Integrand size = 28, antiderivative size = 449 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.40 (sec) , antiderivative size = 449, 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 = {826, 828, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (56 A c e (2 c d-b e)-B \left (-b^2 e^2-72 b c d e+128 c^2 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) (-14 A c e-b B e+16 B c d)\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{35 c^{3/2} e^5 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (3 c e x (-14 A c e-b B e+16 B c d)+7 A c e (8 c d-7 b e)-B \left (b^2 e^2-60 b c d e+64 c^2 d^2\right )\right )}{35 c e^4}+\frac {2 \left (b x+c x^2\right )^{3/2} (-7 A e+8 B d+B e x)}{7 e^2 \sqrt {d+e x}} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 826
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {6 \int \frac {\left (\frac {1}{2} b (8 B d-7 A e)+\frac {1}{2} (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 e^2} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} b d \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )\right )+\frac {1}{4} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-2 (16 B c d-b B e-14 A c e) \left (4 c^2 d^2-\frac {3}{2} b c d e-b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{35 c e^4} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{35 c e^5}-\frac {\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{35 c e^5} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{35 c e^5 \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{35 c e^5 \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {\left (\left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\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}{35 c e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\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}{35 c e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (7 A c e (8 c d-7 b e)-B \left (64 c^2 d^2-60 b c d e+b^2 e^2\right )+3 c e (16 B c d-b B e-14 A c e) x\right ) \sqrt {b x+c x^2}}{35 c e^4}+\frac {2 (8 B d-7 A e+B e x) \left (b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \left (5 b c e (8 B d-7 A e) (2 c d-b e)-(16 B c d-b B e-14 A c e) \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\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 )}{35 c^{3/2} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (56 A c e (2 c d-b e)-B \left (128 c^2 d^2-72 b c d e-b^2 e^2\right )\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 )}{35 c^{3/2} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 24.52 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (x (b+c x))^{3/2} \left (b e x (b+c x) \left (35 c d (B d-A e) (c d-b e)+\left (7 A c e (-3 c d+2 b e)+B \left (29 c^2 d^2-25 b c d e+b^2 e^2\right )\right ) (d+e x)+c e (-13 B c d+8 b B e+7 A c e) x (d+e x)+5 B c^2 e^2 x^2 (d+e x)\right )+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (7 A c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B \left (128 c^3 d^3-136 b c^2 d^2 e+11 b^2 c d e^2+2 b^3 e^3\right )\right ) (b+c x) (d+e x)+i b e \left (7 A c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )-B \left (128 c^3 d^3-136 b c^2 d^2 e+11 b^2 c d e^2+2 b^3 e^3\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (7 A c e (8 c d-b e)+2 B \left (-32 c^2 d^2+6 b c d e+b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{35 b c e^5 x^2 (b+c x)^2 \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1151\) vs. \(2(395)=790\).
Time = 0.89 (sec) , antiderivative size = 1152, normalized size of antiderivative = 2.57
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1152\) |
default | \(\text {Expression too large to display}\) | \(1610\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.86 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (128 \, B c^{4} d^{5} - 8 \, {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{4} e + {\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{3} e^{2} + 2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d^{2} e^{3} + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} d e^{4} + {\left (128 \, B c^{4} d^{4} e - 8 \, {\left (25 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3} e^{2} + {\left (55 \, B b^{2} c^{2} + 168 \, A b c^{3}\right )} d^{2} e^{3} + 2 \, {\left (5 \, B b^{3} c - 21 \, A b^{2} c^{2}\right )} d e^{4} + {\left (2 \, B b^{4} - 7 \, A b^{3} c\right )} e^{5}\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 (128 \, B c^{4} d^{4} e - 8 \, {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{3} e^{2} + {\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d^{2} e^{3} + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} d e^{4} + {\left (128 \, B c^{4} d^{3} e^{2} - 8 \, {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2} e^{3} + {\left (11 \, B b^{2} c^{2} + 112 \, A b c^{3}\right )} d e^{4} + {\left (2 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} e^{5}\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 (5 \, B c^{4} e^{5} x^{3} + 64 \, B c^{4} d^{3} e^{2} - 4 \, {\left (15 \, B b c^{3} + 14 \, A c^{4}\right )} d^{2} e^{3} + {\left (B b^{2} c^{2} + 49 \, A b c^{3}\right )} d e^{4} - {\left (8 \, B c^{4} d e^{4} - {\left (8 \, B b c^{3} + 7 \, A c^{4}\right )} e^{5}\right )} x^{2} + {\left (16 \, B c^{4} d^{2} e^{3} - {\left (17 \, B b c^{3} + 14 \, A c^{4}\right )} d e^{4} + {\left (B b^{2} c^{2} + 14 \, A b c^{3}\right )} e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{105 \, {\left (c^{3} e^{7} x + c^{3} d e^{6}\right )}} \]
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\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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