3.530 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^4} \, dx\)

Optimal. Leaf size=119 \[ -\frac{\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac{b x \sqrt{a+b x^2} (3 a B+2 A b)}{2 a}+\frac{1}{2} \sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.141663, antiderivative size = 119, 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 \[ -\frac{\left (a+b x^2\right )^{3/2} (3 a B+2 A b)}{3 a x}+\frac{b x \sqrt{a+b x^2} (3 a B+2 A b)}{2 a}+\frac{1}{2} \sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{A \left (a+b x^2\right )^{5/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.9124, size = 105, normalized size = 0.88 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{3 a x^{3}} + \frac{\sqrt{b} \left (2 A b + 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2} + \frac{b x \sqrt{a + b x^{2}} \left (2 A b + 3 B a\right )}{2 a} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b + 3 B a\right )}{3 a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.100132, size = 86, normalized size = 0.72 \[ \frac{1}{2} \sqrt{b} (3 a B+2 A b) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{a+b x^2} \left (\frac{-3 a B-4 A b}{3 x}-\frac{a A}{3 x^3}+\frac{b B x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^4,x]

[Out]

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

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Maple [A]  time = 0.013, size = 168, normalized size = 1.4 \[ -{\frac{A}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Ab}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,Ax{b}^{2}}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ax{b}^{2}}{a}\sqrt{b{x}^{2}+a}}+A{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) -{\frac{B}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{bBx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,bBx}{2}\sqrt{b{x}^{2}+a}}+{\frac{3\,Ba}{2}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/x^4,x)

[Out]

-1/3*A*(b*x^2+a)^(5/2)/a/x^3-2/3*A*b/a^2/x*(b*x^2+a)^(5/2)+2/3*A*b^2/a^2*x*(b*x^
2+a)^(3/2)+A*b^2/a*x*(b*x^2+a)^(1/2)+A*b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-B/a
/x*(b*x^2+a)^(5/2)+B*b/a*x*(b*x^2+a)^(3/2)+3/2*B*b*x*(b*x^2+a)^(1/2)+3/2*B*b^(1/
2)*a*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)*(b*x^2 + a)^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228589, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (3 \, B b x^{4} - 2 \,{\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a}}{12 \, x^{3}}, \frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{-b} x^{3} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) +{\left (3 \, B b x^{4} - 2 \,{\left (3 \, B a + 4 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a}}{6 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 17.0085, size = 202, normalized size = 1.7 \[ - \frac{A \sqrt{a} b}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} + A b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{A b^{2} x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{3}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B \sqrt{a} b x \sqrt{1 + \frac{b x^{2}}{a}}}{2} - \frac{B \sqrt{a} b x}{\sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**4,x)

[Out]

-A*sqrt(a)*b/(x*sqrt(1 + b*x**2/a)) - A*a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*x**2)
- A*b**(3/2)*sqrt(a/(b*x**2) + 1)/3 + A*b**(3/2)*asinh(sqrt(b)*x/sqrt(a)) - A*b*
*2*x/(sqrt(a)*sqrt(1 + b*x**2/a)) - B*a**(3/2)/(x*sqrt(1 + b*x**2/a)) + B*sqrt(a
)*b*x*sqrt(1 + b*x**2/a)/2 - B*sqrt(a)*b*x/sqrt(1 + b*x**2/a) + 3*B*a*sqrt(b)*as
inh(sqrt(b)*x/sqrt(a))/2

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GIAC/XCAS [A]  time = 0.242217, size = 279, normalized size = 2.34 \[ \frac{1}{2} \, \sqrt{b x^{2} + a} B b x - \frac{1}{4} \,{\left (3 \, B a \sqrt{b} + 2 \, A b^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{2} \sqrt{b} + 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a b^{\frac{3}{2}} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{3} \sqrt{b} - 6 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{2} b^{\frac{3}{2}} + 3 \, B a^{4} \sqrt{b} + 4 \, A a^{3} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^4,x, algorithm="giac")

[Out]

1/2*sqrt(b*x^2 + a)*B*b*x - 1/4*(3*B*a*sqrt(b) + 2*A*b^(3/2))*ln((sqrt(b)*x - sq
rt(b*x^2 + a))^2) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*sqrt(b) + 6*(sq
rt(b)*x - sqrt(b*x^2 + a))^4*A*a*b^(3/2) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a
^3*sqrt(b) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^2*b^(3/2) + 3*B*a^4*sqrt(b) +
4*A*a^3*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3