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

Optimal. Leaf size=59 \[ A \sqrt{a+b x^2}-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{B \left (a+b x^2\right )^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.133804, antiderivative size = 59, 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 \[ A \sqrt{a+b x^2}-\sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{B \left (a+b x^2\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.6356, size = 49, normalized size = 0.83 \[ - A \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + A \sqrt{a + b x^{2}} + \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.101877, size = 70, normalized size = 1.19 \[ \frac{\sqrt{a+b x^2} \left (a B+3 A b+b B x^2\right )}{3 b}-\sqrt{a} A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} A \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 57, normalized size = 1. \[ -A\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +A\sqrt{b{x}^{2}+a}+{\frac{B}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223702, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, A \sqrt{a} b \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (B b x^{2} + B a + 3 \, A b\right )} \sqrt{b x^{2} + a}}{6 \, b}, -\frac{3 \, A \sqrt{-a} b \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (B b x^{2} + B a + 3 \, A b\right )} \sqrt{b x^{2} + a}}{3 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*A*sqrt(a)*b*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(B*b
*x^2 + B*a + 3*A*b)*sqrt(b*x^2 + a))/b, -1/3*(3*A*sqrt(-a)*b*arctan(a/(sqrt(b*x^
2 + a)*sqrt(-a))) - (B*b*x^2 + B*a + 3*A*b)*sqrt(b*x^2 + a))/b]

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Sympy [A]  time = 12.4458, size = 117, normalized size = 1.98 \[ - A a \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x^{2} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x^{2} \wedge - a < 0 \end{cases}\right ) + A \sqrt{a + b x^{2}} + \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a*Piecewise((-atan(sqrt(a + b*x**2)/sqrt(-a))/sqrt(-a), -a > 0), (acoth(sqrt(
a + b*x**2)/sqrt(a))/sqrt(a), (-a < 0) & (a < a + b*x**2)), (atanh(sqrt(a + b*x*
*2)/sqrt(a))/sqrt(a), (-a < 0) & (a > a + b*x**2))) + A*sqrt(a + b*x**2) + B*(a
+ b*x**2)**(3/2)/(3*b)

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GIAC/XCAS [A]  time = 0.225121, size = 81, normalized size = 1.37 \[ \frac{A a \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} B b^{2} + 3 \, \sqrt{b x^{2} + a} A b^{3}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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