3.46 \(\int \frac{a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=272 \[ -\frac{x (b c-a d)}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)}+\frac{\sqrt{e} \sqrt{c+d x^2} (-2 a d f+b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{f} \sqrt{c+d x^2} (-a c f-a d e+2 b c e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

-(((b*c - a*d)*x)/(c*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (Sqrt[f]*(2
*b*c*e - a*d*e - a*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1
- (d*e)/(c*f)])/(c*Sqrt[e]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*
Sqrt[e + f*x^2]) + (Sqrt[e]*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[
ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*
(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.641017, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{x (b c-a d)}{c \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)}+\frac{\sqrt{e} \sqrt{c+d x^2} (-2 a d f+b c f+b d e) F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac{\sqrt{f} \sqrt{c+d x^2} (-a c f-a d e+2 b c e) E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{e+f x^2} (d e-c f)^2 \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]

[Out]

-(((b*c - a*d)*x)/(c*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) - (Sqrt[f]*(2
*b*c*e - a*d*e - a*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1
- (d*e)/(c*f)])/(c*Sqrt[e]*(d*e - c*f)^2*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*
Sqrt[e + f*x^2]) + (Sqrt[e]*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x^2]*EllipticF[
ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*(d*e - c*f)^2*Sqrt[(e*
(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])

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Rubi in Sympy [A]  time = 71.8301, size = 236, normalized size = 0.87 \[ - \frac{\sqrt{e} \sqrt{c + d x^{2}} \left (2 a d f - b c f - b d e\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{c \sqrt{f} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (c f - d e\right )^{2}} - \frac{x \left (a d - b c\right )}{c \sqrt{c + d x^{2}} \sqrt{e + f x^{2}} \left (c f - d e\right )} + \frac{\sqrt{f} \sqrt{c + d x^{2}} \left (a c f + a d e - 2 b c e\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{f} x}{\sqrt{e}} \right )}\middle | 1 - \frac{d e}{c f}\right )}{c \sqrt{e} \sqrt{\frac{e \left (c + d x^{2}\right )}{c \left (e + f x^{2}\right )}} \sqrt{e + f x^{2}} \left (c f - d e\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(d*x**2+c)**(3/2)/(f*x**2+e)**(3/2),x)

[Out]

-sqrt(e)*sqrt(c + d*x**2)*(2*a*d*f - b*c*f - b*d*e)*elliptic_f(atan(sqrt(f)*x/sq
rt(e)), 1 - d*e/(c*f))/(c*sqrt(f)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2)))*sqrt(e +
f*x**2)*(c*f - d*e)**2) - x*(a*d - b*c)/(c*sqrt(c + d*x**2)*sqrt(e + f*x**2)*(c
*f - d*e)) + sqrt(f)*sqrt(c + d*x**2)*(a*c*f + a*d*e - 2*b*c*e)*elliptic_e(atan(
sqrt(f)*x/sqrt(e)), 1 - d*e/(c*f))/(c*sqrt(e)*sqrt(e*(c + d*x**2)/(c*(e + f*x**2
)))*sqrt(e + f*x**2)*(c*f - d*e)**2)

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Mathematica [C]  time = 1.25059, size = 262, normalized size = 0.96 \[ \frac{\sqrt{\frac{d}{c}} \left (x \sqrt{\frac{d}{c}} \left (a \left (c^2 f^2+c d f^2 x^2+d^2 e \left (e+f x^2\right )\right )-b c e \left (c f+d \left (e+2 f x^2\right )\right )\right )-i e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (b c-a d) (c f-d e) F\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )-i d e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (2 b c e-a (c f+d e)) E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )\right )}{d e \sqrt{c+d x^2} \sqrt{e+f x^2} (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/((c + d*x^2)^(3/2)*(e + f*x^2)^(3/2)),x]

[Out]

(Sqrt[d/c]*(Sqrt[d/c]*x*(a*(c^2*f^2 + c*d*f^2*x^2 + d^2*e*(e + f*x^2)) - b*c*e*(
c*f + d*(e + 2*f*x^2))) - I*d*e*(2*b*c*e - a*(d*e + c*f))*Sqrt[1 + (d*x^2)/c]*Sq
rt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*(b*c - a*d)
*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sq
rt[d/c]*x], (c*f)/(d*e)]))/(d*e*(d*e - c*f)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Maple [A]  time = 0.043, size = 581, normalized size = 2.1 \[{\frac{1}{ce \left ( cf-de \right ) ^{2} \left ( df{x}^{4}+cf{x}^{2}+de{x}^{2}+ce \right ) } \left ({x}^{3}acd{f}^{2}\sqrt{-{\frac{d}{c}}}+{x}^{3}a{d}^{2}ef\sqrt{-{\frac{d}{c}}}-2\,{x}^{3}bcdef\sqrt{-{\frac{d}{c}}}-{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) acdef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) a{d}^{2}{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) b{c}^{2}ef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcd{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) acdef\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}-{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) a{d}^{2}{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+2\,{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{cf}{de}}} \right ) bcd{e}^{2}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}+xa{c}^{2}{f}^{2}\sqrt{-{\frac{d}{c}}}+xa{d}^{2}{e}^{2}\sqrt{-{\frac{d}{c}}}-xb{c}^{2}ef\sqrt{-{\frac{d}{c}}}-xbcd{e}^{2}\sqrt{-{\frac{d}{c}}} \right ) \sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(d*x^2+c)^(3/2)/(f*x^2+e)^(3/2),x)

[Out]

(x^3*a*c*d*f^2*(-d/c)^(1/2)+x^3*a*d^2*e*f*(-d/c)^(1/2)-2*x^3*b*c*d*e*f*(-d/c)^(1
/2)-EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*c*d*e*f*((d*x^2+c)/c)^(1/2)*((f*
x^2+e)/e)^(1/2)+EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*a*d^2*e^2*((d*x^2+c)/c
)^(1/2)*((f*x^2+e)/e)^(1/2)+EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c^2*e*f*
((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)
)*b*c*d*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/c)^(1/2),(c*
f/d/e)^(1/2))*a*c*d*e*f*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)-EllipticE(x*(-d/
c)^(1/2),(c*f/d/e)^(1/2))*a*d^2*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)+2*El
lipticE(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c*d*e^2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)
/e)^(1/2)+x*a*c^2*f^2*(-d/c)^(1/2)+x*a*d^2*e^2*(-d/c)^(1/2)-x*b*c^2*e*f*(-d/c)^(
1/2)-x*b*c*d*e^2*(-d/c)^(1/2))*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/e/c/(-d/c)^(1/2)/
(c*f-d*e)^2/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b x^{2} + a}{{\left (d f x^{4} +{\left (d e + c f\right )} x^{2} + c e\right )} \sqrt{d x^{2} + c} \sqrt{f x^{2} + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)/((d*f*x^4 + (d*e + c*f)*x^2 + c*e)*sqrt(d*x^2 + c)*sqrt(f*x
^2 + e)), x)

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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((b*x**2+a)/(d*x**2+c)**(3/2)/(f*x**2+e)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)/((d*x^2 + c)^(3/2)*(f*x^2 + e)^(3/2)), x)