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

Optimal. Leaf size=122 \[ -\frac{a^2 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}+\frac{a x \sqrt{a+b x^2} (2 A b-a B)}{16 b^2}+\frac{x^3 \sqrt{a+b x^2} (2 A b-a B)}{8 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b} \]

[Out]

(a*(2*A*b - a*B)*x*Sqrt[a + b*x^2])/(16*b^2) + ((2*A*b - a*B)*x^3*Sqrt[a + b*x^2
])/(8*b) + (B*x^3*(a + b*x^2)^(3/2))/(6*b) - (a^2*(2*A*b - a*B)*ArcTanh[(Sqrt[b]
*x)/Sqrt[a + b*x^2]])/(16*b^(5/2))

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Rubi [A]  time = 0.182169, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{a^2 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{5/2}}+\frac{a x \sqrt{a+b x^2} (2 A b-a B)}{16 b^2}+\frac{x^3 \sqrt{a+b x^2} (2 A b-a B)}{8 b}+\frac{B x^3 \left (a+b x^2\right )^{3/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(a*(2*A*b - a*B)*x*Sqrt[a + b*x^2])/(16*b^2) + ((2*A*b - a*B)*x^3*Sqrt[a + b*x^2
])/(8*b) + (B*x^3*(a + b*x^2)^(3/2))/(6*b) - (a^2*(2*A*b - a*B)*ArcTanh[(Sqrt[b]
*x)/Sqrt[a + b*x^2]])/(16*b^(5/2))

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Rubi in Sympy [A]  time = 19.4402, size = 107, normalized size = 0.88 \[ \frac{B x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{6 b} - \frac{a^{2} \left (2 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{5}{2}}} + \frac{a x \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{16 b^{2}} + \frac{x^{3} \sqrt{a + b x^{2}} \left (2 A b - B a\right )}{8 b} \]

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*(a + b*x**2)**(3/2)/(6*b) - a**2*(2*A*b - B*a)*atanh(sqrt(b)*x/sqrt(a + b
*x**2))/(16*b**(5/2)) + a*x*sqrt(a + b*x**2)*(2*A*b - B*a)/(16*b**2) + x**3*sqrt
(a + b*x**2)*(2*A*b - B*a)/(8*b)

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Mathematica [A]  time = 0.0965216, size = 99, normalized size = 0.81 \[ \frac{a^2 (a B-2 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{a x (a B-2 A b)}{16 b^2}+\frac{x^3 (a B+6 A b)}{24 b}+\frac{B x^5}{6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*(-(a*(-2*A*b + a*B)*x)/(16*b^2) + ((6*A*b + a*B)*x^3)/(24*b) + (
B*x^5)/6) + (a^2*(-2*A*b + a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(16*b^(5/2))

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Maple [A]  time = 0.01, size = 139, normalized size = 1.1 \[{\frac{Ax}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{aAx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}+{\frac{B{x}^{3}}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bxa}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bx{a}^{2}}{16\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{B{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

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]

1/4*A*x*(b*x^2+a)^(3/2)/b-1/8*A*a/b*x*(b*x^2+a)^(1/2)-1/8*A*a^2/b^(3/2)*ln(x*b^(
1/2)+(b*x^2+a)^(1/2))+1/6*B*x^3*(b*x^2+a)^(3/2)/b-1/8*B*a/b^2*x*(b*x^2+a)^(3/2)+
1/16*B*a^2/b^2*x*(b*x^2+a)^(1/2)+1/16*B*a^3/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)*sqrt(b*x^2 + a)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257115, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{5} + 2 \,{\left (B a b + 6 \, A b^{2}\right )} x^{3} - 3 \,{\left (B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{5}{2}}}, \frac{{\left (8 \, B b^{2} x^{5} + 2 \,{\left (B a b + 6 \, A b^{2}\right )} x^{3} - 3 \,{\left (B a^{2} - 2 \, A a b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*B*b^2*x^5 + 2*(B*a*b + 6*A*b^2)*x^3 - 3*(B*a^2 - 2*A*a*b)*x)*sqrt(b*
x^2 + a)*sqrt(b) - 3*(B*a^3 - 2*A*a^2*b)*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 +
a)*sqrt(b)))/b^(5/2), 1/48*((8*B*b^2*x^5 + 2*(B*a*b + 6*A*b^2)*x^3 - 3*(B*a^2 -
2*A*a*b)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 3*(B*a^3 - 2*A*a^2*b)*arctan(sqrt(-b)*x/s
qrt(b*x^2 + a)))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 29.6152, size = 226, normalized size = 1.85 \[ \frac{A a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{A b x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x}{16 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{3}{2}} x^{3}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 B \sqrt{a} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{5}{2}}} + \frac{B b x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

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*a**(3/2)*x/(8*b*sqrt(1 + b*x**2/a)) + 3*A*sqrt(a)*x**3/(8*sqrt(1 + b*x**2/a))
- A*a**2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(3/2)) + A*b*x**5/(4*sqrt(a)*sqrt(1 + b*
x**2/a)) - B*a**(5/2)*x/(16*b**2*sqrt(1 + b*x**2/a)) - B*a**(3/2)*x**3/(48*b*sqr
t(1 + b*x**2/a)) + 5*B*sqrt(a)*x**5/(24*sqrt(1 + b*x**2/a)) + B*a**3*asinh(sqrt(
b)*x/sqrt(a))/(16*b**(5/2)) + B*b*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.252508, size = 135, normalized size = 1.11 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, B x^{2} + \frac{B a b^{3} + 6 \, A b^{4}}{b^{4}}\right )} x^{2} - \frac{3 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )}}{b^{4}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (B a^{3} - 2 \, A a^{2} b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*B*x^2 + (B*a*b^3 + 6*A*b^4)/b^4)*x^2 - 3*(B*a^2*b^2 - 2*A*a*b^3)/b^4)
*sqrt(b*x^2 + a)*x - 1/16*(B*a^3 - 2*A*a^2*b)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a
)))/b^(5/2)