Optimal. Leaf size=426 \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (-3 a^2 d f+a b (2 d e-2 c f)+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)} \]
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Rubi [A] time = 1.33234, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{c} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (-3 a^2 d f+2 a b (d e-c f)+b^2 c e\right ) \Pi \left (-\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a^2 \sqrt{d} \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (b e-a f)}+\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (b e-a f)}-\frac{\sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c)}+\frac{b \sqrt{c} \sqrt{d} \sqrt{1-\frac{d x^2}{c}} \sqrt{e+f x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{c f}{d e}\right )}{2 a \sqrt{c-d x^2} \sqrt{\frac{f x^2}{e}+1} (a d+b c) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**2/(-d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
[Out]
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Mathematica [C] time = 6.64912, size = 773, normalized size = 1.81 \[ -\frac{b^2 x \sqrt{c-d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right ) (a d+b c) (a f-b e)}+\frac{\sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )} \left (\frac{i b^2 c e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{a \sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}+\frac{2 i b d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}-\frac{2 i b c f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}-\frac{3 i a d f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}+\frac{i a d f \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}+\frac{i b d e \sqrt{1-\frac{d x^2}{c}} \sqrt{\frac{f x^2}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c f}{d e}\right )\right )}{\sqrt{-\frac{d}{c}} \sqrt{\left (c-d x^2\right ) \left (e+f x^2\right )}}\right )}{2 a \sqrt{c-d x^2} \sqrt{e+f x^2} (a d+b c) (a f-b e)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]),x]
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Maple [B] time = 0.073, size = 1105, normalized size = 2.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^2/(-d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**2/(-d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{2} \sqrt{-d x^{2} + c} \sqrt{f x^{2} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="giac")
[Out]