3.529 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx\)

Optimal. Leaf size=110 \[ \frac{\left (a+b x^2\right )^{3/2} (2 a B+3 A b)}{6 a}+\frac{1}{2} \sqrt{a+b x^2} (2 a B+3 A b)-\frac{1}{2} \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{A \left (a+b x^2\right )^{5/2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.21969, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (a+b x^2\right )^{3/2} (2 a B+3 A b)}{6 a}+\frac{1}{2} \sqrt{a+b x^2} (2 a B+3 A b)-\frac{1}{2} \sqrt{a} (2 a B+3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )-\frac{A \left (a+b x^2\right )^{5/2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.9377, size = 94, normalized size = 0.85 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{2 a x^{2}} - \sqrt{a} \left (\frac{3 A b}{2} + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + \sqrt{a + b x^{2}} \left (\frac{3 A b}{2} + B a\right ) + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (\frac{3 A b}{2} + B a\right )}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.189435, size = 100, normalized size = 0.91 \[ \frac{1}{6} \left (-3 \sqrt{a} (2 a B+3 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\frac{\sqrt{a+b x^2} \left (-3 a A+8 a B x^2+6 A b x^2+2 b B x^4\right )}{x^2}+3 \sqrt{a} \log (x) (2 a B+3 A b)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^3,x]

[Out]

((Sqrt[a + b*x^2]*(-3*a*A + 6*A*b*x^2 + 8*a*B*x^2 + 2*b*B*x^4))/x^2 + 3*Sqrt[a]*
(3*A*b + 2*a*B)*Log[x] - 3*Sqrt[a]*(3*A*b + 2*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^
2]])/6

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Maple [A]  time = 0.013, size = 132, normalized size = 1.2 \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Ab}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{3\,Ab}{2}\sqrt{b{x}^{2}+a}}+{\frac{B}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-B{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +B\sqrt{b{x}^{2}+a}a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^3,x)

[Out]

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

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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)*(b*x^2 + a)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230479, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{a} x^{2} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, B b x^{4} + 2 \,{\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt{b x^{2} + a}}{12 \, x^{2}}, -\frac{3 \,{\left (2 \, B a + 3 \, A b\right )} \sqrt{-a} x^{2} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) -{\left (2 \, B b x^{4} + 2 \,{\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt{b x^{2} + a}}{6 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^3,x, algorithm="fricas")

[Out]

[1/12*(3*(2*B*a + 3*A*b)*sqrt(a)*x^2*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2
*a)/x^2) + 2*(2*B*b*x^4 + 2*(4*B*a + 3*A*b)*x^2 - 3*A*a)*sqrt(b*x^2 + a))/x^2, -
1/6*(3*(2*B*a + 3*A*b)*sqrt(-a)*x^2*arctan(a/(sqrt(b*x^2 + a)*sqrt(-a))) - (2*B*
b*x^4 + 2*(4*B*a + 3*A*b)*x^2 - 3*A*a)*sqrt(b*x^2 + a))/x^2]

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Sympy [A]  time = 42.7624, size = 184, normalized size = 1.67 \[ - \frac{3 A \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{A a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{A a \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} - B a^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{B a^{2}}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B a \sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} + B b \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**3,x)

[Out]

-3*A*sqrt(a)*b*asinh(sqrt(a)/(sqrt(b)*x))/2 - A*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(
2*x) + A*a*sqrt(b)/(x*sqrt(a/(b*x**2) + 1)) + A*b**(3/2)*x/sqrt(a/(b*x**2) + 1)
- B*a**(3/2)*asinh(sqrt(a)/(sqrt(b)*x)) + B*a**2/(sqrt(b)*x*sqrt(a/(b*x**2) + 1)
) + B*a*sqrt(b)*x/sqrt(a/(b*x**2) + 1) + B*b*Piecewise((sqrt(a)*x**2/2, Eq(b, 0)
), ((a + b*x**2)**(3/2)/(3*b), True))

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GIAC/XCAS [A]  time = 0.233839, size = 139, normalized size = 1.26 \[ \frac{2 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B b + 6 \, \sqrt{b x^{2} + a} B a b + 6 \, \sqrt{b x^{2} + a} A b^{2} - \frac{3 \, \sqrt{b x^{2} + a} A a b}{x^{2}} + \frac{3 \,{\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^3,x, algorithm="giac")

[Out]

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