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

Optimal. Leaf size=87 \[ \frac{a (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 A b-a B)}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0817528, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{a (4 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 A b-a B)}{8 b}+\frac{B x \left (a+b x^2\right )^{3/2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.79674, size = 75, normalized size = 0.86 \[ \frac{B x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4 b} + \frac{a \left (4 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{3}{2}}} + \frac{x \sqrt{a + b x^{2}} \left (4 A b - B a\right )}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0695272, size = 78, normalized size = 0.9 \[ \sqrt{a+b x^2} \left (\frac{x (a B+4 A b)}{8 b}+\frac{B x^3}{4}\right )-\frac{a (a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 96, normalized size = 1.1 \[{\frac{Ax}{2}\sqrt{b{x}^{2}+a}}+{\frac{Aa}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{Bx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{Bxa}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225792, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x^{3} +{\left (B a + 4 \, A b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} -{\left (B a^{2} - 4 \, A a b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \, b^{\frac{3}{2}}}, \frac{{\left (2 \, B b x^{3} +{\left (B a + 4 \, A b\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} -{\left (B a^{2} - 4 \, A a b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.382, size = 144, normalized size = 1.66 \[ \frac{A \sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{B a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{B b x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235181, size = 93, normalized size = 1.07 \[ \frac{1}{8} \,{\left (2 \, B x^{2} + \frac{B a b + 4 \, A b^{2}}{b^{2}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (B a^{2} - 4 \, A a b\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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