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

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

[Out]

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

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Rubi [A]  time = 0.229334, 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 \[ \frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{5/2} (b c-a d)^{3/2}}+\frac{x \sqrt{c+d x^2} (3 b c-4 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)}+\frac{b x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.7139, size = 129, normalized size = 0.87 \[ - \frac{b x \left (c + d x^{2}\right )^{\frac{3}{2}}}{4 a \left (a + b x^{2}\right )^{2} \left (a d - b c\right )} + \frac{x \sqrt{c + d x^{2}} \left (4 a d - 3 b c\right )}{8 a^{2} \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{c \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{8 a^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.262205, size = 130, normalized size = 0.87 \[ \frac{\frac{\sqrt{a} x \sqrt{c+d x^2} \left (-4 a^2 d+a b \left (5 c-2 d x^2\right )+3 b^2 c x^2\right )}{\left (a+b x^2\right )^2 (b c-a d)}+\frac{c (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{3/2}}}{8 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.05, size = 5177, normalized size = 34.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.404444, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} +{\left (5 \, a b c - 4 \, a^{2} d\right )} x\right )} \sqrt{-a b c + a^{2} d} \sqrt{d x^{2} + c} +{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d +{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \log \left (\frac{{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d} + 4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{3} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{32 \,{\left (a^{4} b c - a^{5} d +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{4} + 2 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x^{2}\right )} \sqrt{-a b c + a^{2} d}}, \frac{2 \,{\left ({\left (3 \, b^{2} c - 2 \, a b d\right )} x^{3} +{\left (5 \, a b c - 4 \, a^{2} d\right )} x\right )} \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} +{\left (3 \, a^{2} b c^{2} - 4 \, a^{3} c d +{\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{2}\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{a b c - a^{2} d} \sqrt{d x^{2} + c} x}\right )}{16 \,{\left (a^{4} b c - a^{5} d +{\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{4} + 2 \,{\left (a^{3} b^{2} c - a^{4} b d\right )} x^{2}\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/32*(4*((3*b^2*c - 2*a*b*d)*x^3 + (5*a*b*c - 4*a^2*d)*x)*sqrt(-a*b*c + a^2*d)*
sqrt(d*x^2 + c) + (3*a^2*b*c^2 - 4*a^3*c*d + (3*b^3*c^2 - 4*a*b^2*c*d)*x^4 + 2*(
3*a*b^2*c^2 - 4*a^2*b*c*d)*x^2)*log((((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^
2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2)*sqrt(-a*b*c + a^2*d) + 4*((a*b^2*c^2 - 3*
a^2*b*c*d + 2*a^3*d^2)*x^3 - (a^2*b*c^2 - a^3*c*d)*x)*sqrt(d*x^2 + c))/(b^2*x^4
+ 2*a*b*x^2 + a^2)))/((a^4*b*c - a^5*d + (a^2*b^3*c - a^3*b^2*d)*x^4 + 2*(a^3*b^
2*c - a^4*b*d)*x^2)*sqrt(-a*b*c + a^2*d)), 1/16*(2*((3*b^2*c - 2*a*b*d)*x^3 + (5
*a*b*c - 4*a^2*d)*x)*sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c) + (3*a^2*b*c^2 - 4*a^3*
c*d + (3*b^3*c^2 - 4*a*b^2*c*d)*x^4 + 2*(3*a*b^2*c^2 - 4*a^2*b*c*d)*x^2)*arctan(
1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)))/((a^4*b*
c - a^5*d + (a^2*b^3*c - a^3*b^2*d)*x^4 + 2*(a^3*b^2*c - a^4*b*d)*x^2)*sqrt(a*b*
c - a^2*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((d*x**2+c)**(1/2)/(b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.24272, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x