Integrand size = 23, antiderivative size = 451 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.31 (sec) , antiderivative size = 451, 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, 836, 857, 729, 113, 111, 118, 117} \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {4 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \sqrt {d+e x} \left (2 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 754
Rule 836
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+\frac {3}{2} c e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)} \\ & = -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b c d e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-\frac {1}{2} c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2} \\ & = -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2} \\ & = -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 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}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (2 c (2 c d-b e) \left (4 c^2 d^2-4 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}{3 b^4 d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \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 b^4 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 c^2 d^2-5 b c d e-2 b^2 e^2\right )+2 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {4 \sqrt {c} (2 c d-b e) \left (4 c^2 d^2-4 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 )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e-b^2 e^2\right ) \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 (-b)^{7/2} 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 = 10.15 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (b (d+e x) \left (b c^3 d^2 (c d-b e) x^2+2 c^3 d^2 (4 c d-5 b e) x^2 (b+c x)-b d (c d-b e)^2 (b+c x)^2+2 (c d-b e)^2 (4 c d+b e) x (b+c x)^2\right )-\sqrt {\frac {b}{c}} c x (b+c x) \left (2 \sqrt {\frac {b}{c}} \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) (b+c x) (d+e x)+2 i b e \left (8 c^3 d^3-12 b c^2 d^2 e+2 b^2 c d e^2+b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e \left (8 c^3 d^3-13 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^2 (c d-b e)^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
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Time = 2.50 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.45
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 d \,b^{3} x^{2}}+\frac {4 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (b e +4 c d \right )}{3 b^{4} d^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )^{2}}-\frac {4 \left (c e \,x^{2}+c d x \right ) c^{2} \left (5 b e -4 c d \right )}{3 b^{4} \left (b e -c d \right )^{2} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {c e}{3 b^{3} d}-\frac {c^{2} e}{3 \left (b e -c d \right ) b^{3}}+\frac {2 c^{2} \left (5 b e -4 c d \right )}{3 \left (b e -c d \right ) b^{4}}+\frac {2 c^{3} d \left (5 b e -4 c d \right )}{3 b^{4} \left (b e -c d \right )^{2}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {2 c e \left (b e +4 c d \right )}{3 b^{4} d^{2}}+\frac {2 e \,c^{3} \left (5 b e -4 c d \right )}{3 \left (b e -c d \right )^{2} b^{4}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(653\) |
default | \(\text {Expression too large to display}\) | \(1763\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 953, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{6} d^{4} - 32 \, b c^{5} d^{3} e + 13 \, b^{2} c^{4} d^{2} e^{2} + 3 \, b^{3} c^{3} d e^{3} + 2 \, b^{4} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (16 \, b c^{5} d^{4} - 32 \, b^{2} c^{4} d^{3} e + 13 \, b^{3} c^{3} d^{2} e^{2} + 3 \, b^{4} c^{2} d e^{3} + 2 \, b^{5} c e^{4}\right )} x^{3} + {\left (16 \, b^{2} c^{4} d^{4} - 32 \, b^{3} c^{3} d^{3} e + 13 \, b^{4} c^{2} d^{2} e^{2} + 3 \, b^{5} c d e^{3} + 2 \, b^{6} e^{4}\right )} x^{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 ({\left (8 \, c^{6} d^{3} e - 12 \, b c^{5} d^{2} e^{2} + 2 \, b^{2} c^{4} d e^{3} + b^{3} c^{3} e^{4}\right )} x^{4} + 2 \, {\left (8 \, b c^{5} d^{3} e - 12 \, b^{2} c^{4} d^{2} e^{2} + 2 \, b^{3} c^{3} d e^{3} + b^{4} c^{2} e^{4}\right )} x^{3} + {\left (8 \, b^{2} c^{4} d^{3} e - 12 \, b^{3} c^{3} d^{2} e^{2} + 2 \, b^{4} c^{2} d e^{3} + b^{5} c e^{4}\right )} x^{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 ) - 3 \, {\left (b^{3} c^{3} d^{3} e - 2 \, b^{4} c^{2} d^{2} e^{2} + b^{5} c d e^{3} - 2 \, {\left (8 \, c^{6} d^{3} e - 12 \, b c^{5} d^{2} e^{2} + 2 \, b^{2} c^{4} d e^{3} + b^{3} c^{3} e^{4}\right )} x^{3} - {\left (24 \, b c^{5} d^{3} e - 37 \, b^{2} c^{4} d^{2} e^{2} + 7 \, b^{3} c^{3} d e^{3} + 4 \, b^{4} c^{2} e^{4}\right )} x^{2} - 2 \, {\left (3 \, b^{2} c^{4} d^{3} e - 5 \, b^{3} c^{3} d^{2} e^{2} + b^{4} c^{2} d e^{3} + b^{5} c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left ({\left (b^{4} c^{5} d^{4} e - 2 \, b^{5} c^{4} d^{3} e^{2} + b^{6} c^{3} d^{2} e^{3}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} d^{4} e - 2 \, b^{6} c^{3} d^{3} e^{2} + b^{7} c^{2} d^{2} e^{3}\right )} x^{3} + {\left (b^{6} c^{3} d^{4} e - 2 \, b^{7} c^{2} d^{3} e^{2} + b^{8} c d^{2} e^{3}\right )} x^{2}\right )}} \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{5/2}\,\sqrt {d+e\,x}} \,d x \]
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