3.558 \(\int \frac{x^2 \left (A+B x^2\right )}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{a (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}+\frac{x \sqrt{a+b x^2} (4 A b-3 a B)}{8 b^2}+\frac{B x^3 \sqrt{a+b x^2}}{4 b} \]

[Out]

((4*A*b - 3*a*B)*x*Sqrt[a + b*x^2])/(8*b^2) + (B*x^3*Sqrt[a + b*x^2])/(4*b) - (a
*(4*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))

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Rubi [A]  time = 0.127204, antiderivative size = 89, 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 \[ -\frac{a (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{5/2}}+\frac{x \sqrt{a+b x^2} (4 A b-3 a B)}{8 b^2}+\frac{B x^3 \sqrt{a+b x^2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

((4*A*b - 3*a*B)*x*Sqrt[a + b*x^2])/(8*b^2) + (B*x^3*Sqrt[a + b*x^2])/(4*b) - (a
*(4*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(5/2))

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Rubi in Sympy [A]  time = 15.1568, size = 82, normalized size = 0.92 \[ \frac{B x^{3} \sqrt{a + b x^{2}}}{4 b} - \frac{a \left (4 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{5}{2}}} + \frac{x \sqrt{a + b x^{2}} \left (4 A b - 3 B a\right )}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(B*x**2+A)/(b*x**2+a)**(1/2),x)

[Out]

B*x**3*sqrt(a + b*x**2)/(4*b) - a*(4*A*b - 3*B*a)*atanh(sqrt(b)*x/sqrt(a + b*x**
2))/(8*b**(5/2)) + x*sqrt(a + b*x**2)*(4*A*b - 3*B*a)/(8*b**2)

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Mathematica [A]  time = 0.0728713, size = 77, normalized size = 0.87 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (-3 a B+4 A b+2 b B x^2\right )+a (3 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(4*A*b - 3*a*B + 2*b*B*x^2) + a*(-4*A*b + 3*a*B)*Log[
b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(5/2))

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Maple [A]  time = 0.01, size = 101, normalized size = 1.1 \[{\frac{Ax}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{{x}^{3}B}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,Bxa}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*A*x/b*(b*x^2+a)^(1/2)-1/2*A*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/4*B*x^
3*(b*x^2+a)^(1/2)/b-3/8*B*a/b^2*x*(b*x^2+a)^(1/2)+3/8*B*a^2/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/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243589, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x^{3} -{\left (3 \, B a - 4 \, A b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} -{\left (3 \, B a^{2} - 4 \, A a b\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \, b^{\frac{5}{2}}}, \frac{{\left (2 \, B b x^{3} -{\left (3 \, B a - 4 \, A b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (3 \, B a^{2} - 4 \, A a b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(2*(2*B*b*x^3 - (3*B*a - 4*A*b)*x)*sqrt(b*x^2 + a)*sqrt(b) - (3*B*a^2 - 4*
A*a*b)*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/b^(5/2), 1/8*((2*B*b*
x^3 - (3*B*a - 4*A*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) + (3*B*a^2 - 4*A*a*b)*arctan(s
qrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 19.907, size = 150, normalized size = 1.69 \[ \frac{A \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{3 B a^{\frac{3}{2}} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B \sqrt{a} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{B x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(B*x**2+A)/(b*x**2+a)**(1/2),x)

[Out]

A*sqrt(a)*x*sqrt(1 + b*x**2/a)/(2*b) - A*a*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2))
- 3*B*a**(3/2)*x/(8*b**2*sqrt(1 + b*x**2/a)) - B*sqrt(a)*x**3/(8*b*sqrt(1 + b*x
**2/a)) + 3*B*a**2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) + B*x**5/(4*sqrt(a)*sqr
t(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.246939, size = 101, normalized size = 1.13 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (\frac{2 \, B x^{2}}{b} - \frac{3 \, B a b - 4 \, A b^{2}}{b^{3}}\right )} x - \frac{{\left (3 \, B a^{2} - 4 \, A a b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(b*x^2 + a)*(2*B*x^2/b - (3*B*a*b - 4*A*b^2)/b^3)*x - 1/8*(3*B*a^2 - 4*A
*a*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)