Integrand size = 32, antiderivative size = 980 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b (b e-a f)^2 \sqrt {e+f x^2}}+\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{3 e f (b e-a f)^2 \sqrt {e+f x^2}}+\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}-\frac {(b c-a d) \sqrt {e} (b d e+4 b c f-3 a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 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 )}{3 \sqrt {e} f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 b c \sqrt {f} (b e-a 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} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\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 )}{3 f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{a b c \sqrt {f} (b e-a 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 = 0.71 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {558, 557, 553, 542, 545, 429, 506, 422, 540} \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\frac {e^{3/2} \sqrt {d x^2+c} \operatorname {EllipticPi}\left (1-\frac {b e}{a f},\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right ) (b c-a d)^3}{a b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} (b d e+4 b c f-3 a d f) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right ) (b c-a d)}{3 b \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d (5 b c-3 a d) e^{3/2} \sqrt {d x^2+c} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right ) (b c-a d)}{3 b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d x \sqrt {d x^2+c} \sqrt {f x^2+e} (b c-a d)}{3 (b e-a f)^2}+\frac {(b d e+4 b c f-3 a d f) x \sqrt {d x^2+c} (b c-a d)}{3 b (b e-a f)^2 \sqrt {f x^2+e}}-\frac {\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) \sqrt {d x^2+c} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 \sqrt {e} f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}-\frac {\sqrt {e} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d f e-3 c^2 f^2\right )\right ) \sqrt {d x^2+c} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (d x^2+c\right )}{c \left (f x^2+e\right )}} \sqrt {f x^2+e}}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {d x^2+c} \sqrt {f x^2+e}}{3 e f (b e-a f)^2}+\frac {(d e-c f) x \left (d x^2+c\right )^{3/2}}{e (b e-a f) \sqrt {f x^2+e}}+\frac {\left (b e \left (6 d^2 e^2-7 c d f e-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d f e+3 c^2 f^2\right )\right ) x \sqrt {d x^2+c}}{3 e f (b e-a f)^2 \sqrt {f x^2+e}} \]
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Rule 422
Rule 429
Rule 506
Rule 540
Rule 542
Rule 545
Rule 553
Rule 557
Rule 558
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\left (c+d x^2\right )^{3/2} \left (-b d e^2+2 b c e f-a c f^2+(b c-a d) f^2 x^2\right )}{\left (e+f x^2\right )^{3/2}} \, dx}{(b e-a f)^2}+\frac {(b (b c-a d)) \int \frac {\left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{a+b x^2} \, dx}{(b e-a f)^2} \\ & = \frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {(d (b c-a d)) \int \frac {\left (2 b c-a d+b d x^2\right ) \sqrt {e+f x^2}}{\sqrt {c+d x^2}} \, dx}{b (b e-a f)^2}+\frac {(b c-a d)^3 \int \frac {\sqrt {e+f x^2}}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b (b e-a f)^2}+\frac {\int \frac {\sqrt {c+d x^2} \left (-c (b c-a d) e f^2+d f (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x^2\right )}{\sqrt {e+f x^2}} \, dx}{e f (b e-a f)^2} \\ & = \frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d) \int \frac {d (5 b c-3 a d) e+d (b d e+4 b c f-3 a d f) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b (b e-a f)^2}+\frac {\int \frac {-c e f \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\right )\right )+d f \left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f^2 (b e-a f)^2} \\ & = \frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(d (5 b c-3 a d) (b c-a d) e) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b (b e-a f)^2}+\frac {(d (b c-a d) (b d e+4 b c f-3 a d f)) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 b (b e-a f)^2}-\frac {\left (c \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 f (b e-a f)^2}+\frac {\left (d \left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right )\right ) \int \frac {x^2}{\sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx}{3 e f (b e-a f)^2} \\ & = \frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b (b e-a f)^2 \sqrt {e+f x^2}}+\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{3 e f (b e-a f)^2 \sqrt {e+f x^2}}+\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b c \sqrt {f} (b e-a 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} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\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 )}{3 f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b c \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {((b c-a d) e (b d e+4 b c f-3 a d f)) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 b (b e-a f)^2}-\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{3 f (b e-a f)^2} \\ & = \frac {(b c-a d) (b d e+4 b c f-3 a d f) x \sqrt {c+d x^2}}{3 b (b e-a f)^2 \sqrt {e+f x^2}}+\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{3 e f (b e-a f)^2 \sqrt {e+f x^2}}+\frac {(d e-c f) x \left (c+d x^2\right )^{3/2}}{e (b e-a f) \sqrt {e+f x^2}}+\frac {d (b c-a d) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 (b e-a f)^2}+\frac {d (a f (4 d e-3 c f)-b e (3 d e-2 c f)) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{3 e f (b e-a f)^2}-\frac {(b c-a d) \sqrt {e} (b d e+4 b c f-3 a d 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 )}{3 b \sqrt {f} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {\left (b e \left (6 d^2 e^2-7 c d e f-c^2 f^2\right )-a f \left (8 d^2 e^2-13 c d e f+3 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 )}{3 \sqrt {e} f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {d (5 b c-3 a d) (b c-a d) e^{3/2} \sqrt {c+d x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{3 b c \sqrt {f} (b e-a 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} \left (2 a d f (2 d e-3 c f)-b \left (3 d^2 e^2-2 c d e f-3 c^2 f^2\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 )}{3 f^{3/2} (b e-a f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {(b c-a d)^3 e^{3/2} \sqrt {c+d x^2} \Pi \left (1-\frac {b e}{a f};\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{a b c \sqrt {f} (b e-a 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 = 6.83 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.36 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\frac {-i a b d e \left (-a d^2 e f+b \left (2 d^2 e^2-2 c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i a d^2 e (b e-a f) (-2 b d e+3 b c f-a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )-f \left (a b^2 \sqrt {\frac {d}{c}} (d e-c f)^2 x \left (c+d x^2\right )+i (b c-a d)^3 e f \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{a b^2 \sqrt {\frac {d}{c}} e f^2 (b e-a f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
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Time = 4.21 (sec) , antiderivative size = 1063, normalized size of antiderivative = 1.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1063\) |
elliptic | \(\text {Expression too large to display}\) | \(1255\) |
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right ) \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{{\left (b x^{2} + a\right )} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right ) \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{\left (b\,x^2+a\right )\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]
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