Integrand size = 28, antiderivative size = 193 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=-\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (5 b^2 c^2-14 a b c d+21 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {475, 470, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {\left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (21 a^2 d^2-14 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 \sqrt [4]{c} d^{9/4} \sqrt {e} \sqrt {c+d x^2}}-\frac {2 b \sqrt {e x} \sqrt {c+d x^2} (5 b c-14 a d)}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3} \]
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Rule 226
Rule 335
Rule 470
Rule 475
Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {2 \int \frac {\frac {7 a^2 d}{2}-\frac {1}{2} b (5 b c-14 a d) x^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{7 d} \\ & = -\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}-\frac {1}{21} \left (-21 a^2-\frac {b c (5 b c-14 a d)}{d^2}\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx \\ & = -\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (2 \left (21 a^2+\frac {b c (5 b c-14 a d)}{d^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 e} \\ & = -\frac {2 b (5 b c-14 a d) \sqrt {e x} \sqrt {c+d x^2}}{21 d^2 e}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d e^3}+\frac {\left (21 a^2+\frac {b c (5 b c-14 a d)}{d^2}\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{c} \sqrt [4]{d} \sqrt {e} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {2 x \left (-b \left (c+d x^2\right ) \left (5 b c-14 a d-3 b d x^2\right )+\frac {i \left (5 b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 d^2 \sqrt {e x} \sqrt {c+d x^2}} \]
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Time = 3.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {2 b \left (3 b d \,x^{2}+14 a d -5 b c \right ) x \sqrt {d \,x^{2}+c}}{21 d^{2} \sqrt {e x}}+\frac {\left (21 a^{2} d^{2}-14 a b c d +5 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{21 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(198\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {2 b^{2} x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (2 a b -\frac {5 b^{2} c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (a^{2}-\frac {c \left (2 a b -\frac {5 b^{2} c}{7 d}\right )}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(224\) |
default | \(\frac {21 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2}-14 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d +5 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}+6 b^{2} d^{3} x^{5}+28 a b \,d^{3} x^{3}-4 b^{2} c \,d^{2} x^{3}+28 x a b c \,d^{2}-10 x \,b^{2} c^{2} d}{21 \sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{3}}\) | \(350\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left ({\left (5 \, b^{2} c^{2} - 14 \, a b c d + 21 \, a^{2} d^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (3 \, b^{2} d^{2} x^{2} - 5 \, b^{2} c d + 14 \, a b d^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{21 \, d^{3} e} \]
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Result contains complex when optimal does not.
Time = 3.92 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\frac {a^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {a b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \sqrt {e} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \sqrt {e x}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{\sqrt {d x^{2} + c} \sqrt {e x}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {e x} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{\sqrt {e\,x}\,\sqrt {d\,x^2+c}} \,d x \]
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