Integrand size = 23, antiderivative size = 392 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+15 b^2 e^2\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
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Time = 0.30 (sec) , antiderivative size = 392, 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 = {746, 826, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) \left (15 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 \sqrt {c} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {4 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (23 b^2 e^2-128 b c d e+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b x+c x^2} \left (15 b^2 e^2+16 c e x (2 c d-b e)-112 b c d e+128 c^2 d^2\right )}{15 e^5 \sqrt {d+e x}}+\frac {2 \left (b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}} \]
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Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 746
Rule 826
Rule 857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {\int \frac {(b+2 c x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{e} \\ & = \frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {2 \int \frac {\left (\frac {1}{2} b (16 c d-5 b e)+8 c (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^3} \\ & = -\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {4 \int \frac {\frac {1}{4} b \left (128 c^2 d^2-112 b c d e+15 b^2 e^2\right )+\frac {1}{2} c \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 e^5} \\ & = -\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+15 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 e^6}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 e^6} \\ & = -\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+15 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}{15 e^6 \sqrt {b x+c x^2}}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+23 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}{15 e^6 \sqrt {b x+c x^2}} \\ & = -\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {\left (2 c \left (128 c^2 d^2-128 b c d e+23 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}{15 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left ((2 c d-b e) \left (128 c^2 d^2-128 b c d e+15 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}{15 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 \left (128 c^2 d^2-112 b c d e+15 b^2 e^2+16 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (b x+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (b x+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {4 \sqrt {-b} \sqrt {c} \left (128 c^2 d^2-128 b c d e+23 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 )}{15 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^2 d^2-128 b c d e+15 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 )}{15 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 23.68 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.02 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {2 (x (b+c x))^{5/2} \left (\frac {2 \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) (b+c x) (d+e x)}{\sqrt {x}}-\frac {e \sqrt {x} (b+c x) \left (b^2 e^2 \left (15 d^2+35 d e x+23 e^2 x^2\right )-b c e \left (112 d^3+256 d^2 e x+161 d e^2 x^2+11 e^3 x^3\right )+c^2 \left (128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right )\right )}{(d+e x)^2}+2 i \sqrt {\frac {b}{c}} c e \left (128 c^2 d^2-128 b c d e+23 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i \sqrt {\frac {b}{c}} c e \left (128 c^2 d^2-144 b c d e+31 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{15 e^6 x^{5/2} (b+c x)^3 \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(911\) vs. \(2(332)=664\).
Time = 2.03 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.33
method | result | size |
elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (-\frac {2 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {22 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e^{7} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+b e x \right ) \left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right )}{15 e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 c^{2} x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{4}}+\frac {2 \left (\frac {3 c^{2} \left (b e -c d \right )}{e^{4}}-\frac {2 c^{2} \left (2 b e +2 c d \right )}{5 e^{4}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (\frac {b^{3} e^{3}-9 b^{2} d \,e^{2} c +18 b \,c^{2} d^{2} e -10 c^{3} d^{3}}{e^{6}}+\frac {11 c d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{15 e^{6}}-\frac {\left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) \left (b e -c d \right )}{15 e^{6}}+\frac {b \left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right )}{15 e^{5}}-\frac {\left (\frac {3 c^{2} \left (b e -c d \right )}{e^{4}}-\frac {2 c^{2} \left (2 b e +2 c d \right )}{5 e^{4}}\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 (\frac {3 c \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{e^{5}}+\frac {\left (23 b^{2} e^{2}-128 b c d e +128 c^{2} d^{2}\right ) c}{15 e^{5}}-\frac {3 c^{2} b d}{5 e^{4}}-\frac {2 \left (\frac {3 c^{2} \left (b e -c d \right )}{e^{4}}-\frac {2 c^{2} \left (2 b e +2 c d \right )}{5 e^{4}}\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 {e x +d}\, x \left (c x +b \right )}\) | \(912\) |
default | \(\text {Expression too large to display}\) | \(2170\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.36 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.07 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left ({\left (256 \, c^{3} d^{6} - 384 \, b c^{2} d^{5} e + 126 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + {\left (256 \, c^{3} d^{3} e^{3} - 384 \, b c^{2} d^{2} e^{4} + 126 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 3 \, {\left (256 \, c^{3} d^{4} e^{2} - 384 \, b c^{2} d^{3} e^{3} + 126 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (256 \, c^{3} d^{5} e - 384 \, b c^{2} d^{4} e^{2} + 126 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\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 ) + 6 \, {\left (128 \, c^{3} d^{5} e - 128 \, b c^{2} d^{4} e^{2} + 23 \, b^{2} c d^{3} e^{3} + {\left (128 \, c^{3} d^{2} e^{4} - 128 \, b c^{2} d e^{5} + 23 \, b^{2} c e^{6}\right )} x^{3} + 3 \, {\left (128 \, c^{3} d^{3} e^{3} - 128 \, b c^{2} d^{2} e^{4} + 23 \, b^{2} c d e^{5}\right )} x^{2} + 3 \, {\left (128 \, c^{3} d^{4} e^{2} - 128 \, b c^{2} d^{3} e^{3} + 23 \, b^{2} c d^{2} e^{4}\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 (3 \, c^{3} e^{6} x^{4} - 128 \, c^{3} d^{4} e^{2} + 112 \, b c^{2} d^{3} e^{3} - 15 \, b^{2} c d^{2} e^{4} - {\left (10 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{3} - {\left (176 \, c^{3} d^{2} e^{4} - 161 \, b c^{2} d e^{5} + 23 \, b^{2} c e^{6}\right )} x^{2} - {\left (288 \, c^{3} d^{3} e^{3} - 256 \, b c^{2} d^{2} e^{4} + 35 \, b^{2} c d e^{5}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, {\left (c e^{10} x^{3} + 3 \, c d e^{9} x^{2} + 3 \, c d^{2} e^{8} x + c d^{3} e^{7}\right )}} \]
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\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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