Integrand size = 23, antiderivative size = 241 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}+\frac {4 \sqrt {-b} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {756, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 756
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}+\frac {2 \int \frac {\frac {1}{2} d (3 c d-b e)+e (2 c d-b e) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 c} \\ & = \frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}-\frac {(d (c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 c}+\frac {(2 (2 c d-b e)) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 c} \\ & = \frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}-\frac {\left (d (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}{3 c \sqrt {b x+c x^2}}+\frac {\left (2 (2 c d-b e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 c \sqrt {b x+c x^2}} \\ & = \frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}+\frac {\left (2 (2 c d-b 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}{3 c \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (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}{3 c \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}+\frac {4 \sqrt {-b} (2 c d-b 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 )}{3 c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (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 )}{3 c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 14.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {-2 b (b+c x) (d+e x) (2 b e-c (4 d+e x))-4 i b \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i \sqrt {\frac {b}{c}} c \left (3 c^2 d^2-5 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{3 b c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
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Time = 2.08 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.58
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c}+\frac {2 \left (d^{2}-\frac {e b d}{3 c}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (2 d e -\frac {2 e \left (b e +c d \right )}{3 c}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(381\) |
default | \(\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (b^{2} d \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) e c -\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e^{2}-6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e +4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}+c^{3} e^{2} x^{3}+b \,c^{2} e^{2} x^{2}+c^{3} d e \,x^{2}+b \,c^{2} d e x \right )}{3 x \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{3}}\) | \(460\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {e x + d} c^{2} e^{2} + {\left (5 \, c^{2} d^{2} - 5 \, b c d e + 2 \, b^{2} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 6 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )\right )}}{9 \, c^{3} e} \]
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\[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]
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\[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,x^2+b\,x}} \,d x \]
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