Integrand size = 28, antiderivative size = 244 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {2 \left (5 b^2 c^2-22 a b c d+77 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d^2 e}-\frac {2 b (5 b c-22 a d) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac {2 c^{3/4} \left (5 b^2 c^2-22 a b c d+77 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{9/4} \sqrt {e} \sqrt {c+d x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {475, 470, 285, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {2 c^{3/4} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{9/4} \sqrt {e} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \sqrt {c+d x^2} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right )}{231 d^2 e}-\frac {2 b \sqrt {e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3} \]
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Rule 226
Rule 285
Rule 335
Rule 470
Rule 475
Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac {2 \int \frac {\sqrt {c+d x^2} \left (\frac {11 a^2 d}{2}-\frac {1}{2} b (5 b c-22 a d) x^2\right )}{\sqrt {e x}} \, dx}{11 d} \\ & = -\frac {2 b (5 b c-22 a d) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}-\frac {1}{77} \left (-77 a^2-\frac {b c (5 b c-22 a d)}{d^2}\right ) \int \frac {\sqrt {c+d x^2}}{\sqrt {e x}} \, dx \\ & = \frac {2 \left (77 a^2+\frac {b c (5 b c-22 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 e}-\frac {2 b (5 b c-22 a d) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac {1}{231} \left (2 c \left (77 a^2+\frac {b c (5 b c-22 a d)}{d^2}\right )\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx \\ & = \frac {2 \left (77 a^2+\frac {b c (5 b c-22 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 e}-\frac {2 b (5 b c-22 a d) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac {\left (4 c \left (77 a^2+\frac {b c (5 b c-22 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 )}{231 e} \\ & = \frac {2 \left (77 a^2+\frac {b c (5 b c-22 a d)}{d^2}\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 e}-\frac {2 b (5 b c-22 a d) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{77 d^2 e}+\frac {2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3}+\frac {2 c^{3/4} \left (77 a^2+\frac {b c (5 b c-22 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 )}{231 \sqrt [4]{d} \sqrt {e} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {\sqrt {x} \left (\frac {2 \sqrt {x} \left (c+d x^2\right ) \left (77 a^2 d^2+22 a b d \left (2 c+3 d x^2\right )+b^2 \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )\right )}{d^2}+\frac {4 i c \left (5 b^2 c^2-22 a b c d+77 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^2}\right )}{231 \sqrt {e x} \sqrt {c+d x^2}} \]
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Time = 3.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {2 \left (21 b^{2} d^{2} x^{4}+66 x^{2} a b \,d^{2}+6 x^{2} b^{2} c d +77 a^{2} d^{2}+44 a b c d -10 b^{2} c^{2}\right ) x \sqrt {d \,x^{2}+c}}{231 d^{2} \sqrt {e x}}+\frac {2 c \left (77 a^{2} d^{2}-22 a b c d +5 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{3} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(236\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (\frac {2 b^{2} x^{4} \sqrt {d e \,x^{3}+c e x}}{11 e}+\frac {2 \left (2 a b d +\frac {2}{11} b^{2} c \right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (a^{2} d +2 a b c -\frac {5 c \left (2 a b d +\frac {2}{11} b^{2} c \right )}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (a^{2} c -\frac {c \left (a^{2} d +2 a b c -\frac {5 c \left (2 a b d +\frac {2}{11} b^{2} c \right )}{7 d}\right )}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(289\) |
default | \(\frac {\frac {2 b^{2} d^{4} x^{7}}{11}+\frac {2 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}}{3}-\frac {4 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d}{21}+\frac {10 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}}{231}+\frac {4 a b \,d^{4} x^{5}}{7}+\frac {18 b^{2} c \,d^{3} x^{5}}{77}+\frac {2 a^{2} d^{4} x^{3}}{3}+\frac {20 x^{3} d^{3} b a c}{21}-\frac {8 b^{2} c^{2} d^{2} x^{3}}{231}+\frac {2 a^{2} c \,d^{3} x}{3}+\frac {8 a b \,c^{2} d^{2} x}{21}-\frac {20 b^{2} d x \,c^{3}}{231}}{\sqrt {d \,x^{2}+c}\, \sqrt {e x}\, d^{3}}\) | \(401\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {2 \, {\left (2 \, {\left (5 \, b^{2} c^{3} - 22 \, a b c^{2} d + 77 \, a^{2} c d^{2}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (21 \, b^{2} d^{3} x^{4} - 10 \, b^{2} c^{2} d + 44 \, a b c d^{2} + 77 \, a^{2} d^{3} + 6 \, {\left (b^{2} c d^{2} + 11 \, a b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{231 \, d^{3} e} \]
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Result contains complex when optimal does not.
Time = 3.77 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\frac {a^{2} \sqrt {c} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {a b \sqrt {c} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} \sqrt {c} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\sqrt {e x}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{\sqrt {e x}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{\sqrt {e x}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{\sqrt {e\,x}} \,d x \]
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