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

Optimal. Leaf size=113 \[ \frac{x \sqrt{c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac{(a c f-2 a d e+b c e) \tanh ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}} \]

[Out]

((b*e - a*f)*x*Sqrt[c + d*x^2])/(2*e*(d*e - c*f)*(e + f*x^2)) - ((b*c*e - 2*a*d*
e + a*c*f)*ArcTanh[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(2*e^(3/2)*(d
*e - c*f)^(3/2))

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Rubi [A]  time = 0.336599, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x \sqrt{c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac{(a c f-2 a d e+b c e) \tanh ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{e} \sqrt{c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b*e - a*f)*x*Sqrt[c + d*x^2])/(2*e*(d*e - c*f)*(e + f*x^2)) - ((b*c*e - 2*a*d*
e + a*c*f)*ArcTanh[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/(2*e^(3/2)*(d
*e - c*f)^(3/2))

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Rubi in Sympy [A]  time = 34.0952, size = 97, normalized size = 0.86 \[ \frac{x \sqrt{c + d x^{2}} \left (a f - b e\right )}{2 e \left (e + f x^{2}\right ) \left (c f - d e\right )} + \frac{\left (a c f - 2 a d e + b c e\right ) \operatorname{atan}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{e} \sqrt{c + d x^{2}}} \right )}}{2 e^{\frac{3}{2}} \left (c f - d e\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(f*x**2+e)**2/(d*x**2+c)**(1/2),x)

[Out]

x*sqrt(c + d*x**2)*(a*f - b*e)/(2*e*(e + f*x**2)*(c*f - d*e)) + (a*c*f - 2*a*d*e
+ b*c*e)*atan(x*sqrt(c*f - d*e)/(sqrt(e)*sqrt(c + d*x**2)))/(2*e**(3/2)*(c*f -
d*e)**(3/2))

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Mathematica [A]  time = 0.236834, size = 114, normalized size = 1.01 \[ \frac{\frac{\sqrt{e} x \sqrt{c+d x^2} (b e-a f)}{e+f x^2}-\frac{(a c f-2 a d e+b c e) \tan ^{-1}\left (\frac{x \sqrt{c f-d e}}{\sqrt{e} \sqrt{c+d x^2}}\right )}{\sqrt{c f-d e}}}{2 e^{3/2} (d e-c f)} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[e]*(b*e - a*f)*x*Sqrt[c + d*x^2])/(e + f*x^2) - ((b*c*e - 2*a*d*e + a*c*f
)*ArcTan[(Sqrt[-(d*e) + c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/Sqrt[-(d*e) + c*f])/
(2*e^(3/2)*(d*e - c*f))

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Maple [B]  time = 0.049, size = 1622, normalized size = 14.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(f*x^2+e)^2/(d*x^2+c)^(1/2),x)

[Out]

1/4/e/(c*f-d*e)/(x-(-e*f)^(1/2)/f)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x
-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2)*a-1/4/f/(c*f-d*e)/(x-(-e*f)^(1/2)/f)*((x-(-e
*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2)*b-1/4/
e/f*d*(-e*f)^(1/2)/(c*f-d*e)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f+2*d*(-e*f)^(1
/2)/f*(x-(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f
)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*f)^(1/2)/f))*a+1/4/f^2*d
*(-e*f)^(1/2)/(c*f-d*e)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f+2*d*(-e*f)^(1/2)/f
*(x-(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/
2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*f)^(1/2)/f))*b+1/4/e/(c*f-d*e
)/(x+(-e*f)^(1/2)/f)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/
f)+(c*f-d*e)/f)^(1/2)*a-1/4/f/(c*f-d*e)/(x+(-e*f)^(1/2)/f)*((x+(-e*f)^(1/2)/f)^2
*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2)*b+1/4/e/f*d*(-e*f)^(
1/2)/(c*f-d*e)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(x+(-e*f
)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(
-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x+(-e*f)^(1/2)/f))*a-1/4/f^2*d*(-e*f)^(1/2)/
(c*f-d*e)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/
2)/f)+2*((c*f-d*e)/f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)
^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x+(-e*f)^(1/2)/f))*b+1/4/e/(-e*f)^(1/2)/((c*f-d*e
)/f)^(1/2)*ln((2*(c*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+2*((c*f-d*e)/
f)^(1/2)*((x+(-e*f)^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c*f-d*e)
/f)^(1/2))/(x+(-e*f)^(1/2)/f))*a+1/4/(-e*f)^(1/2)/f/((c*f-d*e)/f)^(1/2)*ln((2*(c
*f-d*e)/f-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x+(-e*f)
^(1/2)/f)^2*d-2*d*(-e*f)^(1/2)/f*(x+(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x+(-e*f
)^(1/2)/f))*b-1/4/e/(-e*f)^(1/2)/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f+2*d*(-e*f
)^(1/2)/f*(x-(-e*f)^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(
-e*f)^(1/2)/f*(x-(-e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*f)^(1/2)/f))*a-1/4/(
-e*f)^(1/2)/f/((c*f-d*e)/f)^(1/2)*ln((2*(c*f-d*e)/f+2*d*(-e*f)^(1/2)/f*(x-(-e*f)
^(1/2)/f)+2*((c*f-d*e)/f)^(1/2)*((x-(-e*f)^(1/2)/f)^2*d+2*d*(-e*f)^(1/2)/f*(x-(-
e*f)^(1/2)/f)+(c*f-d*e)/f)^(1/2))/(x-(-e*f)^(1/2)/f))*b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.6132, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{d e^{2} - c e f} \sqrt{d x^{2} + c}{\left (b e - a f\right )} x -{\left (a c e f +{\left (b c - 2 \, a d\right )} e^{2} +{\left (a c f^{2} +{\left (b c - 2 \, a d\right )} e f\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (8 \, d^{2} e^{2} - 8 \, c d e f + c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} + 2 \,{\left (4 \, c d e^{2} - 3 \, c^{2} e f\right )} x^{2}\right )} \sqrt{d e^{2} - c e f} + 4 \,{\left ({\left (2 \, d^{2} e^{3} - 3 \, c d e^{2} f + c^{2} e f^{2}\right )} x^{3} +{\left (c d e^{3} - c^{2} e^{2} f\right )} x\right )} \sqrt{d x^{2} + c}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right )}{8 \,{\left (d e^{3} - c e^{2} f +{\left (d e^{2} f - c e f^{2}\right )} x^{2}\right )} \sqrt{d e^{2} - c e f}}, \frac{2 \, \sqrt{-d e^{2} + c e f} \sqrt{d x^{2} + c}{\left (b e - a f\right )} x -{\left (a c e f +{\left (b c - 2 \, a d\right )} e^{2} +{\left (a c f^{2} +{\left (b c - 2 \, a d\right )} e f\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-d e^{2} + c e f}{\left ({\left (2 \, d e - c f\right )} x^{2} + c e\right )}}{2 \,{\left (d e^{2} - c e f\right )} \sqrt{d x^{2} + c} x}\right )}{4 \,{\left (d e^{3} - c e^{2} f +{\left (d e^{2} f - c e f^{2}\right )} x^{2}\right )} \sqrt{-d e^{2} + c e f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(4*sqrt(d*e^2 - c*e*f)*sqrt(d*x^2 + c)*(b*e - a*f)*x - (a*c*e*f + (b*c - 2*
a*d)*e^2 + (a*c*f^2 + (b*c - 2*a*d)*e*f)*x^2)*log((((8*d^2*e^2 - 8*c*d*e*f + c^2
*f^2)*x^4 + c^2*e^2 + 2*(4*c*d*e^2 - 3*c^2*e*f)*x^2)*sqrt(d*e^2 - c*e*f) + 4*((2
*d^2*e^3 - 3*c*d*e^2*f + c^2*e*f^2)*x^3 + (c*d*e^3 - c^2*e^2*f)*x)*sqrt(d*x^2 +
c))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/((d*e^3 - c*e^2*f + (d*e^2*f - c*e*f^2)*x^2)*s
qrt(d*e^2 - c*e*f)), 1/4*(2*sqrt(-d*e^2 + c*e*f)*sqrt(d*x^2 + c)*(b*e - a*f)*x -
(a*c*e*f + (b*c - 2*a*d)*e^2 + (a*c*f^2 + (b*c - 2*a*d)*e*f)*x^2)*arctan(1/2*sq
rt(-d*e^2 + c*e*f)*((2*d*e - c*f)*x^2 + c*e)/((d*e^2 - c*e*f)*sqrt(d*x^2 + c)*x)
))/((d*e^3 - c*e^2*f + (d*e^2*f - c*e*f^2)*x^2)*sqrt(-d*e^2 + c*e*f))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/(f*x**2+e)**2/(d*x**2+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 1.63653, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x