Optimal. Leaf size=52 \[ \frac {x^4}{6 a (a+b x)^6}+\frac {x^4}{15 a^2 (a+b x)^5}+\frac {x^4}{60 a^3 (a+b x)^4} \]
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Rubi [A]
time = 0.02, 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 = {45}
\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 45
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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Mathics [A]
time = 2.47, size = 97, normalized size = 1.87 \begin {gather*} \frac {-a^3-6 a^2 b x-15 a b^2 x^2-20 b^3 x^3}{60 b^4 \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 57, normalized size = 1.10
method | result | size |
gosper | \(-\frac {20 b^{3} x^{3}+15 a \,b^{2} x^{2}+6 a^{2} b x +a^{3}}{60 \left (b x +a \right )^{6} b^{4}}\) | \(41\) |
norman | \(\frac {-\frac {x^{3}}{3 b}-\frac {a \,x^{2}}{4 b^{2}}-\frac {a^{2} x}{10 b^{3}}-\frac {a^{3}}{60 b^{4}}}{\left (b x +a \right )^{6}}\) | \(44\) |
risch | \(\frac {-\frac {x^{3}}{3 b}-\frac {a \,x^{2}}{4 b^{2}}-\frac {a^{2} x}{10 b^{3}}-\frac {a^{3}}{60 b^{4}}}{\left (b x +a \right )^{6}}\) | \(44\) |
default | \(\frac {a^{3}}{6 b^{4} \left (b x +a \right )^{6}}-\frac {3 a^{2}}{5 b^{4} \left (b x +a \right )^{5}}+\frac {3 a}{4 b^{4} \left (b x +a \right )^{4}}-\frac {1}{3 b^{4} \left (b x +a \right )^{3}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (46) = 92\).
time = 0.25, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs.
\(2 (46) = 92\).
time = 0.31, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (42) = 84\).
time = 0.23, 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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Giac [A]
time = 0.00, size = 46, normalized size = 0.88 \begin {gather*} \frac {-20 x^{3} b^{3}-15 x^{2} b^{2} a-6 x b a^{2}-a^{3}}{60 b^{4} \left (x b+a\right )^{6}} \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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