Integrand size = 28, antiderivative size = 399 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.37 (sec) , antiderivative size = 399, 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 = {832, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 c e (6 A e+13 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-8 b^3 B e^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{3/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (-3 b c (A e+B d)+6 A c^2 d+4 b^2 B e\right )}{3 b^2 c^2} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 832
Rule 846
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {1}{2} b (b B+3 A c) d e+\frac {1}{2} e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{b^2 c} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {4 \int \frac {\frac {1}{4} b d e \left (3 A c^2 d-4 b^2 B e+3 b c (2 B d+A e)\right )+\frac {1}{4} e \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}-\frac {\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^2 c^2}+\frac {\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^2 c^2} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}-\frac {\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 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^2 c^2 \sqrt {b x+c x^2}}+\frac {\left (\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 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^2 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {\left (\left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 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^2 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 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^2 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \left (6 A c^2 d+4 b^2 B e-3 b c (B d+A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{3 b^2 c^2}+\frac {2 \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 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)^{3/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) \left (6 A c^2 d+4 b^2 B e-3 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)^{3/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 24.12 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b (d+e x) \left (3 (b B-A c) (c d-b e)^2 x-3 A c^2 d^2 (b+c x)+b^2 B e^2 x (b+c x)\right )+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) (b+c x) (d+e x)+i b e \left (6 A c^3 d^2-8 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (13 B d+6 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (3 A c^2 d+8 b^2 B e-3 b c (3 B d+2 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^3 c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(801\) vs. \(2(347)=694\).
Time = 1.21 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.01
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 +c d x +b d \right ) A \,d^{2}}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right )}{b^{2} c^{3} \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 B \,e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c^{2}}+\frac {2 \left (-\frac {e \left (A b c \,e^{2}-3 A \,c^{2} d e -B \,b^{2} e^{2}+3 B b c d e -3 B \,c^{2} d^{2}\right )}{c^{3}}+\frac {\left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right ) \left (b e -c d \right )}{c^{3} b^{2}}+\frac {d \left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right )}{c^{2} b^{2}}-\frac {B \,e^{2} b d}{3 c^{2}}\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 {e^{2} \left (A c e -B b e +3 B c d \right )}{c^{2}}+\frac {A c \,d^{2} e}{b^{2}}+\frac {\left (A \,b^{2} c \,e^{2}-2 A b \,c^{2} d e +A \,c^{3} d^{2}-b^{3} B \,e^{2}+2 B \,b^{2} c d e -B b \,c^{2} d^{2}\right ) e}{c^{2} b^{2}}-\frac {2 B \,e^{2} \left (b e +c d \right )}{3 c^{2}}\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}}\) | \(802\) |
default | \(\text {Expression too large to display}\) | \(1324\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.86 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (8 \, B b^{2} c^{3} + 9 \, A b c^{4}\right )} d^{2} e - {\left (17 \, B b^{3} c^{2} - 9 \, A b^{2} c^{3}\right )} d e^{2} + 2 \, {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} e^{3}\right )} x^{2} + {\left (3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + {\left (8 \, B b^{3} c^{2} + 9 \, A b^{2} c^{3}\right )} d^{2} e - {\left (17 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} d e^{2} + 2 \, {\left (4 \, B b^{5} - 3 \, A b^{4} c\right )} 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 ) + 3 \, {\left ({\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} e - {\left (13 \, B b^{2} c^{3} - 6 \, A b c^{4}\right )} d e^{2} + 2 \, {\left (4 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} e^{3}\right )} x^{2} + {\left (3 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{2} e - {\left (13 \, B b^{3} c^{2} - 6 \, A b^{2} c^{3}\right )} d e^{2} + 2 \, {\left (4 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} 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 b^{2} c^{3} e^{3} x^{2} - 3 \, A b c^{4} d^{2} e + {\left (3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{2} e - 6 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d e^{2} + {\left (4 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{2} c^{5} e x^{2} + b^{3} c^{4} e x\right )}} \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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