Optimal. Leaf size=41 \[ -\frac {c^4 (a-b x)^5}{6 x^6}-\frac {7 b c^4 (a-b x)^5}{30 a x^5} \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} -\frac {7 b c^4 (a-b x)^5}{30 a x^5}-\frac {c^4 (a-b x)^5}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx &=-\frac {c^4 (a-b x)^5}{6 x^6}+\frac {1}{6} (7 b) \int \frac {(a c-b c x)^4}{x^6} \, dx\\ &=-\frac {c^4 (a-b x)^5}{6 x^6}-\frac {7 b c^4 (a-b x)^5}{30 a x^5}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 85, normalized size = 2.07 \begin {gather*} -\frac {a^5 c^4}{6 x^6}+\frac {3 a^4 b c^4}{5 x^5}-\frac {a^3 b^2 c^4}{2 x^4}-\frac {2 a^2 b^3 c^4}{3 x^3}+\frac {3 a b^4 c^4}{2 x^2}-\frac {b^5 c^4}{x} \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 {(a+b x) (a c-b c x)^4}{x^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.41, size = 75, normalized size = 1.83 \begin {gather*} -\frac {30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 75, normalized size = 1.83 \begin {gather*} -\frac {30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 62, normalized size = 1.51 \begin {gather*} \left (-\frac {b^{5}}{x}+\frac {3 a \,b^{4}}{2 x^{2}}-\frac {2 a^{2} b^{3}}{3 x^{3}}-\frac {a^{3} b^{2}}{2 x^{4}}+\frac {3 a^{4} b}{5 x^{5}}-\frac {a^{5}}{6 x^{6}}\right ) c^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 75, normalized size = 1.83 \begin {gather*} -\frac {30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 74, normalized size = 1.80 \begin {gather*} -\frac {\frac {a^5\,c^4}{6}-\frac {3\,a^4\,b\,c^4\,x}{5}+\frac {a^3\,b^2\,c^4\,x^2}{2}+\frac {2\,a^2\,b^3\,c^4\,x^3}{3}-\frac {3\,a\,b^4\,c^4\,x^4}{2}+b^5\,c^4\,x^5}{x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.41, size = 80, normalized size = 1.95 \begin {gather*} \frac {- 5 a^{5} c^{4} + 18 a^{4} b c^{4} x - 15 a^{3} b^{2} c^{4} x^{2} - 20 a^{2} b^{3} c^{4} x^{3} + 45 a b^{4} c^{4} x^{4} - 30 b^{5} c^{4} x^{5}}{30 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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