3.47 \(\int \sqrt{a+b x^2} \left (c+d x^2\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0789974, 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 \[ \frac{a (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{x \sqrt{a+b x^2} (4 b c-a d)}{8 b}+\frac{d x \left (a+b x^2\right )^{3/2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.62465, size = 75, normalized size = 0.86 \[ - \frac{a \left (a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{3}{2}}} + \frac{d x \left (a + b x^{2}\right )^{\frac{3}{2}}}{4 b} - \frac{x \sqrt{a + b x^{2}} \left (a d - 4 b c\right )}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)*(d*x**2+c),x)

[Out]

-a*(a*d - 4*b*c)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(8*b**(3/2)) + d*x*(a + b*x**
2)**(3/2)/(4*b) - x*sqrt(a + b*x**2)*(a*d - 4*b*c)/(8*b)

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Mathematica [A]  time = 0.0657133, size = 78, normalized size = 0.9 \[ \sqrt{a+b x^2} \left (\frac{x (a d+4 b c)}{8 b}+\frac{d x^3}{4}\right )-\frac{a (a d-4 b c) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 96, normalized size = 1.1 \[{\frac{cx}{2}\sqrt{b{x}^{2}+a}}+{\frac{ac}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{dx}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{adx}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{2}d}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232479, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b d x^{3} +{\left (4 \, b c + a d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} -{\left (4 \, a b c - a^{2} d\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \, b^{\frac{3}{2}}}, \frac{{\left (2 \, b d x^{3} +{\left (4 \, b c + a d\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} +{\left (4 \, a b c - a^{2} d\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.1212, size = 144, normalized size = 1.66 \[ \frac{a^{\frac{3}{2}} d x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 \sqrt{a} d x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{b}} + \frac{b d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)*(d*x**2+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.232104, size = 95, normalized size = 1.09 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (2 \, d x^{2} + \frac{4 \, b^{2} c + a b d}{b^{2}}\right )} x - \frac{{\left (4 \, a b c - a^{2} d\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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