Integrand size = 23, antiderivative size = 379 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {16 \sqrt {-b} (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.34 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {756, 846, 857, 729, 113, 111, 118, 117} \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {16 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{105 c^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac {12 e \sqrt {b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{5/2}}{7 c} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 756
Rule 846
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d (7 c d-b e)+3 e (2 c d-b e) x\right )}{\sqrt {b x+c x^2}} \, dx}{7 c} \\ & = \frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} d \left (35 c^2 d^2-17 b c d e+6 b^2 e^2\right )+\frac {1}{4} e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{35 c^2} \\ & = \frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {8 \int \frac {\frac {1}{8} d (7 c d-3 b e) \left (15 c^2 d^2-11 b c d e+8 b^2 e^2\right )+e (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^3} \\ & = \frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{105 c^3}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{105 c^3} \\ & = \frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{105 c^3 \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{105 c^3 \sqrt {b x+c x^2}} \\ & = \frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\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}{105 c^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\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}{105 c^3 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {b x+c x^2}}{7 c}+\frac {16 \sqrt {-b} (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\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 )}{105 c^{7/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\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 )}{105 c^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 20.22 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x} \left (\frac {8 \left (22 c^3 d^3-33 b c^2 d^2 e+23 b^2 c d e^2-6 b^3 e^3\right ) (b+c x) (d+e x)}{c \sqrt {x}}+e \sqrt {x} (b+c x) (d+e x) \left (24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+8 i \sqrt {\frac {b}{c}} e \left (22 c^3 d^3-33 b c^2 d^2 e+23 b^2 c d e^2-6 b^3 e^3\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\frac {i \sqrt {\frac {b}{c}} \left (105 c^4 d^4-298 b c^3 d^3 e+353 b^2 c^2 d^2 e^2-208 b^3 c d e^3+48 b^4 e^4\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b}\right )}{105 c^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(696\) vs. \(2(319)=638\).
Time = 2.05 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.84
method | result | size |
elliptic | \(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{7 c}+\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 c e}+\frac {2 \left (6 d^{2} e^{2}-\frac {5 e^{3} b d}{7 c}-\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (d^{4}-\frac {\left (6 d^{2} e^{2}-\frac {5 e^{3} b d}{7 c}-\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) b d}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \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 (\frac {b}{c}+x \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 (4 d^{3} e -\frac {3 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) b d}{5 c e}-\frac {2 \left (6 d^{2} e^{2}-\frac {5 e^{3} b d}{7 c}-\frac {2 \left (4 d \,e^{3}-\frac {2 e^{3} \left (3 b e +3 c d \right )}{7 c}\right ) \left (2 b e +2 c d \right )}{5 c e}\right ) \left (b e +c d \right )}{3 c e}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \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 (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \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}}\) | \(697\) |
default | \(\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (15 c^{5} e^{4} x^{5}+24 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} c d \,e^{3}-95 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c^{2} d^{2} e^{2}+142 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{3} d^{3} e -71 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{4} d^{4}+48 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{5} e^{4}-232 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{4} c d \,e^{3}+448 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} c^{2} d^{2} e^{2}-440 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c^{3} d^{3} e +176 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{4} d^{4}-3 b \,c^{4} e^{4} x^{4}+81 c^{5} d \,e^{3} x^{4}+6 b^{2} c^{3} e^{4} x^{3}-26 b \,c^{4} d \,e^{3} x^{3}+188 c^{5} d^{2} e^{2} x^{3}+24 b^{3} c^{2} e^{4} x^{2}-83 b^{2} c^{3} d \,e^{3} x^{2}+99 b \,c^{4} d^{2} e^{2} x^{2}+122 c^{5} d^{3} e \,x^{2}+24 b^{3} c^{2} d \,e^{3} x -89 b^{2} c^{3} d^{2} e^{2} x +122 b \,c^{4} d^{3} e x \right )}{105 c^{5} x \left (c e \,x^{2}+b e x +c d x +b d \right )}\) | \(918\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (139 \, c^{4} d^{4} - 278 \, b c^{3} d^{3} e + 347 \, b^{2} c^{2} d^{2} e^{2} - 208 \, b^{3} c d e^{3} + 48 \, b^{4} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 24 \, {\left (22 \, c^{4} d^{3} e - 33 \, b c^{3} d^{2} e^{2} + 23 \, b^{2} c^{2} d e^{3} - 6 \, b^{3} c e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (15 \, c^{4} e^{4} x^{2} + 122 \, c^{4} d^{2} e^{2} - 89 \, b c^{3} d e^{3} + 24 \, b^{2} c^{2} e^{4} + 6 \, {\left (11 \, c^{4} d e^{3} - 3 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{315 \, c^{5} e} \]
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\[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]
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\[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]
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\[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{7/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {c\,x^2+b\,x}} \,d x \]
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