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

Optimal. Leaf size=106 \[ -\frac{b (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{5/2}}+\frac{b x \sqrt{c+d x^2} (3 b c-2 a d)}{2 c d^2}-\frac{x \left (a+b x^2\right ) (b c-a d)}{c d \sqrt{c+d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.142381, antiderivative size = 106, 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 \[ -\frac{b (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 d^{5/2}}+\frac{b x \sqrt{c+d x^2} (3 b c-2 a d)}{2 c d^2}-\frac{x \left (a+b x^2\right ) (b c-a d)}{c d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.6608, size = 95, normalized size = 0.9 \[ \frac{b \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 d^{\frac{5}{2}}} - \frac{b x \sqrt{c + d x^{2}} \left (2 a d - 3 b c\right )}{2 c d^{2}} + \frac{x \left (a + b x^{2}\right ) \left (a d - b c\right )}{c d \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.162037, size = 93, normalized size = 0.88 \[ \sqrt{c+d x^2} \left (\frac{x (b c-a d)^2}{c d^2 \left (c+d x^2\right )}+\frac{b^2 x}{2 d^2}\right )-\frac{b (3 b c-4 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{2 d^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 123, normalized size = 1.2 \[{\frac{{a}^{2}x}{c}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{{b}^{2}{x}^{3}}{2\,d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{3\,{b}^{2}cx}{2\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{3\,{b}^{2}c}{2}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{5}{2}}}}-2\,{\frac{abx}{d\sqrt{d{x}^{2}+c}}}+2\,{\frac{ab\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }{{d}^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*x/c/(d*x^2+c)^(1/2)+1/2*b^2*x^3/d/(d*x^2+c)^(1/2)+3/2*b^2*c/d^2*x/(d*x^2+c)^
(1/2)-3/2*b^2*c/d^(5/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-2*a*b*x/d/(d*x^2+c)^(1/2)+
2*a*b/d^(3/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))

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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)^2/(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233874, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b^{2} c d x^{3} +{\left (3 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{d} -{\left (3 \, b^{2} c^{3} - 4 \, a b c^{2} d +{\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right )}{4 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \sqrt{d}}, \frac{{\left (b^{2} c d x^{3} +{\left (3 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2}\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-d} -{\left (3 \, b^{2} c^{3} - 4 \, a b c^{2} d +{\left (3 \, b^{2} c^{2} d - 4 \, a b c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right )}{2 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**2)**2/(c + d*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.24809, size = 124, normalized size = 1.17 \[ \frac{{\left (\frac{b^{2} x^{2}}{d} + \frac{3 \, b^{2} c^{2} d - 4 \, a b c d^{2} + 2 \, a^{2} d^{3}}{c d^{3}}\right )} x}{2 \, \sqrt{d x^{2} + c}} + \frac{{\left (3 \, b^{2} c - 4 \, a b d\right )}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right )}{2 \, d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*x^2/d + (3*b^2*c^2*d - 4*a*b*c*d^2 + 2*a^2*d^3)/(c*d^3))*x/sqrt(d*x^2 +
c) + 1/2*(3*b^2*c - 4*a*b*d)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))/d^(5/2)