3.573 \(\int \frac{x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ \frac{(2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{5/2}}-\frac{x (A b-a B)}{b^2 \sqrt{a+b x^2}}+\frac{B x \sqrt{a+b x^2}}{2 b^2} \]

[Out]

-(((A*b - a*B)*x)/(b^2*Sqrt[a + b*x^2])) + (B*x*Sqrt[a + b*x^2])/(2*b^2) + ((2*A
*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))

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Rubi [A]  time = 0.164013, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(2 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{5/2}}-\frac{x (A b-a B)}{b^2 \sqrt{a+b x^2}}+\frac{B x \sqrt{a+b x^2}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(((A*b - a*B)*x)/(b^2*Sqrt[a + b*x^2])) + (B*x*Sqrt[a + b*x^2])/(2*b^2) + ((2*A
*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*b^(5/2))

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Rubi in Sympy [A]  time = 22.5442, size = 75, normalized size = 0.9 \[ \frac{B x \sqrt{a + b x^{2}}}{2 b^{2}} - \frac{x \left (A b - B a\right )}{b^{2} \sqrt{a + b x^{2}}} + \frac{\left (2 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

B*x*sqrt(a + b*x**2)/(2*b**2) - x*(A*b - B*a)/(b**2*sqrt(a + b*x**2)) + (2*A*b -
3*B*a)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2*b**(5/2))

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Mathematica [A]  time = 0.103335, size = 75, normalized size = 0.9 \[ \frac{\frac{\sqrt{b} x \left (3 a B-2 A b+b B x^2\right )}{\sqrt{a+b x^2}}+(2 A b-3 a B) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*x*(-2*A*b + 3*a*B + b*B*x^2))/Sqrt[a + b*x^2] + (2*A*b - 3*a*B)*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]])/(2*b^(5/2))

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Maple [A]  time = 0.01, size = 97, normalized size = 1.2 \[ -{\frac{Ax}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{A\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{x}^{3}B}{2\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Bxa}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Ba}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*x/b/(b*x^2+a)^(1/2)+A/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*B*x^3/b/(b*x^
2+a)^(1/2)+3/2*B*a/b^2*x/(b*x^2+a)^(1/2)-3/2*B*a/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^
(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)*x^2/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23767, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (B b x^{3} +{\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} -{\left (3 \, B a^{2} - 2 \, A a b +{\left (3 \, B a b - 2 \, A b^{2}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{b}}, \frac{{\left (B b x^{3} +{\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} -{\left (3 \, B a^{2} - 2 \, A a b +{\left (3 \, B a b - 2 \, A b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 20.5681, size = 114, normalized size = 1.37 \[ A \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + B \left (\frac{3 \sqrt{a} x}{2 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{5}{2}}} + \frac{x^{3}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

A*(asinh(sqrt(b)*x/sqrt(a))/b**(3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a))) + B*(3*
sqrt(a)*x/(2*b**2*sqrt(1 + b*x**2/a)) - 3*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(5/2)
) + x**3/(2*sqrt(a)*b*sqrt(1 + b*x**2/a)))

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GIAC/XCAS [A]  time = 0.259891, size = 95, normalized size = 1.14 \[ \frac{{\left (\frac{B x^{2}}{b} + \frac{3 \, B a b - 2 \, A b^{2}}{b^{3}}\right )} x}{2 \, \sqrt{b x^{2} + a}} + \frac{{\left (3 \, B a - 2 \, A b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*x^2/b + (3*B*a*b - 2*A*b^2)/b^3)*x/sqrt(b*x^2 + a) + 1/2*(3*B*a - 2*A*b)*
ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)