Optimal. Leaf size=76 \[ \frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {(a+b x)^{11}}{14 a x^{14}} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {45, 37} \begin {gather*} \frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {(a+b x)^{11}}{14 a x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10}}{x^{15}} \, dx &=-\frac {(a+b x)^{11}}{14 a x^{14}}-\frac {(3 b) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{14 a}\\ &=-\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}+\frac {\left (3 b^2\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{91 a^2}\\ &=-\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}-\frac {b^3 \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{364 a^3}\\ &=-\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 128, normalized size = 1.68 \begin {gather*} -\frac {a^{10}}{14 x^{14}}-\frac {10 a^9 b}{13 x^{13}}-\frac {15 a^8 b^2}{4 x^{12}}-\frac {120 a^7 b^3}{11 x^{11}}-\frac {21 a^6 b^4}{x^{10}}-\frac {28 a^5 b^5}{x^9}-\frac {105 a^4 b^6}{4 x^8}-\frac {120 a^3 b^7}{7 x^7}-\frac {15 a^2 b^8}{2 x^6}-\frac {2 a b^9}{x^5}-\frac {b^{10}}{4 x^4} \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)^{10}}{x^{15}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.29, size = 112, normalized size = 1.47 \begin {gather*} -\frac {1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 112, normalized size = 1.47 \begin {gather*} -\frac {1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 113, normalized size = 1.49 \begin {gather*} -\frac {b^{10}}{4 x^{4}}-\frac {2 a \,b^{9}}{x^{5}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {120 a^{3} b^{7}}{7 x^{7}}-\frac {105 a^{4} b^{6}}{4 x^{8}}-\frac {28 a^{5} b^{5}}{x^{9}}-\frac {21 a^{6} b^{4}}{x^{10}}-\frac {120 a^{7} b^{3}}{11 x^{11}}-\frac {15 a^{8} b^{2}}{4 x^{12}}-\frac {10 a^{9} b}{13 x^{13}}-\frac {a^{10}}{14 x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 112, normalized size = 1.47 \begin {gather*} -\frac {1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 112, normalized size = 1.47 \begin {gather*} -\frac {\frac {a^{10}}{14}+\frac {10\,a^9\,b\,x}{13}+\frac {15\,a^8\,b^2\,x^2}{4}+\frac {120\,a^7\,b^3\,x^3}{11}+21\,a^6\,b^4\,x^4+28\,a^5\,b^5\,x^5+\frac {105\,a^4\,b^6\,x^6}{4}+\frac {120\,a^3\,b^7\,x^7}{7}+\frac {15\,a^2\,b^8\,x^8}{2}+2\,a\,b^9\,x^9+\frac {b^{10}\,x^{10}}{4}}{x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 121, normalized size = 1.59 \begin {gather*} \frac {- 286 a^{10} - 3080 a^{9} b x - 15015 a^{8} b^{2} x^{2} - 43680 a^{7} b^{3} x^{3} - 84084 a^{6} b^{4} x^{4} - 112112 a^{5} b^{5} x^{5} - 105105 a^{4} b^{6} x^{6} - 68640 a^{3} b^{7} x^{7} - 30030 a^{2} b^{8} x^{8} - 8008 a b^{9} x^{9} - 1001 b^{10} x^{10}}{4004 x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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