Optimal. Leaf size=76 \[ -\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}} \]
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Rubi [A]
time = 0.01, 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 = {47, 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 47
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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Mathics [A]
time = 2.91, size = 112, normalized size = 1.47 \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*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 113, normalized size = 1.49
method | result | size |
norman | \(\frac {-\frac {1}{14} a^{10}-\frac {10}{13} a^{9} b x -\frac {15}{4} a^{8} b^{2} x^{2}-\frac {120}{11} a^{7} b^{3} x^{3}-21 a^{6} b^{4} x^{4}-28 a^{5} b^{5} x^{5}-\frac {105}{4} a^{4} b^{6} x^{6}-\frac {120}{7} a^{3} b^{7} x^{7}-\frac {15}{2} a^{2} b^{8} x^{8}-2 a \,b^{9} x^{9}-\frac {1}{4} b^{10} x^{10}}{x^{14}}\) | \(112\) |
risch | \(\frac {-\frac {1}{14} a^{10}-\frac {10}{13} a^{9} b x -\frac {15}{4} a^{8} b^{2} x^{2}-\frac {120}{11} a^{7} b^{3} x^{3}-21 a^{6} b^{4} x^{4}-28 a^{5} b^{5} x^{5}-\frac {105}{4} a^{4} b^{6} x^{6}-\frac {120}{7} a^{3} b^{7} x^{7}-\frac {15}{2} a^{2} b^{8} x^{8}-2 a \,b^{9} x^{9}-\frac {1}{4} b^{10} x^{10}}{x^{14}}\) | \(112\) |
gosper | \(-\frac {1001 b^{10} x^{10}+8008 a \,b^{9} x^{9}+30030 a^{2} b^{8} x^{8}+68640 a^{3} b^{7} x^{7}+105105 a^{4} b^{6} x^{6}+112112 a^{5} b^{5} x^{5}+84084 a^{6} b^{4} x^{4}+43680 a^{7} b^{3} x^{3}+15015 a^{8} b^{2} x^{2}+3080 a^{9} b x +286 a^{10}}{4004 x^{14}}\) | \(113\) |
default | \(-\frac {21 a^{6} b^{4}}{x^{10}}-\frac {120 a^{7} b^{3}}{11 x^{11}}-\frac {a^{10}}{14 x^{14}}-\frac {b^{10}}{4 x^{4}}-\frac {15 a^{8} b^{2}}{4 x^{12}}-\frac {28 a^{5} b^{5}}{x^{9}}-\frac {105 a^{4} b^{6}}{4 x^{8}}-\frac {2 a \,b^{9}}{x^{5}}-\frac {10 a^{9} b}{13 x^{13}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {120 a^{3} b^{7}}{7 x^{7}}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.24, 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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Fricas [A]
time = 0.30, 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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Sympy [A]
time = 0.51, 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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Giac [A]
time = 0.00, size = 125, normalized size = 1.64 \begin {gather*} \frac {-1001 x^{10} b^{10}-8008 x^{9} b^{9} a-30030 x^{8} b^{8} a^{2}-68640 x^{7} b^{7} a^{3}-105105 x^{6} b^{6} a^{4}-112112 x^{5} b^{5} a^{5}-84084 x^{4} b^{4} a^{6}-43680 x^{3} b^{3} a^{7}-15015 x^{2} b^{2} a^{8}-3080 x b a^{9}-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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