Optimal. Leaf size=218 \[ -\frac {a^{10} A}{6 x^6}-\frac {a^9 (10 A b+a B)}{5 x^5}-\frac {5 a^8 b (9 A b+2 a B)}{4 x^4}-\frac {5 a^7 b^2 (8 A b+3 a B)}{x^3}-\frac {15 a^6 b^3 (7 A b+4 a B)}{x^2}-\frac {42 a^5 b^4 (6 A b+5 a B)}{x}+30 a^3 b^6 (4 A b+7 a B) x+\frac {15}{2} a^2 b^7 (3 A b+8 a B) x^2+\frac {5}{3} a b^8 (2 A b+9 a B) x^3+\frac {1}{4} b^9 (A b+10 a B) x^4+\frac {1}{5} b^{10} B x^5+42 a^4 b^5 (5 A b+6 a B) \log (x) \]
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Rubi [A]
time = 0.10, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} -\frac {a^{10} A}{6 x^6}-\frac {a^9 (a B+10 A b)}{5 x^5}-\frac {5 a^8 b (2 a B+9 A b)}{4 x^4}-\frac {5 a^7 b^2 (3 a B+8 A b)}{x^3}-\frac {15 a^6 b^3 (4 a B+7 A b)}{x^2}-\frac {42 a^5 b^4 (5 a B+6 A b)}{x}+42 a^4 b^5 \log (x) (6 a B+5 A b)+30 a^3 b^6 x (7 a B+4 A b)+\frac {15}{2} a^2 b^7 x^2 (8 a B+3 A b)+\frac {1}{4} b^9 x^4 (10 a B+A b)+\frac {5}{3} a b^8 x^3 (9 a B+2 A b)+\frac {1}{5} b^{10} B x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^{10} (A+B x)}{x^7} \, dx &=\int \left (30 a^3 b^6 (4 A b+7 a B)+\frac {a^{10} A}{x^7}+\frac {a^9 (10 A b+a B)}{x^6}+\frac {5 a^8 b (9 A b+2 a B)}{x^5}+\frac {15 a^7 b^2 (8 A b+3 a B)}{x^4}+\frac {30 a^6 b^3 (7 A b+4 a B)}{x^3}+\frac {42 a^5 b^4 (6 A b+5 a B)}{x^2}+\frac {42 a^4 b^5 (5 A b+6 a B)}{x}+15 a^2 b^7 (3 A b+8 a B) x+5 a b^8 (2 A b+9 a B) x^2+b^9 (A b+10 a B) x^3+b^{10} B x^4\right ) \, dx\\ &=-\frac {a^{10} A}{6 x^6}-\frac {a^9 (10 A b+a B)}{5 x^5}-\frac {5 a^8 b (9 A b+2 a B)}{4 x^4}-\frac {5 a^7 b^2 (8 A b+3 a B)}{x^3}-\frac {15 a^6 b^3 (7 A b+4 a B)}{x^2}-\frac {42 a^5 b^4 (6 A b+5 a B)}{x}+30 a^3 b^6 (4 A b+7 a B) x+\frac {15}{2} a^2 b^7 (3 A b+8 a B) x^2+\frac {5}{3} a b^8 (2 A b+9 a B) x^3+\frac {1}{4} b^9 (A b+10 a B) x^4+\frac {1}{5} b^{10} B x^5+42 a^4 b^5 (5 A b+6 a B) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 210, normalized size = 0.96 \begin {gather*} -\frac {252 a^5 A b^5}{x}+210 a^4 b^6 B x+60 a^3 b^7 x (2 A+B x)-\frac {105 a^6 b^4 (A+2 B x)}{x^2}+\frac {15}{2} a^2 b^8 x^2 (3 A+2 B x)-\frac {20 a^7 b^3 (2 A+3 B x)}{x^3}+\frac {5}{6} a b^9 x^3 (4 A+3 B x)-\frac {15 a^8 b^2 (3 A+4 B x)}{4 x^4}+\frac {1}{20} b^{10} x^4 (5 A+4 B x)-\frac {a^9 b (4 A+5 B x)}{2 x^5}-\frac {a^{10} (5 A+6 B x)}{30 x^6}+42 a^4 b^5 (5 A b+6 a B) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 218, normalized size = 1.00
method | result | size |
default | \(\frac {b^{10} B \,x^{5}}{5}+\frac {A \,b^{10} x^{4}}{4}+\frac {5 B a \,b^{9} x^{4}}{2}+\frac {10 A a \,b^{9} x^{3}}{3}+15 B \,a^{2} b^{8} x^{3}+\frac {45 A \,a^{2} b^{8} x^{2}}{2}+60 B \,a^{3} b^{7} x^{2}+120 a^{3} b^{7} A x +210 a^{4} b^{6} B x -\frac {42 a^{5} b^{4} \left (6 A b +5 B a \right )}{x}-\frac {a^{9} \left (10 A b +B a \right )}{5 x^{5}}-\frac {5 a^{8} b \left (9 A b +2 B a \right )}{4 x^{4}}-\frac {a^{10} A}{6 x^{6}}-\frac {15 a^{6} b^{3} \left (7 A b +4 B a \right )}{x^{2}}+42 a^{4} b^{5} \left (5 A b +6 B a \right ) \ln \left (x \right )-\frac {5 a^{7} b^{2} \left (8 A b +3 B a \right )}{x^{3}}\) | \(218\) |
norman | \(\frac {\left (\frac {1}{4} b^{10} A +\frac {5}{2} a \,b^{9} B \right ) x^{10}+\left (\frac {10}{3} a \,b^{9} A +15 a^{2} b^{8} B \right ) x^{9}+\left (\frac {45}{2} a^{2} b^{8} A +60 a^{3} b^{7} B \right ) x^{8}+\left (-\frac {45}{4} a^{8} b^{2} A -\frac {5}{2} a^{9} b B \right ) x^{2}+\left (-2 a^{9} b A -\frac {1}{5} a^{10} B \right ) x +\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{7}+\left (-252 a^{5} b^{5} A -210 a^{6} b^{4} B \right ) x^{5}+\left (-105 a^{6} b^{4} A -60 a^{7} b^{3} B \right ) x^{4}+\left (-40 a^{7} b^{3} A -15 a^{8} b^{2} B \right ) x^{3}-\frac {a^{10} A}{6}+\frac {b^{10} B \,x^{11}}{5}}{x^{6}}+\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) \ln \left (x \right )\) | \(235\) |
risch | \(\frac {b^{10} B \,x^{5}}{5}+\frac {A \,b^{10} x^{4}}{4}+\frac {5 B a \,b^{9} x^{4}}{2}+\frac {10 A a \,b^{9} x^{3}}{3}+15 B \,a^{2} b^{8} x^{3}+\frac {45 A \,a^{2} b^{8} x^{2}}{2}+60 B \,a^{3} b^{7} x^{2}+120 a^{3} b^{7} A x +210 a^{4} b^{6} B x +\frac {\left (-252 a^{5} b^{5} A -210 a^{6} b^{4} B \right ) x^{5}+\left (-105 a^{6} b^{4} A -60 a^{7} b^{3} B \right ) x^{4}+\left (-40 a^{7} b^{3} A -15 a^{8} b^{2} B \right ) x^{3}+\left (-\frac {45}{4} a^{8} b^{2} A -\frac {5}{2} a^{9} b B \right ) x^{2}+\left (-2 a^{9} b A -\frac {1}{5} a^{10} B \right ) x -\frac {a^{10} A}{6}}{x^{6}}+210 A \ln \left (x \right ) a^{4} b^{6}+252 B \ln \left (x \right ) a^{5} b^{5}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 241, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, B b^{10} x^{5} + \frac {1}{4} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{4} + \frac {5}{3} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{3} + \frac {15}{2} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{2} + 30 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x + 42 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} \log \left (x\right ) - \frac {10 \, A a^{10} + 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 900 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 300 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 75 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 245, normalized size = 1.12 \begin {gather*} \frac {12 \, B b^{10} x^{11} - 10 \, A a^{10} + 15 \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 100 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 450 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 1800 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 2520 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} \log \left (x\right ) - 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} - 900 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} - 300 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} - 75 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} - 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.03, size = 253, normalized size = 1.16 \begin {gather*} \frac {B b^{10} x^{5}}{5} + 42 a^{4} b^{5} \cdot \left (5 A b + 6 B a\right ) \log {\left (x \right )} + x^{4} \left (\frac {A b^{10}}{4} + \frac {5 B a b^{9}}{2}\right ) + x^{3} \cdot \left (\frac {10 A a b^{9}}{3} + 15 B a^{2} b^{8}\right ) + x^{2} \cdot \left (\frac {45 A a^{2} b^{8}}{2} + 60 B a^{3} b^{7}\right ) + x \left (120 A a^{3} b^{7} + 210 B a^{4} b^{6}\right ) + \frac {- 10 A a^{10} + x^{5} \left (- 15120 A a^{5} b^{5} - 12600 B a^{6} b^{4}\right ) + x^{4} \left (- 6300 A a^{6} b^{4} - 3600 B a^{7} b^{3}\right ) + x^{3} \left (- 2400 A a^{7} b^{3} - 900 B a^{8} b^{2}\right ) + x^{2} \left (- 675 A a^{8} b^{2} - 150 B a^{9} b\right ) + x \left (- 120 A a^{9} b - 12 B a^{10}\right )}{60 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 241, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, B b^{10} x^{5} + \frac {5}{2} \, B a b^{9} x^{4} + \frac {1}{4} \, A b^{10} x^{4} + 15 \, B a^{2} b^{8} x^{3} + \frac {10}{3} \, A a b^{9} x^{3} + 60 \, B a^{3} b^{7} x^{2} + \frac {45}{2} \, A a^{2} b^{8} x^{2} + 210 \, B a^{4} b^{6} x + 120 \, A a^{3} b^{7} x + 42 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} \log \left ({\left | x \right |}\right ) - \frac {10 \, A a^{10} + 2520 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 900 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 300 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 75 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 12 \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 224, normalized size = 1.03 \begin {gather*} x^4\,\left (\frac {A\,b^{10}}{4}+\frac {5\,B\,a\,b^9}{2}\right )-\frac {x\,\left (\frac {B\,a^{10}}{5}+2\,A\,b\,a^9\right )+\frac {A\,a^{10}}{6}+x^2\,\left (\frac {5\,B\,a^9\,b}{2}+\frac {45\,A\,a^8\,b^2}{4}\right )+x^3\,\left (15\,B\,a^8\,b^2+40\,A\,a^7\,b^3\right )+x^4\,\left (60\,B\,a^7\,b^3+105\,A\,a^6\,b^4\right )+x^5\,\left (210\,B\,a^6\,b^4+252\,A\,a^5\,b^5\right )}{x^6}+\ln \left (x\right )\,\left (252\,B\,a^5\,b^5+210\,A\,a^4\,b^6\right )+\frac {B\,b^{10}\,x^5}{5}+\frac {15\,a^2\,b^7\,x^2\,\left (3\,A\,b+8\,B\,a\right )}{2}+30\,a^3\,b^6\,x\,\left (4\,A\,b+7\,B\,a\right )+\frac {5\,a\,b^8\,x^3\,\left (2\,A\,b+9\,B\,a\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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