Integrand size = 24, antiderivative size = 89 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=-\frac {2 (B d-A e)}{e (b d-a e) \sqrt {a+b x} \sqrt {d+e x}}+\frac {2 (b B d-2 A b e+a B e) \sqrt {d+e x}}{e (b d-a e)^2 \sqrt {a+b x}} \]
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Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {79, 37} \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=\frac {2 \sqrt {d+e x} (a B e-2 A b e+b B d)}{e \sqrt {a+b x} (b d-a e)^2}-\frac {2 (B d-A e)}{e \sqrt {a+b x} \sqrt {d+e x} (b d-a e)} \]
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Rule 37
Rule 79
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (B d-A e)}{e (b d-a e) \sqrt {a+b x} \sqrt {d+e x}}-\frac {(b B d-2 A b e+a B e) \int \frac {1}{(a+b x)^{3/2} \sqrt {d+e x}} \, dx}{e (b d-a e)} \\ & = -\frac {2 (B d-A e)}{e (b d-a e) \sqrt {a+b x} \sqrt {d+e x}}+\frac {2 (b B d-2 A b e+a B e) \sqrt {d+e x}}{e (b d-a e)^2 \sqrt {a+b x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=\frac {2 B (2 a d+b d x+a e x)-2 A (a e+b (d+2 e x))}{(b d-a e)^2 \sqrt {a+b x} \sqrt {d+e x}} \]
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Time = 1.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {2 \left (2 A b e x -B a e x -B b d x +A a e +A b d -2 B a d \right )}{\left (a e -b d \right )^{2} \sqrt {b x +a}\, \sqrt {e x +d}}\) | \(59\) |
gosper | \(-\frac {2 \left (2 A b e x -B a e x -B b d x +A a e +A b d -2 B a d \right )}{\sqrt {b x +a}\, \sqrt {e x +d}\, \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\) | \(72\) |
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none
Time = 0.40 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.66 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (A a e - {\left (2 \, B a - A b\right )} d - {\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + {\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \]
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\[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=\int \frac {A + B x}{\left (a + b x\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (81) = 162\).
Time = 0.34 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.04 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B b^{2} d - A b^{2} e\right )} \sqrt {b x + a}}{{\left (b^{2} d^{2} {\left | b \right |} - 2 \, a b d e {\left | b \right |} + a^{2} e^{2} {\left | b \right |}\right )} \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} + \frac {4 \, {\left (B^{2} a^{2} b^{3} e - 2 \, A B a b^{4} e + A^{2} b^{5} e\right )}}{{\left (\sqrt {b e} B a b^{3} d - \sqrt {b e} A b^{4} d - \sqrt {b e} B a^{2} b^{2} e + \sqrt {b e} A a b^{3} e - \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a b + \sqrt {b e} {\left (\sqrt {b e} \sqrt {b x + a} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{2}\right )} {\left (b d {\left | b \right |} - a e {\left | b \right |}\right )}} \]
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Time = 2.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx=-\frac {\left (\frac {2\,A\,a\,e+2\,A\,b\,d-4\,B\,a\,d}{e\,{\left (a\,e-b\,d\right )}^2}-\frac {x\,\left (2\,B\,a\,e-4\,A\,b\,e+2\,B\,b\,d\right )}{e\,{\left (a\,e-b\,d\right )}^2}\right )\,\sqrt {d+e\,x}}{x\,\sqrt {a+b\,x}+\frac {d\,\sqrt {a+b\,x}}{e}} \]
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