Integrand size = 30, antiderivative size = 272 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {(b c-a d) x}{c (d e-c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {\sqrt {f} (2 b c e-a d e-a c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c \sqrt {e} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b d e+b c f-2 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {541, 539, 429, 422} \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {e} \sqrt {c+d x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {f} \sqrt {c+d x^2} (-a c f-a d e+2 b c e) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c \sqrt {e} \sqrt {e+f x^2} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {x (b c-a d)}{c \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)} \]
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Rule 422
Rule 429
Rule 539
Rule 541
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x}{c (d e-c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {\int \frac {-c (b e-a f)+(b c-a d) f x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx}{c (d e-c f)} \\ & = -\frac {(b c-a d) x}{c (d e-c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {(f (2 b c e-a d e-a c f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{c (d e-c f)^2}+\frac {(b d e+b c f-2 a d f) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{(d e-c f)^2} \\ & = -\frac {(b c-a d) x}{c (d e-c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {\sqrt {f} (2 b c e-a d e-a c f) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c \sqrt {e} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b d e+b c f-2 a d f) \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c \sqrt {f} (d e-c f)^2 \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 = 10.50 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {d}{c}} \left (\sqrt {\frac {d}{c}} x \left (a \left (c^2 f^2+c d f^2 x^2+d^2 e \left (e+f x^2\right )\right )-b c e \left (c f+d \left (e+2 f x^2\right )\right )\right )-i d e (2 b c e-a (d e+c f)) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i (b c-a d) e (-d e+c f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{d e (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
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Time = 4.98 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.89
method | result | size |
elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {2 d f \left (-\frac {\left (a c f +a d e -2 b c e \right ) x^{3}}{2 c e \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}-\frac {\left (c^{2} a \,f^{2}+a \,d^{2} e^{2}-b \,c^{2} e f -b c d \,e^{2}\right ) x}{2 c e \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) d f}\right )}{\sqrt {\left (x^{4}+\frac {\left (c f +d e \right ) x^{2}}{d f}+\frac {c e}{d f}\right ) d f}}+\frac {\left (\frac {a}{c e}-\frac {c^{2} a \,f^{2}+a \,d^{2} e^{2}-b \,c^{2} e f -b c d \,e^{2}}{c e \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d \left (a c f +a d e -2 b c e \right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{c \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(515\) |
default | \(\frac {\left (\sqrt {-\frac {d}{c}}\, a c d \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, a \,d^{2} e f \,x^{3}-2 \sqrt {-\frac {d}{c}}\, b c d e f \,x^{3}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2}+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2}+\sqrt {-\frac {d}{c}}\, a \,c^{2} f^{2} x +\sqrt {-\frac {d}{c}}\, a \,d^{2} e^{2} x -\sqrt {-\frac {d}{c}}\, b \,c^{2} e f x -\sqrt {-\frac {d}{c}}\, b c d \,e^{2} x \right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{c \sqrt {-\frac {d}{c}}\, e \left (c f -d e \right )^{2} \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(581\) |
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Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (266) = 532\).
Time = 0.11 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.27 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {{\left (a c^{2} d^{2} e f + {\left (a c d^{3} f^{2} - {\left (2 \, b c d^{3} - a d^{4}\right )} e f\right )} x^{4} - {\left (2 \, b c^{2} d^{2} - a c d^{3}\right )} e^{2} + {\left (a c^{2} d^{2} f^{2} - {\left (2 \, b c d^{3} - a d^{4}\right )} e^{2} - 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} e f\right )} x^{2}\right )} \sqrt {c e} \sqrt {-\frac {d}{c}} E(\arcsin \left (x \sqrt {-\frac {d}{c}}\right )\,|\,\frac {c f}{d e}) + {\left ({\left ({\left (b c^{2} d^{2} + 2 \, b c d^{3} - a d^{4}\right )} e f + {\left (b c^{3} d - 2 \, a c^{2} d^{2} - a c d^{3}\right )} f^{2}\right )} x^{4} + {\left (b c^{3} d + 2 \, b c^{2} d^{2} - a c d^{3}\right )} e^{2} + {\left (b c^{4} - 2 \, a c^{3} d - a c^{2} d^{2}\right )} e f + {\left ({\left (b c^{2} d^{2} + 2 \, b c d^{3} - a d^{4}\right )} e^{2} + 2 \, {\left (b c^{3} d - {\left (a - b\right )} c^{2} d^{2} - a c d^{3}\right )} e f + {\left (b c^{4} - 2 \, a c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} x^{2}\right )} \sqrt {c e} \sqrt {-\frac {d}{c}} F(\arcsin \left (x \sqrt {-\frac {d}{c}}\right )\,|\,\frac {c f}{d e}) - {\left ({\left (a c^{2} d^{2} f^{2} - {\left (2 \, b c^{2} d^{2} - a c d^{3}\right )} e f\right )} x^{3} - {\left (b c^{3} d e f - a c^{3} d f^{2} + {\left (b c^{2} d^{2} - a c d^{3}\right )} e^{2}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{c^{3} d^{3} e^{4} - 2 \, c^{4} d^{2} e^{3} f + c^{5} d e^{2} f^{2} + {\left (c^{2} d^{4} e^{3} f - 2 \, c^{3} d^{3} e^{2} f^{2} + c^{4} d^{2} e f^{3}\right )} x^{4} + {\left (c^{2} d^{4} e^{4} - c^{3} d^{3} e^{3} f - c^{4} d^{2} e^{2} f^{2} + c^{5} d e f^{3}\right )} x^{2}} \]
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\[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {a + b x^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {b\,x^2+a}{{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]
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