Optimal. Leaf size=80 \[ -\frac {2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac {5 a^2 c^4 (a-b x)^6}{6 b^3}+\frac {c^4 (a-b x)^8}{8 b^3}-\frac {4 a c^4 (a-b x)^7}{7 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \[ \frac {5 a^2 c^4 (a-b x)^6}{6 b^3}-\frac {2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac {c^4 (a-b x)^8}{8 b^3}-\frac {4 a c^4 (a-b x)^7}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 75
Rubi steps
\begin {align*} \int x^2 (a+b x) (a c-b c x)^4 \, dx &=\int \left (\frac {2 a^3 (a c-b c x)^4}{b^2}-\frac {5 a^2 (a c-b c x)^5}{b^2 c}+\frac {4 a (a c-b c x)^6}{b^2 c^2}-\frac {(a c-b c x)^7}{b^2 c^3}\right ) \, dx\\ &=-\frac {2 a^3 c^4 (a-b x)^5}{5 b^3}+\frac {5 a^2 c^4 (a-b x)^6}{6 b^3}-\frac {4 a c^4 (a-b x)^7}{7 b^3}+\frac {c^4 (a-b x)^8}{8 b^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 87, normalized size = 1.09 \[ \frac {1}{3} a^5 c^4 x^3-\frac {3}{4} a^4 b c^4 x^4+\frac {2}{5} a^3 b^2 c^4 x^5+\frac {1}{3} a^2 b^3 c^4 x^6-\frac {3}{7} a b^4 c^4 x^7+\frac {1}{8} b^5 c^4 x^8 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 75, normalized size = 0.94 \[ \frac {1}{8} x^{8} c^{4} b^{5} - \frac {3}{7} x^{7} c^{4} b^{4} a + \frac {1}{3} x^{6} c^{4} b^{3} a^{2} + \frac {2}{5} x^{5} c^{4} b^{2} a^{3} - \frac {3}{4} x^{4} c^{4} b a^{4} + \frac {1}{3} x^{3} c^{4} a^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 75, normalized size = 0.94 \[ \frac {1}{8} \, b^{5} c^{4} x^{8} - \frac {3}{7} \, a b^{4} c^{4} x^{7} + \frac {1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac {2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac {3}{4} \, a^{4} b c^{4} x^{4} + \frac {1}{3} \, a^{5} c^{4} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 0.95 \[ \frac {1}{8} b^{5} c^{4} x^{8}-\frac {3}{7} a \,b^{4} c^{4} x^{7}+\frac {1}{3} a^{2} b^{3} c^{4} x^{6}+\frac {2}{5} a^{3} b^{2} c^{4} x^{5}-\frac {3}{4} a^{4} b \,c^{4} x^{4}+\frac {1}{3} a^{5} c^{4} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 75, normalized size = 0.94 \[ \frac {1}{8} \, b^{5} c^{4} x^{8} - \frac {3}{7} \, a b^{4} c^{4} x^{7} + \frac {1}{3} \, a^{2} b^{3} c^{4} x^{6} + \frac {2}{5} \, a^{3} b^{2} c^{4} x^{5} - \frac {3}{4} \, a^{4} b c^{4} x^{4} + \frac {1}{3} \, a^{5} c^{4} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 75, normalized size = 0.94 \[ \frac {a^5\,c^4\,x^3}{3}-\frac {3\,a^4\,b\,c^4\,x^4}{4}+\frac {2\,a^3\,b^2\,c^4\,x^5}{5}+\frac {a^2\,b^3\,c^4\,x^6}{3}-\frac {3\,a\,b^4\,c^4\,x^7}{7}+\frac {b^5\,c^4\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 85, normalized size = 1.06 \[ \frac {a^{5} c^{4} x^{3}}{3} - \frac {3 a^{4} b c^{4} x^{4}}{4} + \frac {2 a^{3} b^{2} c^{4} x^{5}}{5} + \frac {a^{2} b^{3} c^{4} x^{6}}{3} - \frac {3 a b^{4} c^{4} x^{7}}{7} + \frac {b^{5} c^{4} x^{8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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