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

Optimal. Leaf size=109 \[ \frac{x \left (a+b x^2\right )^{3/2} (a B+4 A b)}{4 a}+\frac{3}{8} x \sqrt{a+b x^2} (a B+4 A b)+\frac{3 a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b}}-\frac{A \left (a+b x^2\right )^{5/2}}{a x} \]

[Out]

(3*(4*A*b + a*B)*x*Sqrt[a + b*x^2])/8 + ((4*A*b + a*B)*x*(a + b*x^2)^(3/2))/(4*a
) - (A*(a + b*x^2)^(5/2))/(a*x) + (3*a*(4*A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a
+ b*x^2]])/(8*Sqrt[b])

_______________________________________________________________________________________

Rubi [A]  time = 0.125961, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \left (a+b x^2\right )^{3/2} (a B+4 A b)}{4 a}+\frac{3}{8} x \sqrt{a+b x^2} (a B+4 A b)+\frac{3 a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b}}-\frac{A \left (a+b x^2\right )^{5/2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

(3*(4*A*b + a*B)*x*Sqrt[a + b*x^2])/8 + ((4*A*b + a*B)*x*(a + b*x^2)^(3/2))/(4*a
) - (A*(a + b*x^2)^(5/2))/(a*x) + (3*a*(4*A*b + a*B)*ArcTanh[(Sqrt[b]*x)/Sqrt[a
+ b*x^2]])/(8*Sqrt[b])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.6289, size = 100, normalized size = 0.92 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{a x} + \frac{3 a \left (4 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 \sqrt{b}} + x \sqrt{a + b x^{2}} \left (\frac{3 A b}{2} + \frac{3 B a}{8}\right ) + \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (4 A b + B a\right )}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.125061, size = 84, normalized size = 0.77 \[ \frac{3 a (a B+4 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 \sqrt{b}}+\sqrt{a+b x^2} \left (\frac{1}{8} x (5 a B+4 A b)-\frac{a A}{x}+\frac{1}{4} b B x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*(-((a*A)/x) + ((4*A*b + 5*a*B)*x)/8 + (b*B*x^3)/4) + (3*a*(4*A*b
+ a*B)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*Sqrt[b])

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 125, normalized size = 1.2 \[{\frac{Bx}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bxa}{8}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}B}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}-{\frac{A}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Axb}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Axb}{2}\sqrt{b{x}^{2}+a}}+{\frac{3\,Aa}{2}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229091, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} x \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, B b x^{4} +{\left (5 \, B a + 4 \, A b\right )} x^{2} - 8 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{b}}{16 \, \sqrt{b} x}, \frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (2 \, B b x^{4} +{\left (5 \, B a + 4 \, A b\right )} x^{2} - 8 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{8 \, \sqrt{-b} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(B*a^2 + 4*A*a*b)*x*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b))
+ 2*(2*B*b*x^4 + (5*B*a + 4*A*b)*x^2 - 8*A*a)*sqrt(b*x^2 + a)*sqrt(b))/(sqrt(b)
*x), 1/8*(3*(B*a^2 + 4*A*a*b)*x*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (2*B*b*x^4
+ (5*B*a + 4*A*b)*x^2 - 8*A*a)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*x)]

_______________________________________________________________________________________

Sympy [A]  time = 26.9625, size = 216, normalized size = 1.98 \[ - \frac{A a^{\frac{3}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{A \sqrt{a} b x \sqrt{1 + \frac{b x^{2}}{a}}}{2} - \frac{A \sqrt{a} b x}{\sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} + \frac{B a^{\frac{3}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{B a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B \sqrt{a} b x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{b}} + \frac{B b^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.246829, size = 154, normalized size = 1.41 \[ \frac{2 \, A a^{2} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} + \frac{1}{8} \,{\left (2 \, B b x^{2} + \frac{5 \, B a b^{2} + 4 \, A b^{3}}{b^{2}}\right )} \sqrt{b x^{2} + a} x - \frac{3 \,{\left (B a^{2} \sqrt{b} + 4 \, A a b^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{16 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a^2*sqrt(b)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a) + 1/8*(2*B*b*x^2 + (5*B*a*
b^2 + 4*A*b^3)/b^2)*sqrt(b*x^2 + a)*x - 3/16*(B*a^2*sqrt(b) + 4*A*a*b^(3/2))*ln(
(sqrt(b)*x - sqrt(b*x^2 + a))^2)/b