Integrand size = 24, antiderivative size = 147 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {4 b (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 47, 37} \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {4 b (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{105 e (d+e x)^{3/2} (b d-a e)^3}+\frac {2 (a+b x)^{3/2} (-7 a B e+4 A b e+3 b B d)}{35 e (d+e x)^{5/2} (b d-a e)^2}-\frac {2 (a+b x)^{3/2} (B d-A e)}{7 e (d+e x)^{7/2} (b d-a e)} \]
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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)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {(3 b B d+4 A b e-7 a B e) \int \frac {\sqrt {a+b x}}{(d+e x)^{7/2}} \, dx}{7 e (b d-a e)} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {(2 b (3 b B d+4 A b e-7 a B e)) \int \frac {\sqrt {a+b x}}{(d+e x)^{5/2}} \, dx}{35 e (b d-a e)^2} \\ & = -\frac {2 (B d-A e) (a+b x)^{3/2}}{7 e (b d-a e) (d+e x)^{7/2}}+\frac {2 (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{35 e (b d-a e)^2 (d+e x)^{5/2}}+\frac {4 b (3 b B d+4 A b e-7 a B e) (a+b x)^{3/2}}{105 e (b d-a e)^3 (d+e x)^{3/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (35 A b^2-35 a b B-\frac {15 B d e (a+b x)^2}{(d+e x)^2}+\frac {15 A e^2 (a+b x)^2}{(d+e x)^2}+\frac {21 b B d (a+b x)}{d+e x}-\frac {42 A b e (a+b x)}{d+e x}+\frac {21 a B e (a+b x)}{d+e x}\right )}{105 (b d-a e)^3 (d+e x)^{3/2}} \]
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Time = 3.52 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (8 A \,b^{2} e^{2} x^{2}-14 B a b \,e^{2} x^{2}+6 B \,b^{2} d e \,x^{2}-12 A a b \,e^{2} x +28 A \,b^{2} d e x +21 B \,a^{2} e^{2} x -58 B a b d e x +21 b^{2} B \,d^{2} x +15 a^{2} A \,e^{2}-42 A a b d e +35 A \,b^{2} d^{2}+6 B \,a^{2} d e -14 B a b \,d^{2}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (a e -b d \right )^{3}}\) | \(149\) |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (8 A \,b^{2} e^{2} x^{2}-14 B a b \,e^{2} x^{2}+6 B \,b^{2} d e \,x^{2}-12 A a b \,e^{2} x +28 A \,b^{2} d e x +21 B \,a^{2} e^{2} x -58 B a b d e x +21 b^{2} B \,d^{2} x +15 a^{2} A \,e^{2}-42 A a b d e +35 A \,b^{2} d^{2}+6 B \,a^{2} d e -14 B a b \,d^{2}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(177\) |
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Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (129) = 258\).
Time = 5.63 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.99 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (15 \, A a^{3} e^{2} + 2 \, {\left (3 \, B b^{3} d e - {\left (7 \, B a b^{2} - 4 \, A b^{3}\right )} e^{2}\right )} x^{3} - 7 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} d^{2} + 6 \, {\left (B a^{3} - 7 \, A a^{2} b\right )} d e + {\left (21 \, B b^{3} d^{2} - 4 \, {\left (13 \, B a b^{2} - 7 \, A b^{3}\right )} d e + {\left (7 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{2}\right )} x^{2} + {\left (7 \, {\left (B a b^{2} + 5 \, A b^{3}\right )} d^{2} - 2 \, {\left (26 \, B a^{2} b + 7 \, A a b^{2}\right )} d e + 3 \, {\left (7 \, B a^{3} + A a^{2} b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{105 \, {\left (b^{3} d^{7} - 3 \, a b^{2} d^{6} e + 3 \, a^{2} b d^{5} e^{2} - a^{3} d^{4} e^{3} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{4} + 4 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{3} + 6 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x^{2} + 4 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4}\right )} x\right )}} \]
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\[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {a + b x}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (129) = 258\).
Time = 0.42 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.58 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B b^{8} d e^{4} {\left | b \right |} - 7 \, B a b^{7} e^{5} {\left | b \right |} + 4 \, A b^{8} e^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}} + \frac {7 \, {\left (3 \, B b^{9} d^{2} e^{3} {\left | b \right |} - 10 \, B a b^{8} d e^{4} {\left | b \right |} + 4 \, A b^{9} d e^{4} {\left | b \right |} + 7 \, B a^{2} b^{7} e^{5} {\left | b \right |} - 4 \, A a b^{8} e^{5} {\left | b \right |}\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}}\right )} - \frac {35 \, {\left (B a b^{9} d^{2} e^{3} {\left | b \right |} - A b^{10} d^{2} e^{3} {\left | b \right |} - 2 \, B a^{2} b^{8} d e^{4} {\left | b \right |} + 2 \, A a b^{9} d e^{4} {\left | b \right |} + B a^{3} b^{7} e^{5} {\left | b \right |} - A a^{2} b^{8} e^{5} {\left | b \right |}\right )}}{b^{5} d^{3} e^{3} - 3 \, a b^{4} d^{2} e^{4} + 3 \, a^{2} b^{3} d e^{5} - a^{3} b^{2} e^{6}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{105 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {7}{2}}} \]
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Time = 2.13 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.01 \[ \int \frac {\sqrt {a+b x} (A+B x)}{(d+e x)^{9/2}} \, dx=-\frac {\sqrt {d+e\,x}\,\left (\frac {\sqrt {a+b\,x}\,\left (12\,B\,a^3\,d\,e+30\,A\,a^3\,e^2-28\,B\,a^2\,b\,d^2-84\,A\,a^2\,b\,d\,e+70\,A\,a\,b^2\,d^2\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^3}+\frac {x\,\sqrt {a+b\,x}\,\left (42\,B\,a^3\,e^2-104\,B\,a^2\,b\,d\,e+6\,A\,a^2\,b\,e^2+14\,B\,a\,b^2\,d^2-28\,A\,a\,b^2\,d\,e+70\,A\,b^3\,d^2\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b^2\,x^3\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-7\,B\,a\,e+3\,B\,b\,d\right )}{105\,e^3\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,b\,x^2\,\left (a\,e-7\,b\,d\right )\,\sqrt {a+b\,x}\,\left (4\,A\,b\,e-7\,B\,a\,e+3\,B\,b\,d\right )}{105\,e^4\,{\left (a\,e-b\,d\right )}^3}\right )}{x^4+\frac {d^4}{e^4}+\frac {4\,d\,x^3}{e}+\frac {4\,d^3\,x}{e^3}+\frac {6\,d^2\,x^2}{e^2}} \]
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