Integrand size = 28, antiderivative size = 262 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {c} (B d-A e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{d e (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} B \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {848, 857, 729, 113, 111, 118, 117} \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (B d-A e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{d e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 \sqrt {b x+c x^2} (B d-A e)}{d \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {-b} B \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 848
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}-\frac {2 \int \frac {\frac {1}{2} (b B-A c) d+\frac {1}{2} c (B d-A e) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{d (c d-b e)} \\ & = \frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}+\frac {B \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{e}-\frac {(c (B d-A e)) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{d e (c d-b e)} \\ & = \frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}+\frac {\left (B \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{e \sqrt {b x+c x^2}}-\frac {\left (c (B d-A e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{d e (c d-b e) \sqrt {b x+c x^2}} \\ & = \frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}-\frac {\left (c (B d-A e) \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}{d e (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (B \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}{e \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 (B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {c} (B d-A e) \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 )}{d e (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} B \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 )}{\sqrt {c} e \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 16.73 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {-2 \sqrt {\frac {b}{c}} d (B d-A e) (b+c x)+2 i b e (-B d+A e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i A e (c d-b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{\sqrt {\frac {b}{c}} d e (c d-b e) \sqrt {x (b+c x)} \sqrt {d+e x}} \]
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Time = 1.18 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.68
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 \left (c e \,x^{2}+b e x \right ) \left (A e -B d \right )}{e d \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {B}{e}+\frac {A e -B d}{e d}-\frac {\left (A e -B d \right ) b}{d \left (b e -c d \right )}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {2 \left (A e -B d \right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{d \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(441\) |
default | \(\frac {2 \left (A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} e^{2}-A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b c d e +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} d e -B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b c \,d^{2}-B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} d e +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b c \,d^{2}+A \,c^{2} e^{2} x^{2}-B \,c^{2} d e \,x^{2}+A b c \,e^{2} x -B b c d e x \right ) \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}{d e c \left (b e -c d \right ) x \left (c e \,x^{2}+b e x +c d x +b d \right )}\) | \(547\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.73 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (B c d^{3} - A b d e^{2} - 2 \, {\left (B b - A c\right )} d^{2} e + {\left (B c d^{2} e - A b e^{3} - 2 \, {\left (B b - A c\right )} d e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (B c d^{2} e - A c d e^{2} + {\left (B c d e^{2} - A c e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (B c d e^{2} - A c e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (c^{2} d^{3} e^{2} - b c d^{2} e^{3} + {\left (c^{2} d^{2} e^{3} - b c d e^{4}\right )} x\right )}} \]
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\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]
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