Optimal. Leaf size=247 \[ \frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}} \]
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Rubi [A] time = 0.746629, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226 \[ \frac{x \sqrt{a-b x^2} (d e-c f)}{c \sqrt{c+d x^2} (a d+b c)}+\frac{\sqrt{a} \sqrt{b} \sqrt{1-\frac{b x^2}{a}} \sqrt{c+d x^2} (d e-c f) E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{c d \sqrt{a-b x^2} \sqrt{\frac{d x^2}{c}+1} (a d+b c)}+\frac{\sqrt{a} f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a-b x^2} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
[In] Int[(e + f*x^2)/(Sqrt[a - b*x^2]*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 113.146, size = 209, normalized size = 0.85 \[ - \frac{\sqrt{a} \sqrt{b} \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{c + d x^{2}} \left (c f - d e\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{c d \sqrt{1 + \frac{d x^{2}}{c}} \sqrt{a - b x^{2}} \left (a d + b c\right )} + \frac{\sqrt{a} f \sqrt{1 - \frac{b x^{2}}{a}} \sqrt{1 + \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c}\right )}{\sqrt{b} d \sqrt{a - b x^{2}} \sqrt{c + d x^{2}}} - \frac{x \sqrt{a - b x^{2}} \left (c f - d e\right )}{c \sqrt{c + d x^{2}} \left (a d + b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**2+e)/(d*x**2+c)**(3/2)/(-b*x**2+a)**(1/2),x)
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Mathematica [C] time = 1.23432, size = 220, normalized size = 0.89 \[ \frac{d x \sqrt{-\frac{b}{a}} \left (a-b x^2\right ) (d e-c f)+i b c \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (c f-d e) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )-i c f \sqrt{1-\frac{b x^2}{a}} \sqrt{\frac{d x^2}{c}+1} (a d+b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{c d \sqrt{-\frac{b}{a}} \sqrt{a-b x^2} \sqrt{c+d x^2} (a d+b c)} \]
Antiderivative was successfully verified.
[In] Integrate[(e + f*x^2)/(Sqrt[a - b*x^2]*(c + d*x^2)^(3/2)),x]
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Maple [A] time = 0.073, size = 359, normalized size = 1.5 \[{\frac{1}{cd \left ( ad+bc \right ) \left ( bd{x}^{4}-ad{x}^{2}+c{x}^{2}b-ac \right ) } \left ( -{x}^{3}bcdf\sqrt{{\frac{b}{a}}}+{x}^{3}b{d}^{2}e\sqrt{{\frac{b}{a}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) acdf\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) b{c}^{2}f\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}-{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ) bcde\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}+xacdf\sqrt{{\frac{b}{a}}}-xa{d}^{2}e\sqrt{{\frac{b}{a}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{b}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^2+e)/(d*x^2+c)^(3/2)/(-b*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{e + f x^{2}}{\sqrt{a - b x^{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**2+e)/(d*x**2+c)**(3/2)/(-b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f x^{2} + e}{\sqrt{-b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e)/(sqrt(-b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="giac")
[Out]