Optimal. Leaf size=52 \[ \frac {x^7}{252 a^3 (a+b x)^7}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {x^7}{9 a (a+b x)^9} \]
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Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac {x^7}{252 a^3 (a+b x)^7}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {x^7}{9 a (a+b x)^9} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {x^6}{(a+b x)^{10}} \, dx &=\frac {x^7}{9 a (a+b x)^9}+\frac {2 \int \frac {x^6}{(a+b x)^9} \, dx}{9 a}\\ &=\frac {x^7}{9 a (a+b x)^9}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {\int \frac {x^6}{(a+b x)^8} \, dx}{36 a^2}\\ &=\frac {x^7}{9 a (a+b x)^9}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {x^7}{252 a^3 (a+b x)^7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 1.44 \[ -\frac {a^6+9 a^5 b x+36 a^4 b^2 x^2+84 a^3 b^3 x^3+126 a^2 b^4 x^4+126 a b^5 x^5+84 b^6 x^6}{252 b^7 (a+b x)^9} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 164, normalized size = 3.15 \[ -\frac {84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \, {\left (b^{16} x^{9} + 9 \, a b^{15} x^{8} + 36 \, a^{2} b^{14} x^{7} + 84 \, a^{3} b^{13} x^{6} + 126 \, a^{4} b^{12} x^{5} + 126 \, a^{5} b^{11} x^{4} + 84 \, a^{6} b^{10} x^{3} + 36 \, a^{7} b^{9} x^{2} + 9 \, a^{8} b^{8} x + a^{9} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 73, normalized size = 1.40 \[ -\frac {84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \, {\left (b x + a\right )}^{9} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 102, normalized size = 1.96 \[ -\frac {a^{6}}{9 \left (b x +a \right )^{9} b^{7}}+\frac {3 a^{5}}{4 \left (b x +a \right )^{8} b^{7}}-\frac {15 a^{4}}{7 \left (b x +a \right )^{7} b^{7}}+\frac {10 a^{3}}{3 \left (b x +a \right )^{6} b^{7}}-\frac {3 a^{2}}{\left (b x +a \right )^{5} b^{7}}+\frac {3 a}{2 \left (b x +a \right )^{4} b^{7}}-\frac {1}{3 \left (b x +a \right )^{3} b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.49, size = 164, normalized size = 3.15 \[ -\frac {84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \, {\left (b^{16} x^{9} + 9 \, a b^{15} x^{8} + 36 \, a^{2} b^{14} x^{7} + 84 \, a^{3} b^{13} x^{6} + 126 \, a^{4} b^{12} x^{5} + 126 \, a^{5} b^{11} x^{4} + 84 \, a^{6} b^{10} x^{3} + 36 \, a^{7} b^{9} x^{2} + 9 \, a^{8} b^{8} x + a^{9} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 85, normalized size = 1.63 \[ -\frac {\frac {1}{3\,{\left (a+b\,x\right )}^3}-\frac {3\,a}{2\,{\left (a+b\,x\right )}^4}+\frac {3\,a^2}{{\left (a+b\,x\right )}^5}-\frac {10\,a^3}{3\,{\left (a+b\,x\right )}^6}+\frac {15\,a^4}{7\,{\left (a+b\,x\right )}^7}-\frac {3\,a^5}{4\,{\left (a+b\,x\right )}^8}+\frac {a^6}{9\,{\left (a+b\,x\right )}^9}}{b^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.91, size = 175, normalized size = 3.37 \[ \frac {- a^{6} - 9 a^{5} b x - 36 a^{4} b^{2} x^{2} - 84 a^{3} b^{3} x^{3} - 126 a^{2} b^{4} x^{4} - 126 a b^{5} x^{5} - 84 b^{6} x^{6}}{252 a^{9} b^{7} + 2268 a^{8} b^{8} x + 9072 a^{7} b^{9} x^{2} + 21168 a^{6} b^{10} x^{3} + 31752 a^{5} b^{11} x^{4} + 31752 a^{4} b^{12} x^{5} + 21168 a^{3} b^{13} x^{6} + 9072 a^{2} b^{14} x^{7} + 2268 a b^{15} x^{8} + 252 b^{16} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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