Integrand size = 23, antiderivative size = 537 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
[Out]
Time = 0.40 (sec) , antiderivative size = 537, 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 = {748, 828, 857, 729, 113, 111, 118, 117} \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {4 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (2 b^4 e^4+5 b^3 c d e^3+123 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \left (8 b^4 e^4+29 b^3 c d e^3+99 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {10 \left (b x+c x^2\right )^{3/2} \sqrt {d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} \left (-4 b^4 e^4-7 b^3 c d e^3-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}+\frac {2 \left (b x+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e} \]
[In]
[Out]
Rule 111
Rule 113
Rule 117
Rule 118
Rule 729
Rule 748
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {5 \int \frac {(b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{11 e} \\ & = \frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac {10 \int \frac {\left (-\frac {1}{2} b d \left (16 c^2 d^2-23 b c d e+3 b^2 e^2\right )-2 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx}{231 c e^3} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {4 \int \frac {\frac {1}{4} b d \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4\right )+\frac {1}{4} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{693 c^2 e^5} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac {\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{693 c^2 e^6}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{693 c^2 e^6} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac {\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{693 c^2 e^6 \sqrt {b x+c x^2}}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{693 c^2 e^6 \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\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}{693 c^2 e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\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}{693 c^2 e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = \frac {2 \sqrt {d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt {b x+c x^2}}{693 c^2 e^5}+\frac {10 \sqrt {d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac {2 \sqrt {d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac {2 \sqrt {-b} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\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 )}{693 c^{5/2} e^6 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {4 \sqrt {-b} d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\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 )}{693 c^{5/2} e^6 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 22.34 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.04 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (-4 b^4 e^4+b^3 c e^3 (-7 d+3 e x)+b^2 c^2 e^2 \left (195 d^2-139 d e x+113 e^2 x^2\right )+b c^3 e \left (-304 d^3+224 d^2 e x-185 d e^2 x^2+161 e^3 x^3\right )+c^4 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (-256 c^5 d^5+640 b c^4 d^4 e-454 b^2 c^3 d^3 e^2+41 b^3 c^2 d^2 e^3+13 b^4 c d e^4+8 b^5 e^5\right ) (b+c x) (d+e x)-i b e \left (256 c^5 d^5-640 b c^4 d^4 e+454 b^2 c^3 d^3 e^2-41 b^3 c^2 d^2 e^3-13 b^4 c d e^4-8 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e \left (128 c^5 d^5-336 b c^4 d^4 e+259 b^2 c^3 d^3 e^2-34 b^3 c^2 d^2 e^3-9 b^4 c d e^4-8 b^5 e^5\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{693 b c^2 e^6 x^3 (b+c x)^3 \sqrt {d+e x}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1440\) vs. \(2(477)=954\).
Time = 1.95 (sec) , antiderivative size = 1441, normalized size of antiderivative = 2.68
method | result | size |
default | \(\text {Expression too large to display}\) | \(1441\) |
elliptic | \(\text {Expression too large to display}\) | \(1442\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.27 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.19 \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left ({\left (256 \, c^{6} d^{6} - 768 \, b c^{5} d^{5} e + 726 \, b^{2} c^{4} d^{4} e^{2} - 172 \, b^{3} c^{3} d^{3} e^{3} - 33 \, b^{4} c^{2} d^{2} e^{4} - 9 \, b^{5} c d e^{5} - 8 \, b^{6} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (256 \, c^{6} d^{5} e - 640 \, b c^{5} d^{4} e^{2} + 454 \, b^{2} c^{4} d^{3} e^{3} - 41 \, b^{3} c^{3} d^{2} e^{4} - 13 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (63 \, c^{6} e^{6} x^{4} + 128 \, c^{6} d^{4} e^{2} - 304 \, b c^{5} d^{3} e^{3} + 195 \, b^{2} c^{4} d^{2} e^{4} - 7 \, b^{3} c^{3} d e^{5} - 4 \, b^{4} c^{2} e^{6} - 7 \, {\left (10 \, c^{6} d e^{5} - 23 \, b c^{5} e^{6}\right )} x^{3} + {\left (80 \, c^{6} d^{2} e^{4} - 185 \, b c^{5} d e^{5} + 113 \, b^{2} c^{4} e^{6}\right )} x^{2} - {\left (96 \, c^{6} d^{3} e^{3} - 224 \, b c^{5} d^{2} e^{4} + 139 \, b^{2} c^{4} d e^{5} - 3 \, b^{3} c^{3} e^{6}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{2079 \, c^{4} e^{7}} \]
[In]
[Out]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}{\sqrt {d + e x}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{\sqrt {e x + d}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}{\sqrt {e x + d}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}}{\sqrt {d+e\,x}} \,d x \]
[In]
[Out]