Integrand size = 20, antiderivative size = 213 \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}-\frac {256 b^3 (12 A b-11 a B) \sqrt {a+b x}}{693 a^6 x^{3/2}}+\frac {512 b^4 (12 A b-11 a B) \sqrt {a+b x}}{693 a^7 \sqrt {x}} \]
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Time = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37} \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=\frac {512 b^4 \sqrt {a+b x} (12 A b-11 a B)}{693 a^7 \sqrt {x}}-\frac {256 b^3 \sqrt {a+b x} (12 A b-11 a B)}{693 a^6 x^{3/2}}+\frac {64 b^2 \sqrt {a+b x} (12 A b-11 a B)}{231 a^5 x^{5/2}}-\frac {160 b \sqrt {a+b x} (12 A b-11 a B)}{693 a^4 x^{7/2}}+\frac {20 \sqrt {a+b x} (12 A b-11 a B)}{99 a^3 x^{9/2}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}-\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}+\frac {\left (2 \left (-6 A b+\frac {11 a B}{2}\right )\right ) \int \frac {1}{x^{11/2} (a+b x)^{3/2}} \, dx}{11 a} \\ & = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}-\frac {(10 (12 A b-11 a B)) \int \frac {1}{x^{11/2} \sqrt {a+b x}} \, dx}{11 a^2} \\ & = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}+\frac {(80 b (12 A b-11 a B)) \int \frac {1}{x^{9/2} \sqrt {a+b x}} \, dx}{99 a^3} \\ & = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}-\frac {\left (160 b^2 (12 A b-11 a B)\right ) \int \frac {1}{x^{7/2} \sqrt {a+b x}} \, dx}{231 a^4} \\ & = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}+\frac {\left (128 b^3 (12 A b-11 a B)\right ) \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{231 a^5} \\ & = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}-\frac {256 b^3 (12 A b-11 a B) \sqrt {a+b x}}{693 a^6 x^{3/2}}-\frac {\left (256 b^4 (12 A b-11 a B)\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{693 a^6} \\ & = -\frac {2 A}{11 a x^{11/2} \sqrt {a+b x}}-\frac {2 (12 A b-11 a B)}{11 a^2 x^{9/2} \sqrt {a+b x}}+\frac {20 (12 A b-11 a B) \sqrt {a+b x}}{99 a^3 x^{9/2}}-\frac {160 b (12 A b-11 a B) \sqrt {a+b x}}{693 a^4 x^{7/2}}+\frac {64 b^2 (12 A b-11 a B) \sqrt {a+b x}}{231 a^5 x^{5/2}}-\frac {256 b^3 (12 A b-11 a B) \sqrt {a+b x}}{693 a^6 x^{3/2}}+\frac {512 b^4 (12 A b-11 a B) \sqrt {a+b x}}{693 a^7 \sqrt {x}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=-\frac {2 \left (-3072 A b^6 x^6+256 a b^5 x^5 (-6 A+11 B x)+128 a^2 b^4 x^4 (3 A+11 B x)-32 a^3 b^3 x^3 (6 A+11 B x)+7 a^6 (9 A+11 B x)+8 a^4 b^2 x^2 (15 A+22 B x)-2 a^5 b x (42 A+55 B x)\right )}{693 a^7 x^{11/2} \sqrt {a+b x}} \]
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Time = 0.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(-\frac {2 \left (-3072 A \,b^{6} x^{6}+2816 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1408 B \,a^{2} b^{4} x^{5}+384 A \,a^{2} b^{4} x^{4}-352 B \,a^{3} b^{3} x^{4}-192 A \,a^{3} b^{3} x^{3}+176 B \,a^{4} b^{2} x^{3}+120 A \,a^{4} b^{2} x^{2}-110 B \,a^{5} b \,x^{2}-84 A \,a^{5} b x +77 B \,a^{6} x +63 A \,a^{6}\right )}{693 x^{\frac {11}{2}} \sqrt {b x +a}\, a^{7}}\) | \(149\) |
default | \(-\frac {2 \left (-3072 A \,b^{6} x^{6}+2816 B a \,b^{5} x^{6}-1536 A a \,b^{5} x^{5}+1408 B \,a^{2} b^{4} x^{5}+384 A \,a^{2} b^{4} x^{4}-352 B \,a^{3} b^{3} x^{4}-192 A \,a^{3} b^{3} x^{3}+176 B \,a^{4} b^{2} x^{3}+120 A \,a^{4} b^{2} x^{2}-110 B \,a^{5} b \,x^{2}-84 A \,a^{5} b x +77 B \,a^{6} x +63 A \,a^{6}\right )}{693 x^{\frac {11}{2}} \sqrt {b x +a}\, a^{7}}\) | \(149\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-2379 A \,b^{5} x^{5}+2123 B a \,b^{4} x^{5}+843 a A \,b^{4} x^{4}-715 B \,a^{2} b^{3} x^{4}-459 a^{2} A \,b^{3} x^{3}+363 B \,a^{3} b^{2} x^{3}+267 a^{3} A \,b^{2} x^{2}-187 B \,a^{4} b \,x^{2}-147 a^{4} A b x +77 a^{5} B x +63 a^{5} A \right )}{693 a^{7} x^{\frac {11}{2}}}+\frac {2 b^{5} \left (A b -B a \right ) \sqrt {x}}{a^{7} \sqrt {b x +a}}\) | \(152\) |
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Time = 0.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left (63 \, A a^{6} + 256 \, {\left (11 \, B a b^{5} - 12 \, A b^{6}\right )} x^{6} + 128 \, {\left (11 \, B a^{2} b^{4} - 12 \, A a b^{5}\right )} x^{5} - 32 \, {\left (11 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4}\right )} x^{4} + 16 \, {\left (11 \, B a^{4} b^{2} - 12 \, A a^{3} b^{3}\right )} x^{3} - 10 \, {\left (11 \, B a^{5} b - 12 \, A a^{4} b^{2}\right )} x^{2} + 7 \, {\left (11 \, B a^{6} - 12 \, A a^{5} b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{693 \, {\left (a^{7} b x^{7} + a^{8} x^{6}\right )}} \]
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Timed out. \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31 \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=-\frac {512 \, B b^{5} x}{63 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {2048 \, A b^{6} x}{231 \, \sqrt {b x^{2} + a x} a^{7}} - \frac {256 \, B b^{4}}{63 \, \sqrt {b x^{2} + a x} a^{5}} + \frac {1024 \, A b^{5}}{231 \, \sqrt {b x^{2} + a x} a^{6}} + \frac {64 \, B b^{3}}{63 \, \sqrt {b x^{2} + a x} a^{4} x} - \frac {256 \, A b^{4}}{231 \, \sqrt {b x^{2} + a x} a^{5} x} - \frac {32 \, B b^{2}}{63 \, \sqrt {b x^{2} + a x} a^{3} x^{2}} + \frac {128 \, A b^{3}}{231 \, \sqrt {b x^{2} + a x} a^{4} x^{2}} + \frac {20 \, B b}{63 \, \sqrt {b x^{2} + a x} a^{2} x^{3}} - \frac {80 \, A b^{2}}{231 \, \sqrt {b x^{2} + a x} a^{3} x^{3}} - \frac {2 \, B}{9 \, \sqrt {b x^{2} + a x} a x^{4}} + \frac {8 \, A b}{33 \, \sqrt {b x^{2} + a x} a^{2} x^{4}} - \frac {2 \, A}{11 \, \sqrt {b x^{2} + a x} a x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (171) = 342\).
Time = 0.37 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.73 \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left ({\left (b x + a\right )} {\left ({\left (b x + a\right )} {\left (\frac {{\left (2123 \, B a^{21} b^{15} {\left | b \right |} - 2379 \, A a^{20} b^{16} {\left | b \right |}\right )} {\left (b x + a\right )}}{a^{27} b^{6}} - \frac {22 \, {\left (515 \, B a^{22} b^{15} {\left | b \right |} - 579 \, A a^{21} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} + \frac {99 \, {\left (247 \, B a^{23} b^{15} {\left | b \right |} - 279 \, A a^{22} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} - \frac {924 \, {\left (29 \, B a^{24} b^{15} {\left | b \right |} - 33 \, A a^{23} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} {\left (b x + a\right )} + \frac {1155 \, {\left (13 \, B a^{25} b^{15} {\left | b \right |} - 15 \, A a^{24} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} {\left (b x + a\right )} - \frac {693 \, {\left (5 \, B a^{26} b^{15} {\left | b \right |} - 6 \, A a^{25} b^{16} {\left | b \right |}\right )}}{a^{27} b^{6}}\right )} \sqrt {b x + a}}{693 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}}} - \frac {4 \, {\left (B^{2} a^{2} b^{13} - 2 \, A B a b^{14} + A^{2} b^{15}\right )}}{{\left (B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {13}{2}} + B a^{2} b^{\frac {15}{2}} - A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {15}{2}} - A a b^{\frac {17}{2}}\right )} a^{6} {\left | b \right |}} \]
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Time = 0.99 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x}{x^{13/2} (a+b x)^{3/2}} \, dx=-\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{11\,a\,b}+\frac {20\,x^2\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^3}+\frac {64\,b^2\,x^4\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^5}-\frac {256\,b^3\,x^5\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^6}-\frac {512\,b^4\,x^6\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^7}-\frac {32\,b\,x^3\,\left (12\,A\,b-11\,B\,a\right )}{693\,a^4}+\frac {x\,\left (154\,B\,a^6-168\,A\,a^5\,b\right )}{693\,a^7\,b}\right )}{x^{13/2}+\frac {a\,x^{11/2}}{b}} \]
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