}^5}+\frac {a^3}{6\,{\left (a+b\,x\right )}^6}}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 104, normalized size = 2.00 \begin {gather*} \frac {- a^{3} - 6 a^{2} b x - 15 a b^{2} x^{2} - 20 b^{3} x^{3}}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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