Integrand size = 28, antiderivative size = 494 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {2 \sqrt {-b} \sqrt {c} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^3 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d (8 c d-9 b e)+A e (2 c d-b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d e^3 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.42 (sec) , antiderivative size = 494, 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 = {824, 848, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^2 e^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^2}+\frac {2 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (A e (2 c d-b e)+B d (8 c d-9 b e)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d e^3 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (2 A e \left (b^2 e^2-b c d e+c^2 d^2\right )+B d \left (3 b^2 e^2-13 b c d e+8 c^2 d^2\right )\right )}{15 d^2 e^2 \sqrt {d+e x} (c d-b e)^2}-\frac {2 \sqrt {b x+c x^2} (e x (B d (7 c d-6 b e)-A e (2 c d-b e))+d (A e (c d-2 b e)+B d (4 c d-3 b e)))}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 824
Rule 848
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {2 \int \frac {\frac {1}{2} b \left (2 b e \left (\frac {3 B d}{2}+A e\right )-c d (4 B d+A e)\right )-\frac {1}{2} c (2 c d (4 B d+A e)-b e (9 B d+A e)) x}{(d+e x)^{3/2} \sqrt {b x+c x^2}} \, dx}{15 d e^2 (c d-b e)} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {4 \int \frac {-\frac {1}{4} b c d \left (A e (c d+b e)+B \left (4 c d^2-6 b d e\right )\right )-\frac {1}{4} c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d^2 e^2 (c d-b e)^2} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {(c (B d (8 c d-9 b e)+A e (2 c d-b e))) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 d e^3 (c d-b e)}-\frac {\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 d^2 e^3 (c d-b e)^2} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}+\frac {\left (c (B d (8 c d-9 b e)+A e (2 c d-b e)) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{15 d e^3 (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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}{15 d^2 e^3 (c d-b e)^2 \sqrt {b x+c x^2}} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {\left (c \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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}{15 d^2 e^3 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c (B d (8 c d-9 b e)+A e (2 c d-b e)) \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}{15 d e^3 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^2 e^2 (c d-b e)^2 \sqrt {d+e x}}-\frac {2 (d (B d (4 c d-3 b e)+A e (c d-2 b e))+e (B d (7 c d-6 b e)-A e (2 c d-b e)) x) \sqrt {b x+c x^2}}{15 d e^2 (c d-b e) (d+e x)^{5/2}}-\frac {2 \sqrt {-b} \sqrt {c} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 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 )}{15 d^2 e^3 (c d-b e)^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {c} (B d (8 c d-9 b e)+A e (2 c d-b e)) \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 )}{15 d e^3 (c d-b e) \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 19.60 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (b e x (b+c x) \left (3 d^2 (B d-A e) (c d-b e)^2-d (c d-b e) (B d (7 c d-6 b e)+A e (-2 c d+b e)) (d+e x)+\left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) (d+e x)^2\right )-\sqrt {\frac {b}{c}} c (d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) (b+c x) (d+e x)+i b e \left (2 A e \left (c^2 d^2-b c d e+b^2 e^2\right )+B d \left (8 c^2 d^2-13 b c d e+3 b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) (B d (4 c d-3 b e)+A e (c d-2 b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 b d^2 e^3 (c d-b e)^2 \sqrt {x (b+c x)} (d+e x)^{5/2}} \]
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Time = 2.26 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.69
method | result | size |
elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{5} \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (A b \,e^{2}-2 A c d e -6 B b d e +7 B c \,d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e^{4} d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 \left (c e \,x^{2}+b e x \right ) \left (2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}\right )}{15 d^{2} \left (b e -c d \right )^{2} e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {B c}{e^{3}}+\frac {c \left (A b \,e^{2}-2 A c d e -6 B b d e +7 B c \,d^{2}\right )}{15 e^{3} d \left (b e -c d \right )}+\frac {2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}}{15 e^{3} \left (b e -c d \right ) d^{2}}-\frac {b \left (2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}\right )}{15 e^{2} d^{2} \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 (2 A \,b^{2} e^{3}-2 A b c d \,e^{2}+2 A \,c^{2} d^{2} e +3 B \,b^{2} d \,e^{2}-13 B b c \,d^{2} e +8 B \,c^{2} d^{3}\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 )}{15 e^{2} d^{2} \left (b e -c d \right )^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(835\) |
default | \(\text {Expression too large to display}\) | \(3831\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 1269, normalized size of antiderivative = 2.57 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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