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

Optimal. Leaf size=82 \[ \frac{a \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} \sqrt{b c-a d}}+\frac{x \sqrt{a+b x^2}}{2 c \left (c+d x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0978889, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{a \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} \sqrt{b c-a d}}+\frac{x \sqrt{a+b x^2}}{2 c \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.2215, size = 68, normalized size = 0.83 \[ \frac{a \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{2 c^{\frac{3}{2}} \sqrt{a d - b c}} + \frac{x \sqrt{a + b x^{2}}}{2 c \left (c + d x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.151674, size = 82, normalized size = 1. \[ \frac{a \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} \sqrt{a d-b c}}+\frac{x \sqrt{a+b x^2}}{2 c \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.039, size = 2521, normalized size = 30.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/c/(a*d-b*c)/(x-(-c*d)^(1/2)/d)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x
-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)-1/4/c/d*b*(-c*d)^(1/2)/(a*d-b*c)*((x-(-c*d)^
(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+1/4/d*b^(3
/2)/(a*d-b*c)*ln((b*(-c*d)^(1/2)/d+(x-(-c*d)^(1/2)/d)*b)/b^(1/2)+((x-(-c*d)^(1/2
)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))+1/4/c/d*b*(-c
*d)^(1/2)/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-
(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d
*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*a-1/4/d^2*b^2*(-c*d)
^(1/2)/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c
*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x
-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))-1/4/c*b/(a*d-b*c)*((x-(
-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-1/
4/c*b^(1/2)/(a*d-b*c)*ln((b*(-c*d)^(1/2)/d+(x-(-c*d)^(1/2)/d)*b)/b^(1/2)+((x-(-c
*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))*a+1/4
/c/(a*d-b*c)/(x+(-c*d)^(1/2)/d)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-
c*d)^(1/2)/d)+(a*d-b*c)/d)^(3/2)+1/4/c/d*b*(-c*d)^(1/2)/(a*d-b*c)*((x+(-c*d)^(1/
2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+1/4/d*b^(3/2)
/(a*d-b*c)*ln((-b*(-c*d)^(1/2)/d+(x+(-c*d)^(1/2)/d)*b)/b^(1/2)+((x+(-c*d)^(1/2)/
d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))-1/4/c/d*b*(-c*d
)^(1/2)/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-
c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(
x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))*a+1/4/d^2*b^2*(-c*d)^(
1/2)/(a*d-b*c)/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d
)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(
-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))-1/4/c*b/(a*d-b*c)*((x+(-c
*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)*x-1/4/
c*b^(1/2)/(a*d-b*c)*ln((-b*(-c*d)^(1/2)/d+(x+(-c*d)^(1/2)/d)*b)/b^(1/2)+((x+(-c*
d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))*a+1/4/
(-c*d)^(1/2)/c*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*
d-b*c)/d)^(1/2)+1/4/c*b^(1/2)/d*ln((b*(-c*d)^(1/2)/d+(x-(-c*d)^(1/2)/d)*b)/b^(1/
2)+((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1
/2))-1/4/(-c*d)^(1/2)/c/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d
*(x-(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/
2)/d*(x-(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*a+1/4/(-c*d)^(1/
2)/d/((a*d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2)/d)
+2*((a*d-b*c)/d)^(1/2)*((x-(-c*d)^(1/2)/d)^2*b+2*b*(-c*d)^(1/2)/d*(x-(-c*d)^(1/2
)/d)+(a*d-b*c)/d)^(1/2))/(x-(-c*d)^(1/2)/d))*b-1/4/(-c*d)^(1/2)/c*((x+(-c*d)^(1/
2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2)+1/4/c*b^(1/2)
/d*ln((-b*(-c*d)^(1/2)/d+(x+(-c*d)^(1/2)/d)*b)/b^(1/2)+((x+(-c*d)^(1/2)/d)^2*b-2
*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))+1/4/(-c*d)^(1/2)/c/((a*
d-b*c)/d)^(1/2)*ln((2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-
b*c)/d)^(1/2)*((x+(-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d
-b*c)/d)^(1/2))/(x+(-c*d)^(1/2)/d))*a-1/4/(-c*d)^(1/2)/d/((a*d-b*c)/d)^(1/2)*ln(
(2*(a*d-b*c)/d-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+2*((a*d-b*c)/d)^(1/2)*((x+(
-c*d)^(1/2)/d)^2*b-2*b*(-c*d)^(1/2)/d*(x+(-c*d)^(1/2)/d)+(a*d-b*c)/d)^(1/2))/(x+
(-c*d)^(1/2)/d))*b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280321, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a} x +{\left (a d x^{2} + a c\right )} \log \left (\frac{{\left ({\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} + 4 \,{\left ({\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{8 \,{\left (c d x^{2} + c^{2}\right )} \sqrt{b c^{2} - a c d}}, \frac{2 \, \sqrt{-b c^{2} + a c d} \sqrt{b x^{2} + a} x +{\left (a d x^{2} + a c\right )} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )}}{2 \,{\left (b c^{2} - a c d\right )} \sqrt{b x^{2} + a} x}\right )}{4 \,{\left (c d x^{2} + c^{2}\right )} \sqrt{-b c^{2} + a c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 2.64885, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x