Integrand size = 24, antiderivative size = 201 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{9009 e (b d-a e)^4 (d+e x)^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {16 b^2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{9009 e (d+e x)^{7/2} (b d-a e)^4}+\frac {8 b (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{1287 e (d+e x)^{9/2} (b d-a e)^3}+\frac {2 (a+b x)^{7/2} (-13 a B e+6 A b e+7 b B d)}{143 e (d+e x)^{11/2} (b d-a e)^2}-\frac {2 (a+b x)^{7/2} (B d-A e)}{13 e (d+e x)^{13/2} (b d-a e)} \]
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Rule 37
Rule 47
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {(7 b B d+6 A b e-13 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{13/2}} \, dx}{13 e (b d-a e)} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {(4 b (7 b B d+6 A b e-13 a B e)) \int \frac {(a+b x)^{5/2}}{(d+e x)^{11/2}} \, dx}{143 e (b d-a e)^2} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {\left (8 b^2 (7 b B d+6 A b e-13 a B e)\right ) \int \frac {(a+b x)^{5/2}}{(d+e x)^{9/2}} \, dx}{1287 e (b d-a e)^3} \\ & = -\frac {2 (B d-A e) (a+b x)^{7/2}}{13 e (b d-a e) (d+e x)^{13/2}}+\frac {2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{143 e (b d-a e)^2 (d+e x)^{11/2}}+\frac {8 b (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{1287 e (b d-a e)^3 (d+e x)^{9/2}}+\frac {16 b^2 (7 b B d+6 A b e-13 a B e) (a+b x)^{7/2}}{9009 e (b d-a e)^4 (d+e x)^{7/2}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {2 (a+b x)^{7/2} \left (693 B d e^2 (a+b x)^3-693 A e^3 (a+b x)^3-1638 b B d e (a+b x)^2 (d+e x)+2457 A b e^2 (a+b x)^2 (d+e x)-819 a B e^2 (a+b x)^2 (d+e x)+1001 b^2 B d (a+b x) (d+e x)^2-3003 A b^2 e (a+b x) (d+e x)^2+2002 a b B e (a+b x) (d+e x)^2+1287 A b^3 (d+e x)^3-1287 a b^2 B (d+e x)^3\right )}{9009 (b d-a e)^4 (d+e x)^{13/2}} \]
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Time = 1.10 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.60
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-48 A \,b^{3} e^{3} x^{3}+104 B a \,b^{2} e^{3} x^{3}-56 B \,b^{3} d \,e^{2} x^{3}+168 A a \,b^{2} e^{3} x^{2}-312 A \,b^{3} d \,e^{2} x^{2}-364 B \,a^{2} b \,e^{3} x^{2}+872 B a \,b^{2} d \,e^{2} x^{2}-364 B \,b^{3} d^{2} e \,x^{2}-378 A \,a^{2} b \,e^{3} x +1092 A a \,b^{2} d \,e^{2} x -858 A \,b^{3} d^{2} e x +819 B \,a^{3} e^{3} x -2807 B \,a^{2} b d \,e^{2} x +3133 B a \,b^{2} d^{2} e x -1001 b^{3} B \,d^{3} x +693 a^{3} A \,e^{3}-2457 A \,a^{2} b d \,e^{2}+3003 A a \,b^{2} d^{2} e -1287 A \,b^{3} d^{3}+126 B \,a^{3} d \,e^{2}-364 B \,a^{2} b \,d^{2} e +286 B a \,b^{2} d^{3}\right )}{9009 \left (e x +d \right )^{\frac {13}{2}} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(322\) |
default | \(-\frac {2 \left (-48 A \,b^{5} e^{3} x^{5}+104 B a \,b^{4} e^{3} x^{5}-56 B \,b^{5} d \,e^{2} x^{5}+72 A a \,b^{4} e^{3} x^{4}-312 A \,b^{5} d \,e^{2} x^{4}-156 B \,a^{2} b^{3} e^{3} x^{4}+760 B a \,b^{4} d \,e^{2} x^{4}-364 B \,b^{5} d^{2} e \,x^{4}-90 A \,a^{2} b^{3} e^{3} x^{3}+468 A a \,b^{4} d \,e^{2} x^{3}-858 A \,b^{5} d^{2} e \,x^{3}+195 B \,a^{3} b^{2} e^{3} x^{3}-1119 B \,a^{2} b^{3} d \,e^{2} x^{3}+2405 B a \,b^{4} d^{2} e \,x^{3}-1001 B \,b^{5} d^{3} x^{3}+105 A \,a^{3} b^{2} e^{3} x^{2}-585 A \,a^{2} b^{3} d \,e^{2} x^{2}+1287 A a \,b^{4} d^{2} e \,x^{2}-1287 A \,b^{5} d^{3} x^{2}+1274 B \,a^{4} b \,e^{3} x^{2}-4616 B \,a^{3} b^{2} d \,e^{2} x^{2}+5538 B \,a^{2} b^{3} d^{2} e \,x^{2}-1716 B a \,b^{4} d^{3} x^{2}+1008 A \,a^{4} b \,e^{3} x -3822 A \,a^{3} b^{2} d \,e^{2} x +5148 A \,a^{2} b^{3} d^{2} e x -2574 A a \,b^{4} d^{3} x +819 B \,a^{5} e^{3} x -2555 B \,a^{4} b d \,e^{2} x +2405 B \,a^{3} b^{2} d^{2} e x -429 B \,a^{2} b^{3} d^{3} x +693 A \,a^{5} e^{3}-2457 A \,a^{4} b d \,e^{2}+3003 A \,a^{3} b^{2} d^{2} e -1287 A \,a^{2} b^{3} d^{3}+126 B \,a^{5} d \,e^{2}-364 B \,a^{4} b \,d^{2} e +286 B \,a^{3} b^{2} d^{3}\right ) \left (b x +a \right )^{\frac {3}{2}}}{9009 \left (e x +d \right )^{\frac {13}{2}} \left (a e -b d \right )^{4}}\) | \(525\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1047 vs. \(2 (177) = 354\).
Time = 133.28 (sec) , antiderivative size = 1047, normalized size of antiderivative = 5.21 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (693 \, A a^{6} e^{3} - 8 \, {\left (7 \, B b^{6} d e^{2} - {\left (13 \, B a b^{5} - 6 \, A b^{6}\right )} e^{3}\right )} x^{6} - 4 \, {\left (91 \, B b^{6} d^{2} e - 2 \, {\left (88 \, B a b^{5} - 39 \, A b^{6}\right )} d e^{2} + {\left (13 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )} e^{3}\right )} x^{5} - {\left (1001 \, B b^{6} d^{3} - 13 \, {\left (157 \, B a b^{5} - 66 \, A b^{6}\right )} d^{2} e + {\left (359 \, B a^{2} b^{4} - 156 \, A a b^{5}\right )} d e^{2} - 3 \, {\left (13 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} e^{3}\right )} x^{4} + 143 \, {\left (2 \, B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} d^{3} - 91 \, {\left (4 \, B a^{5} b - 33 \, A a^{4} b^{2}\right )} d^{2} e + 63 \, {\left (2 \, B a^{6} - 39 \, A a^{5} b\right )} d e^{2} - {\left (143 \, {\left (19 \, B a b^{5} + 9 \, A b^{6}\right )} d^{3} - 13 \, {\left (611 \, B a^{2} b^{4} + 33 \, A a b^{5}\right )} d^{2} e + {\left (5735 \, B a^{3} b^{3} + 117 \, A a^{2} b^{4}\right )} d e^{2} - {\left (1469 \, B a^{4} b^{2} + 15 \, A a^{3} b^{3}\right )} e^{3}\right )} x^{3} - {\left (429 \, {\left (5 \, B a^{2} b^{4} + 9 \, A a b^{5}\right )} d^{3} - 13 \, {\left (611 \, B a^{3} b^{3} + 495 \, A a^{2} b^{4}\right )} d^{2} e + {\left (7171 \, B a^{4} b^{2} + 4407 \, A a^{3} b^{3}\right )} d e^{2} - 7 \, {\left (299 \, B a^{5} b + 159 \, A a^{4} b^{2}\right )} e^{3}\right )} x^{2} - {\left (143 \, {\left (B a^{3} b^{3} + 27 \, A a^{2} b^{4}\right )} d^{3} - 13 \, {\left (157 \, B a^{4} b^{2} + 627 \, A a^{3} b^{3}\right )} d^{2} e + 7 \, {\left (347 \, B a^{5} b + 897 \, A a^{4} b^{2}\right )} d e^{2} - 63 \, {\left (13 \, B a^{6} + 27 \, A a^{5} b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{9009 \, {\left (b^{4} d^{11} - 4 \, a b^{3} d^{10} e + 6 \, a^{2} b^{2} d^{9} e^{2} - 4 \, a^{3} b d^{8} e^{3} + a^{4} d^{7} e^{4} + {\left (b^{4} d^{4} e^{7} - 4 \, a b^{3} d^{3} e^{8} + 6 \, a^{2} b^{2} d^{2} e^{9} - 4 \, a^{3} b d e^{10} + a^{4} e^{11}\right )} x^{7} + 7 \, {\left (b^{4} d^{5} e^{6} - 4 \, a b^{3} d^{4} e^{7} + 6 \, a^{2} b^{2} d^{3} e^{8} - 4 \, a^{3} b d^{2} e^{9} + a^{4} d e^{10}\right )} x^{6} + 21 \, {\left (b^{4} d^{6} e^{5} - 4 \, a b^{3} d^{5} e^{6} + 6 \, a^{2} b^{2} d^{4} e^{7} - 4 \, a^{3} b d^{3} e^{8} + a^{4} d^{2} e^{9}\right )} x^{5} + 35 \, {\left (b^{4} d^{7} e^{4} - 4 \, a b^{3} d^{6} e^{5} + 6 \, a^{2} b^{2} d^{5} e^{6} - 4 \, a^{3} b d^{4} e^{7} + a^{4} d^{3} e^{8}\right )} x^{4} + 35 \, {\left (b^{4} d^{8} e^{3} - 4 \, a b^{3} d^{7} e^{4} + 6 \, a^{2} b^{2} d^{6} e^{5} - 4 \, a^{3} b d^{5} e^{6} + a^{4} d^{4} e^{7}\right )} x^{3} + 21 \, {\left (b^{4} d^{9} e^{2} - 4 \, a b^{3} d^{8} e^{3} + 6 \, a^{2} b^{2} d^{7} e^{4} - 4 \, a^{3} b d^{6} e^{5} + a^{4} d^{5} e^{6}\right )} x^{2} + 7 \, {\left (b^{4} d^{10} e - 4 \, a b^{3} d^{9} e^{2} + 6 \, a^{2} b^{2} d^{8} e^{3} - 4 \, a^{3} b d^{7} e^{4} + a^{4} d^{6} e^{5}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (177) = 354\).
Time = 1.05 (sec) , antiderivative size = 1003, normalized size of antiderivative = 4.99 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\text {Too large to display} \]
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Time = 3.53 (sec) , antiderivative size = 706, normalized size of antiderivative = 3.51 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{15/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {x^3\,\sqrt {a+b\,x}\,\left (-2938\,B\,a^4\,b^2\,e^3+11470\,B\,a^3\,b^3\,d\,e^2-30\,A\,a^3\,b^3\,e^3-15886\,B\,a^2\,b^4\,d^2\,e+234\,A\,a^2\,b^4\,d\,e^2+5434\,B\,a\,b^5\,d^3-858\,A\,a\,b^5\,d^2\,e+2574\,A\,b^6\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}-\frac {\sqrt {a+b\,x}\,\left (252\,B\,a^6\,d\,e^2+1386\,A\,a^6\,e^3-728\,B\,a^5\,b\,d^2\,e-4914\,A\,a^5\,b\,d\,e^2+572\,B\,a^4\,b^2\,d^3+6006\,A\,a^4\,b^2\,d^2\,e-2574\,A\,a^3\,b^3\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}-\frac {x\,\sqrt {a+b\,x}\,\left (1638\,B\,a^6\,e^3-4858\,B\,a^5\,b\,d\,e^2+3402\,A\,a^5\,b\,e^3+4082\,B\,a^4\,b^2\,d^2\,e-12558\,A\,a^4\,b^2\,d\,e^2-286\,B\,a^3\,b^3\,d^3+16302\,A\,a^3\,b^3\,d^2\,e-7722\,A\,a^2\,b^4\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}+\frac {x^2\,\sqrt {a+b\,x}\,\left (-4186\,B\,a^5\,b\,e^3+14342\,B\,a^4\,b^2\,d\,e^2-2226\,A\,a^4\,b^2\,e^3-15886\,B\,a^3\,b^3\,d^2\,e+8814\,A\,a^3\,b^3\,d\,e^2+4290\,B\,a^2\,b^4\,d^3-12870\,A\,a^2\,b^4\,d^2\,e+7722\,A\,a\,b^5\,d^3\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}+\frac {16\,b^5\,x^6\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^5\,{\left (a\,e-b\,d\right )}^4}-\frac {8\,b^4\,x^5\,\left (a\,e-13\,b\,d\right )\,\sqrt {a+b\,x}\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^6\,{\left (a\,e-b\,d\right )}^4}+\frac {2\,b^3\,x^4\,\sqrt {a+b\,x}\,\left (3\,a^2\,e^2-26\,a\,b\,d\,e+143\,b^2\,d^2\right )\,\left (6\,A\,b\,e-13\,B\,a\,e+7\,B\,b\,d\right )}{9009\,e^7\,{\left (a\,e-b\,d\right )}^4}\right )}{x^7+\frac {d^7}{e^7}+\frac {7\,d\,x^6}{e}+\frac {7\,d^6\,x}{e^6}+\frac {21\,d^2\,x^5}{e^2}+\frac {35\,d^3\,x^4}{e^3}+\frac {35\,d^4\,x^3}{e^4}+\frac {21\,d^5\,x^2}{e^5}} \]
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