Integrand size = 28, antiderivative size = 543 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.51 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {836, 857, 729, 113, 111, 118, 117} \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (b^2 e (9 B d-A e)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}-\frac {2 \sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (3 B d-2 A e)-b c d (5 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-2 A e)+b^2 c d e (4 A e+13 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (c d-b e)^2} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 836
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A 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 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )-\frac {3}{2} c e (b B d-2 A c d+A 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} (A b (c d-b e)+c (2 A c d-b (B d+A 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 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\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 A c^2 d^2+b^2 e (6 B d+A e)-b c d (4 B d+11 A e)\right )-\frac {1}{4} c e \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\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} (A b (c d-b e)+c (2 A c d-b (B d+A 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 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 A e)\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 (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\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} (A b (c d-b e)+c (2 A c d-b (B d+A 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 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 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 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 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^2 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A 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 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 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^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 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 d (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} (A b (c d-b e)+c (2 A c d-b (B d+A 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 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 A c^3 d^3-b^3 e^2 (3 B d-2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 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^2 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 A c^2 d^2+b^2 e (9 B d-A e)-8 b c d (B d+2 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} 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 = 24.73 (sec) , antiderivative size = 514, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b (d+e x) \left (b c^2 (b B-A c) d^2 (c d-b e) x^2+c^2 d^2 \left (-8 A c^2 d-7 b^2 B e+5 b c (B d+2 A e)\right ) x^2 (b+c x)+A b d (c d-b e)^2 (b+c x)^2+(c d-b e)^2 (3 b B d-8 A c d-2 A b e) x (b+c x)^2\right )+\sqrt {\frac {b}{c}} c x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^3 d^3+b^3 e^2 (-3 B d+2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^3 d^3+b^3 e^2 (-3 B d+2 A e)-8 b c^2 d^2 (B d+3 A e)+b^2 c d e (13 B d+4 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-2 A e)-b c d (4 B d+5 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^2 (c d-b e)^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]
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Time = 1.23 (sec) , antiderivative size = 752, normalized size of antiderivative = 1.38
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} d \,x^{2}}+\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (2 A b e +8 A c d -3 B b 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 \left (A c -B b \right ) \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 (x +\frac {b}{c}\right )^{2}}-\frac {2 \left (c e \,x^{2}+c d x \right ) c \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right )^{2} \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} d}-\frac {c e \left (A c -B b \right )}{3 b^{3} \left (b e -c d \right )}+\frac {c \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 \left (b e -c d \right ) b^{4}}+\frac {c^{2} d \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right )^{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 {c e \left (2 A b e +8 A c d -3 B b d \right )}{3 d^{2} b^{4}}+\frac {e \,c^{2} \left (10 A b c e -8 A \,c^{2} d -7 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e -c d \right )^{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}}\) | \(752\) |
default | \(\text {Expression too large to display}\) | \(3140\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 1324, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
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\[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{5/2}\,\sqrt {d+e\,x}} \,d x \]
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