3.51 \(\int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=149 \[ \frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(d*x*(a + b*x^2)^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2) + ((4*b*c - 3*a*d)*x*Sq
rt[a + b*x^2])/(8*c^2*(b*c - a*d)*(c + d*x^2)) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqr
t[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.262617, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-3 a d)}{8 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{d x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x*(a + b*x^2)^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2) + ((4*b*c - 3*a*d)*x*Sq
rt[a + b*x^2])/(8*c^2*(b*c - a*d)*(c + d*x^2)) + (a*(4*b*c - 3*a*d)*ArcTanh[(Sqr
t[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(8*c^(5/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 31.1512, size = 129, normalized size = 0.87 \[ \frac{a \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{8 c^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{d x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4 c \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{x \sqrt{a + b x^{2}} \left (3 a d - 4 b c\right )}{8 c^{2} \left (c + d x^{2}\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/(d*x**2+c)**3,x)

[Out]

a*(3*a*d - 4*b*c)*atan(x*sqrt(a*d - b*c)/(sqrt(c)*sqrt(a + b*x**2)))/(8*c**(5/2)
*(a*d - b*c)**(3/2)) + d*x*(a + b*x**2)**(3/2)/(4*c*(c + d*x**2)**2*(a*d - b*c))
+ x*sqrt(a + b*x**2)*(3*a*d - 4*b*c)/(8*c**2*(c + d*x**2)*(a*d - b*c))

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Mathematica [A]  time = 0.197114, size = 128, normalized size = 0.86 \[ -\frac{a (4 b c-3 a d) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} (a d-b c)^{3/2}}-\frac{x \sqrt{a+b x^2} \left (a d \left (5 c+3 d x^2\right )-2 b c \left (2 c+d x^2\right )\right )}{8 c^2 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[a + b*x^2]*(-2*b*c*(2*c + d*x^2) + a*d*(5*c + 3*d*x^2)))/(8*c^2*(b*c -
a*d)*(c + d*x^2)^2) - (a*(4*b*c - 3*a*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*
Sqrt[a + b*x^2])])/(8*c^(5/2)*(-(b*c) + a*d)^(3/2))

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Maple [B]  time = 0.037, size = 5101, normalized size = 34.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/(d*x^2+c)^3,x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.356658, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (2 \, b c d - 3 \, a d^{2}\right )} x^{3} +{\left (4 \, b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a} +{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d +{\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} + 4 \,{\left ({\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{32 \,{\left (b c^{5} - a c^{4} d +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{4} + 2 \,{\left (b c^{4} d - a c^{3} d^{2}\right )} x^{2}\right )} \sqrt{b c^{2} - a c d}}, \frac{2 \,{\left ({\left (2 \, b c d - 3 \, a d^{2}\right )} x^{3} +{\left (4 \, b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{-b c^{2} + a c d} \sqrt{b x^{2} + a} +{\left (4 \, a b c^{3} - 3 \, a^{2} c^{2} d +{\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )}}{2 \,{\left (b c^{2} - a c d\right )} \sqrt{b x^{2} + a} x}\right )}{16 \,{\left (b c^{5} - a c^{4} d +{\left (b c^{3} d^{2} - a c^{2} d^{3}\right )} x^{4} + 2 \,{\left (b c^{4} d - a c^{3} d^{2}\right )} x^{2}\right )} \sqrt{-b c^{2} + a c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/32*(4*((2*b*c*d - 3*a*d^2)*x^3 + (4*b*c^2 - 5*a*c*d)*x)*sqrt(b*c^2 - a*c*d)*s
qrt(b*x^2 + a) + (4*a*b*c^3 - 3*a^2*c^2*d + (4*a*b*c*d^2 - 3*a^2*d^3)*x^4 + 2*(4
*a*b*c^2*d - 3*a^2*c*d^2)*x^2)*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)))/((b*c^5 - a*c^4*d + (b*c^3*d^2 - a*c^2*d^3)*x^4 + 2*(b*c^4*d
- a*c^3*d^2)*x^2)*sqrt(b*c^2 - a*c*d)), 1/16*(2*((2*b*c*d - 3*a*d^2)*x^3 + (4*b*
c^2 - 5*a*c*d)*x)*sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a) + (4*a*b*c^3 - 3*a^2*c^2*
d + (4*a*b*c*d^2 - 3*a^2*d^3)*x^4 + 2*(4*a*b*c^2*d - 3*a^2*c*d^2)*x^2)*arctan(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)))/((b*c^5 - a*c^4*d + (b*c^3*d^2 - a*c^2*d^3)*x^4 + 2*(b*c^4*d - a*c^3*d^2)
*x^2)*sqrt(-b*c^2 + a*c*d))]

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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)**(1/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 3.94901, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x