Integrand size = 28, antiderivative size = 420 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {834, 836, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (-8 b c (A e+B d)+16 A c^2 d+3 b^2 B e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (A b e-8 A c d+4 b B d)-c x \left (b^2 e (A e+7 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right )\right )}{3 b^4 d \sqrt {b x+c x^2} (c d-b e)} \]
[In]
[Out]
Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 834
Rule 836
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} (8 A c d-b (4 B d+A e))-\frac {3}{2} (b B-2 A c) e x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = -\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} b d e \left (4 b B c d-8 A c^2 d-3 b^2 B e+7 A b c e\right )-\frac {1}{4} c e \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)} \\ & = -\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4}-\frac {\left (c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)} \\ & = -\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (\left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\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 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\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 (c d-b e) \sqrt {b x+c x^2}} \\ & = -\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\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 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (\left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\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 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (b (c d-b e) (4 b B d-8 A c d+A b e)-c \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\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 (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \left (16 A c^2 d+3 b^2 B e-8 b c (B d+A e)\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} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 16.57 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (\sqrt {\frac {b}{c}} (d+e x) \left (b c (b B-A c) d (c d-b e) x^2+c d \left (-8 A c^2 d-4 b^2 B e+b c (5 B d+7 A e)\right ) x^2 (b+c x)+A b d (c d-b e) (b+c x)^2+(c d-b e) (3 b B d-8 A c d+A b e) x (b+c x)^2\right )+x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^2 d^2+b^2 e (7 B d+A e)-8 b c d (B d+2 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) (8 A c d-b (4 B d+A e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^4 \sqrt {\frac {b}{c}} d (c d-b e) (x (b+c x))^{3/2} \sqrt {d+e x}} \]
[In]
[Out]
Time = 1.21 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.69
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 A \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (A b e -8 A c d +3 B b d \right )}{3 b^{4} d \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (A c -B b \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c \,b^{3} \left (x +\frac {b}{c}\right )^{2}}+\frac {2 \left (c e \,x^{2}+c d x \right ) \left (7 A b c e -8 A \,c^{2} d -4 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right ) \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (-\frac {A c e}{3 b^{3}}+\frac {\left (A c -B b \right ) e}{3 b^{3}}-\frac {7 A b c e -8 A \,c^{2} d -4 b^{2} B e +5 B b c d}{3 b^{4}}-\frac {c d \left (7 A b c e -8 A \,c^{2} d -4 b^{2} B e +5 B b c d \right )}{3 b^{4} \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 (\frac {c e \left (A b e -8 A c d +3 B b d \right )}{3 d \,b^{4}}-\frac {c e \left (7 A b c e -8 A \,c^{2} d -4 b^{2} B e +5 B b c d \right )}{3 b^{4} \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}}\, \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 )}{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}}\) | \(710\) |
default | \(\text {Expression too large to display}\) | \(2503\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 1013, normalized size of antiderivative = 2.41 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (A b^{3} c^{2} e^{3} - 8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (11 \, B b^{2} c^{3} - 24 \, A b c^{4}\right )} d^{2} e - 2 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d e^{2}\right )} x^{4} + 2 \, {\left (A b^{4} c e^{3} - 8 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + {\left (11 \, B b^{3} c^{2} - 24 \, A b^{2} c^{3}\right )} d^{2} e - 2 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2}\right )} x^{3} + {\left (A b^{5} e^{3} - 8 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3} + {\left (11 \, B b^{4} c - 24 \, A b^{3} c^{2}\right )} d^{2} e - 2 \, {\left (B b^{5} - 3 \, A b^{4} c\right )} d e^{2}\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 ) + 3 \, {\left ({\left (A b^{2} c^{3} e^{3} - 8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} e + {\left (7 \, B b^{2} c^{3} - 16 \, A b c^{4}\right )} d e^{2}\right )} x^{4} + 2 \, {\left (A b^{3} c^{2} e^{3} - 8 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} e + {\left (7 \, B b^{3} c^{2} - 16 \, A b^{2} c^{3}\right )} d e^{2}\right )} x^{3} + {\left (A b^{4} c e^{3} - 8 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} e + {\left (7 \, B b^{4} c - 16 \, A b^{3} c^{2}\right )} d e^{2}\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 (A b^{3} c^{2} d^{2} e - A b^{4} c d e^{2} - {\left (A b^{2} c^{3} e^{3} - 8 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} e + {\left (7 \, B b^{2} c^{3} - 16 \, A b c^{4}\right )} d e^{2}\right )} x^{3} - {\left (2 \, A b^{3} c^{2} e^{3} - 12 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} e + {\left (11 \, B b^{3} c^{2} - 25 \, A b^{2} c^{3}\right )} d e^{2}\right )} x^{2} - {\left (A b^{4} c e^{3} - 3 \, {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{2} e + {\left (3 \, B b^{4} c - 7 \, A b^{3} c^{2}\right )} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left ({\left (b^{4} c^{4} d^{2} e - b^{5} c^{3} d e^{2}\right )} x^{4} + 2 \, {\left (b^{5} c^{3} d^{2} e - b^{6} c^{2} d e^{2}\right )} x^{3} + {\left (b^{6} c^{2} d^{2} e - b^{7} c d e^{2}\right )} x^{2}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
[In]
[Out]