Integrand size = 27, antiderivative size = 550 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=-\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )} \]
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Time = 0.43 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1587, 748, 846, 857, 732, 435, 430} \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (a x^2+b x+c\right )}-\frac {2 \sqrt {2} x \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \left (a^2 d^2-a e (b d+3 c e)+b^2 e^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \left (a x^2+b x+c\right ) \sqrt {\frac {a (d+e x)}{2 a d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 x (d+e x)^{3/2} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}}}{5 e}-\frac {2 x \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} (2 a d-b e)}{15 a e} \]
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Rule 430
Rule 435
Rule 732
Rule 748
Rule 846
Rule 857
Rule 1587
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \sqrt {d+e x} \sqrt {c+b x+a x^2} \, dx}{\sqrt {c+b x+a x^2}} \\ & = \frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {\left (\sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\sqrt {d+e x} (b d-2 c e+(2 a d-b e) x)}{\sqrt {c+b x+a x^2}} \, dx}{5 e \sqrt {c+b x+a x^2}} \\ & = -\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {\left (2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\frac {1}{2} (a d (b d-8 c e)+b e (b d+c e))+\left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) x}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{15 a e \sqrt {c+b x+a x^2}} \\ & = -\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}+\frac {\left ((2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c+b x+a x^2}} \, dx}{15 a e^2 \sqrt {c+b x+a x^2}}-\frac {\left (2 \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c+b x+a x^2}} \, dx}{15 a e^2 \sqrt {c+b x+a x^2}} \\ & = -\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a 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 a 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 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \left (c+b x+a x^2\right )}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {a \left (c+b x+a 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 a d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )} \\ & = -\frac {2 (2 a d-b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{15 a e}+\frac {2 \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{5 e}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 a d-b e) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 a^2 e^2 \sqrt {d+e x} \left (c+b x+a x^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 32.19 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.91 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\frac {1}{15} x \sqrt {d+e x} \sqrt {a+\frac {c+b x}{x^2}} \left (\frac {2 b}{a}+\frac {2 d}{e}+6 x-\frac {(d+e x) \left (\frac {4 e^2 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) (c+x (b+a x))}{(d+e x)^2}-\frac {i \sqrt {2} \left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (a^2 d^2+b^2 e^2-a e (b d+3 c e)\right ) \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}+\frac {i \sqrt {2} \left (b^2 e^2 \left (-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+a^2 d \left (-8 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a e \left (2 b^2 d e+4 b c e^2-b d \sqrt {\left (b^2-4 a c\right ) e^2}-3 c e \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right ) \sqrt {\frac {-2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 c e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 a d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a d^2-b d e+c e^2}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}\right )}{a^2 e^3 \sqrt {\frac {a d^2+e (-b d+c e)}{-2 a d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (c+x (b+a x))}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1710\) vs. \(2(486)=972\).
Time = 1.67 (sec) , antiderivative size = 1711, normalized size of antiderivative = 3.11
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1711\) |
default | \(\text {Expression too large to display}\) | \(4361\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.89 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\frac {2 \, {\left ({\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} \sqrt {a e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right ) + 6 \, {\left (a^{3} d^{2} e - a^{2} b d e^{2} + {\left (a b^{2} - 3 \, a^{2} c\right )} e^{3}\right )} \sqrt {a e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} d^{2} - a b d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, a^{2} e^{2}}, -\frac {4 \, {\left (2 \, a^{3} d^{3} - 3 \, a^{2} b d^{2} e - 3 \, {\left (a b^{2} - 6 \, a^{2} c\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, a^{3} e^{3}}, \frac {3 \, a e x + a d + b e}{3 \, a e}\right )\right ) + 3 \, {\left (3 \, a^{3} e^{3} x^{2} + {\left (a^{3} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right )}}{45 \, a^{3} e^{3}} \]
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\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int x \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \]
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\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x \,d x } \]
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\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x \,d x } \]
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Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \, dx=\int x\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \]
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