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

Optimal. Leaf size=91 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right )}{d \sqrt{f}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} d \sqrt{d e-c f}} \]

[Out]

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

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Rubi [A]  time = 0.198055, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e+f x^2}}\right )}{d \sqrt{f}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} d \sqrt{d e-c f}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.6409, size = 78, normalized size = 0.86 \[ \frac{b \operatorname{atanh}{\left (\frac{\sqrt{f} x}{\sqrt{e + f x^{2}}} \right )}}{d \sqrt{f}} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{c f - d e}}{\sqrt{c} \sqrt{e + f x^{2}}} \right )}}{\sqrt{c} d \sqrt{c f - d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.147799, size = 93, normalized size = 1.02 \[ \frac{(a d-b c) \tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right )}{\sqrt{c} d \sqrt{d e-c f}}+\frac{b \log \left (\sqrt{f} \sqrt{e+f x^2}+f x\right )}{d \sqrt{f}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 646, normalized size = 7.1 \[{\frac{b}{d}\ln \left ( x\sqrt{f}+\sqrt{f{x}^{2}+e} \right ){\frac{1}{\sqrt{f}}}}+{\frac{a}{2}\ln \left ({1 \left ( -2\,{\frac{cf-de}{d}}-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}-{\frac{bc}{2\,d}\ln \left ({1 \left ( -2\,{\frac{cf-de}{d}}-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f-2\,{\frac{f\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}-{\frac{a}{2}\ln \left ({1 \left ( -2\,{\frac{cf-de}{d}}+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}}+{\frac{bc}{2\,d}\ln \left ({1 \left ( -2\,{\frac{cf-de}{d}}+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{-{\frac{cf-de}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}f+2\,{\frac{f\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }-{\frac{cf-de}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{-{\frac{cf-de}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  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)/((d*x^2 + c)*sqrt(f*x^2 + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.07409, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-c d e + c^{2} f} b \log \left (-2 \, \sqrt{f x^{2} + e} f x -{\left (2 \, f x^{2} + e\right )} \sqrt{f}\right ) -{\left (b c - a d\right )} \sqrt{f} \log \left (\frac{{\left ({\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \,{\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2}\right )} \sqrt{-c d e + c^{2} f} + 4 \,{\left ({\left (c d^{2} e^{2} - 3 \, c^{2} d e f + 2 \, c^{3} f^{2}\right )} x^{3} -{\left (c^{2} d e^{2} - c^{3} e f\right )} x\right )} \sqrt{f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, \sqrt{-c d e + c^{2} f} d \sqrt{f}}, -\frac{{\left (b c - a d\right )} \sqrt{f} \arctan \left (\frac{{\left (d e - 2 \, c f\right )} x^{2} - c e}{2 \, \sqrt{c d e - c^{2} f} \sqrt{f x^{2} + e} x}\right ) - \sqrt{c d e - c^{2} f} b \log \left (-2 \, \sqrt{f x^{2} + e} f x -{\left (2 \, f x^{2} + e\right )} \sqrt{f}\right )}{2 \, \sqrt{c d e - c^{2} f} d \sqrt{f}}, \frac{4 \, \sqrt{-c d e + c^{2} f} b \arctan \left (\frac{\sqrt{-f} x}{\sqrt{f x^{2} + e}}\right ) -{\left (b c - a d\right )} \sqrt{-f} \log \left (\frac{{\left ({\left (d^{2} e^{2} - 8 \, c d e f + 8 \, c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} - 2 \,{\left (3 \, c d e^{2} - 4 \, c^{2} e f\right )} x^{2}\right )} \sqrt{-c d e + c^{2} f} + 4 \,{\left ({\left (c d^{2} e^{2} - 3 \, c^{2} d e f + 2 \, c^{3} f^{2}\right )} x^{3} -{\left (c^{2} d e^{2} - c^{3} e f\right )} x\right )} \sqrt{f x^{2} + e}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, \sqrt{-c d e + c^{2} f} d \sqrt{-f}}, \frac{2 \, \sqrt{c d e - c^{2} f} b \arctan \left (\frac{\sqrt{-f} x}{\sqrt{f x^{2} + e}}\right ) -{\left (b c - a d\right )} \sqrt{-f} \arctan \left (\frac{{\left (d e - 2 \, c f\right )} x^{2} - c e}{2 \, \sqrt{c d e - c^{2} f} \sqrt{f x^{2} + e} x}\right )}{2 \, \sqrt{c d e - c^{2} f} d \sqrt{-f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269784, size = 159, normalized size = 1.75 \[ \frac{{\left (b c \sqrt{f} - a d \sqrt{f}\right )} \arctan \left (\frac{{\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2} d + 2 \, c f - d e}{2 \, \sqrt{-c^{2} f^{2} + c d f e}}\right )}{\sqrt{-c^{2} f^{2} + c d f e} d} - \frac{b{\rm ln}\left ({\left (\sqrt{f} x - \sqrt{f x^{2} + e}\right )}^{2}\right )}{2 \, d \sqrt{f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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