Integrand size = 23, antiderivative size = 370 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \sqrt {c} \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.27 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {754, 848, 857, 729, 113, 111, 118, 117} \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {4 e \sqrt {b x+c x^2} \left (b^2 e^2-b c d e+c^2 d^2\right )}{b^2 d^2 \sqrt {d+e x} (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 754
Rule 848
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b e (c d-2 b e)+\frac {1}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} b c d e (c d+b e)+\frac {1}{2} c e \left (c^2 d^2-b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2 d^2 (c d-b e)^2} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {(c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)}+\frac {\left (2 c \left (c^2 d^2-b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{b^2 d^2 (c d-b e)^2} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {\left (c (2 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (2 c \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{b^2 d^2 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {\left (2 c \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{b^2 d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (c (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}}-\frac {4 e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {4 \sqrt {c} \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.76 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 b d \left (b^2 e^2+b c e^2 x+c^2 d (d+e x)\right )+4 i \sqrt {\frac {b}{c}} c e \left (c^2 d^2-b c d e+b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-2 i \sqrt {\frac {b}{c}} c e \left (c^2 d^2-3 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b^2 d^2 (c d-b e)^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
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Time = 2.14 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.75
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right )}{d^{2} b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 c e x \left (\frac {\left (b^{2} e^{2}+c^{2} d^{2}\right ) x}{d^{2} b^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}+\frac {\left (b e +c d \right ) \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2} d^{2} c e}\right )}{\sqrt {\left (\frac {\left (b e +c d \right ) x}{c e}+x^{2}+\frac {b d}{c e}\right ) c e x}}+\frac {2 \left (-\frac {b e +c d}{b^{2} d^{2}}+\frac {\left (b e +c d \right ) \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2} d^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {c e}{b^{2} d^{2}}+\frac {\left (b^{2} e^{2}+c^{2} d^{2}\right ) c e}{d^{2} b^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(649\) |
default | \(-\frac {2 \left (\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c d \,e^{2}-3 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{2} d^{2} e +2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{3} d^{3}+2 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{4} e^{3}-4 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{3} c d \,e^{2}+4 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b^{2} c^{2} d^{2} e -2 E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, b \,c^{3} d^{3}+2 b^{2} c^{2} e^{3} x^{2}-2 b \,c^{3} d \,e^{2} x^{2}+2 c^{4} d^{2} e \,x^{2}+2 x \,b^{3} c \,e^{3}-b^{2} c^{2} d \,e^{2} x -b \,c^{3} d^{2} e x +2 c^{4} d^{3} x +b^{3} c d \,e^{2}-2 b^{2} d^{2} e \,c^{2}+c^{3} b \,d^{3}\right ) \sqrt {x \left (c x +b \right )}}{x \left (c x +b \right ) \left (b e -c d \right )^{2} c \,b^{2} d^{2} \sqrt {e x +d}}\) | \(698\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} - 3 \, b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x^{3} + {\left (2 \, c^{4} d^{4} - b c^{3} d^{3} e - 6 \, b^{2} c^{2} d^{2} e^{2} - b^{3} c d e^{3} + 2 \, b^{4} e^{4}\right )} x^{2} + {\left (2 \, b c^{3} d^{4} - 3 \, b^{2} c^{2} d^{3} e - 3 \, b^{3} c d^{2} e^{2} + 2 \, b^{4} d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left ({\left (c^{4} d^{2} e^{2} - b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} + {\left (c^{4} d^{3} e + b^{3} c e^{4}\right )} x^{2} + {\left (b c^{3} d^{3} e - b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (b c^{3} d^{3} e - 2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} + 2 \, {\left (c^{4} d^{2} e^{2} - b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{2} + {\left (2 \, c^{4} d^{3} e - b c^{3} d^{2} e^{2} - b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left ({\left (b^{2} c^{4} d^{4} e^{2} - 2 \, b^{3} c^{3} d^{3} e^{3} + b^{4} c^{2} d^{2} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{5} e - b^{3} c^{3} d^{4} e^{2} - b^{4} c^{2} d^{3} e^{3} + b^{5} c d^{2} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{5} e - 2 \, b^{4} c^{2} d^{4} e^{2} + b^{5} c d^{3} e^{3}\right )} x\right )}} \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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