Integrand size = 30, antiderivative size = 400 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {e+f x^2}}+\frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}-\frac {\sqrt {e} \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c d^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Time = 0.27 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {542, 545, 429, 506, 422} \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {e^{3/2} \sqrt {c+d x^2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 c d^2 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right ) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^3 \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {x \sqrt {c+d x^2} \left (10 a d f (2 d e-c f)+b \left (8 c^2 f^2-13 c d e f+3 d^2 e^2\right )\right )}{15 d^3 \sqrt {e+f x^2}}+\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-4 b c f+3 b d e)}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d} \]
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Rule 422
Rule 429
Rule 506
Rule 542
Rule 545
Rubi steps \begin{align*} \text {integral}& = \frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}+\frac {\int \frac {\sqrt {e+f x^2} \left (-((b c-5 a d) e)+(3 b d e-4 b c f+5 a d f) x^2\right )}{\sqrt {c+d x^2}} \, dx}{5 d} \\ & = \frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}+\frac {\int \frac {-e (2 b c (3 d e-2 c f)-5 a d (3 d e-c f))+\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d^2} \\ & = \frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}-\frac {(e (2 b c (3 d e-2 c f)-5 a d (3 d e-c f))) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d^2}+\frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{15 d^2} \\ & = \frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {e+f x^2}}+\frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}+\frac {e^{3/2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c d^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (e \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{15 d^3} \\ & = \frac {\left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^3 \sqrt {e+f x^2}}+\frac {(3 b d e-4 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d^2}+\frac {b x \sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}}{5 d}-\frac {\sqrt {e} \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^3 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {e^{3/2} \left (5 a d (3 d e-c f)-b \left (6 c d e-4 c^2 f\right )\right ) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 c d^2 \sqrt {f} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.10 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (4 b c f-5 a d f-3 b d \left (2 e+f x^2\right )\right )-i e \left (10 a d f (2 d e-c f)+b \left (3 d^2 e^2-13 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i e (-d e+c f) (-3 b d e+4 b c f-5 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{15 c^2 \left (\frac {d}{c}\right )^{5/2} f \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
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Time = 6.17 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.12
method | result | size |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b f \,x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5 d}+\frac {\left (a \,f^{2}+2 b f e -\frac {b f \left (4 c f +4 d e \right )}{5 d}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (e^{2} a -\frac {\left (a \,f^{2}+2 b f e -\frac {b f \left (4 c f +4 d e \right )}{5 d}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (2 a e f +e^{2} b -\frac {3 b f c e}{5 d}-\frac {\left (a \,f^{2}+2 b f e -\frac {b f \left (4 c f +4 d e \right )}{5 d}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(448\) |
risch | \(\frac {x \left (3 b d f \,x^{2}+5 a d f -4 b c f +6 b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 d^{2}}-\frac {\left (-\frac {15 a \,d^{2} e^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {4 b \,c^{2} e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {6 b c d \,e^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {5 a c d e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (10 a c d \,f^{2}-20 a \,d^{2} e f -8 b \,c^{2} f^{2}+13 b c d e f -3 b \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 d^{2} \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(628\) |
default | \(\frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, \left (3 \sqrt {-\frac {d}{c}}\, b \,d^{2} f^{3} x^{7}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} f^{3} x^{5}-\sqrt {-\frac {d}{c}}\, b c d \,f^{3} x^{5}+9 \sqrt {-\frac {d}{c}}\, b \,d^{2} e \,f^{2} x^{5}+5 \sqrt {-\frac {d}{c}}\, a c d \,f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} e \,f^{2} x^{3}-4 \sqrt {-\frac {d}{c}}\, b \,c^{2} f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, b c d e \,f^{2} x^{3}+6 \sqrt {-\frac {d}{c}}\, b \,d^{2} e^{2} f \,x^{3}+5 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e \,f^{2}-5 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2} f -4 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e \,f^{2}+7 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2} f -3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,d^{2} e^{3}-10 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e \,f^{2}+20 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2} f +8 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e \,f^{2}-13 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2} f +3 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,d^{2} e^{3}+5 \sqrt {-\frac {d}{c}}\, a c d e \,f^{2} x -4 \sqrt {-\frac {d}{c}}\, b \,c^{2} e \,f^{2} x +6 \sqrt {-\frac {d}{c}}\, b c d \,e^{2} f x \right )}{15 d^{2} f \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) \sqrt {-\frac {d}{c}}}\) | \(870\) |
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Time = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (3 \, b d^{2} e^{3} - {\left (13 \, b c d - 20 \, a d^{2}\right )} e^{2} f + 2 \, {\left (4 \, b c^{2} - 5 \, a c d\right )} e f^{2}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (3 \, b d^{2} e^{3} - {\left (13 \, b c d - 20 \, a d^{2}\right )} e^{2} f + {\left (8 \, b c^{2} - 2 \, {\left (5 \, a + 3 \, b\right )} c d + 15 \, a d^{2}\right )} e f^{2} + {\left (4 \, b c^{2} - 5 \, a c d\right )} f^{3}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (3 \, b d^{2} f^{3} x^{4} + 3 \, b d^{2} e^{2} f - {\left (13 \, b c d - 20 \, a d^{2}\right )} e f^{2} + 2 \, {\left (4 \, b c^{2} - 5 \, a c d\right )} f^{3} + {\left (6 \, b d^{2} e f^{2} - {\left (4 \, b c d - 5 \, a d^{2}\right )} f^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{15 \, d^{3} f^{2} x} \]
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\[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}}{\sqrt {d x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \]
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