Integrand size = 24, antiderivative size = 509 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {8 e (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
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Time = 0.39 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {756, 846, 857, 732, 435, 430} \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {8 e \sqrt {d+e x} \sqrt {a+b x+c x^2} (2 c d-b e)}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c} \]
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Rule 430
Rule 435
Rule 732
Rule 756
Rule 846
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {2 \int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (5 c d^2-e (b d+3 a e)\right )+2 e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{5 c} \\ & = \frac {8 e (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {4 \int \frac {\frac {1}{4} \left (15 c^2 d^3+4 b e^2 (b d+a e)-c d e (11 b d+17 a e)\right )+\frac {1}{4} e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2} \\ & = \frac {8 e (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}-\frac {\left (4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2}+\frac {\left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c^2} \\ & = \frac {8 e (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 c^3 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = \frac {8 e (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{15 c^2}+\frac {2 e (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 28.47 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.44 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {d+e x} \left (c e^2 (11 c d-4 b e+3 c e x) (a+x (b+c x))+(d+e x) \left (\frac {e^2 \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) (a+x (b+c x))}{(d+e x)^2}-\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (-30 c^3 d^3+8 b^2 e^2 \left (b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )-c^2 d \left (-45 b d e-34 a e^2+23 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c e \left (-31 b^2 d e-17 a b e^2+23 b d \sqrt {\left (b^2-4 a c\right ) e^2}+9 a e \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )\right )}{15 c^3 e \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(966\) vs. \(2(445)=890\).
Time = 2.28 (sec) , antiderivative size = 967, normalized size of antiderivative = 1.90
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{2} x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d}}{5 c}+\frac {2 \left (3 d \,e^{2}-\frac {2 e^{2} \left (2 b e +2 c d \right )}{5 c}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d}}{3 c e}+\frac {2 \left (d^{3}-\frac {2 e^{2} a d}{5 c}-\frac {2 \left (3 d \,e^{2}-\frac {2 e^{2} \left (2 b e +2 c d \right )}{5 c}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d}}+\frac {2 \left (3 d^{2} e -\frac {2 e^{2} \left (\frac {3 a e}{2}+\frac {3 b d}{2}\right )}{5 c}-\frac {2 \left (3 d \,e^{2}-\frac {2 e^{2} \left (2 b e +2 c d \right )}{5 c}\right ) \left (b e +c d \right )}{3 c e}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +x d b +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\) | \(967\) |
risch | \(\text {Expression too large to display}\) | \(1983\) |
default | \(\text {Expression too large to display}\) | \(4786\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \, {\left ({\left (22 \, c^{3} d^{3} - 33 \, b c^{2} d^{2} e + 3 \, {\left (9 \, b^{2} c - 14 \, a c^{2}\right )} d e^{2} - {\left (8 \, b^{3} - 21 \, a b c\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 3 \, {\left (23 \, c^{3} d^{2} e - 23 \, b c^{2} d e^{2} + {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (3 \, c^{3} e^{3} x + 11 \, c^{3} d e^{2} - 4 \, b c^{2} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{45 \, c^{4} e} \]
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\[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \]
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\[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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\[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{5/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
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