3.580 \(\int \frac{A+B x^2}{x^5 \left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=120 \[ -\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{3 \sqrt{a+b x^2} (5 A b-4 a B)}{8 a^3 x^2}-\frac{5 A b-4 a B}{4 a^2 x^2 \sqrt{a+b x^2}}-\frac{A}{4 a x^4 \sqrt{a+b x^2}} \]

[Out]

-A/(4*a*x^4*Sqrt[a + b*x^2]) - (5*A*b - 4*a*B)/(4*a^2*x^2*Sqrt[a + b*x^2]) + (3*
(5*A*b - 4*a*B)*Sqrt[a + b*x^2])/(8*a^3*x^2) - (3*b*(5*A*b - 4*a*B)*ArcTanh[Sqrt
[a + b*x^2]/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.242107, antiderivative size = 120, 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 \[ -\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{3 \sqrt{a+b x^2} (5 A b-4 a B)}{8 a^3 x^2}-\frac{5 A b-4 a B}{4 a^2 x^2 \sqrt{a+b x^2}}-\frac{A}{4 a x^4 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)),x]

[Out]

-A/(4*a*x^4*Sqrt[a + b*x^2]) - (5*A*b - 4*a*B)/(4*a^2*x^2*Sqrt[a + b*x^2]) + (3*
(5*A*b - 4*a*B)*Sqrt[a + b*x^2])/(8*a^3*x^2) - (3*b*(5*A*b - 4*a*B)*ArcTanh[Sqrt
[a + b*x^2]/Sqrt[a]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 20.4755, size = 114, normalized size = 0.95 \[ - \frac{A}{4 a x^{4} \sqrt{a + b x^{2}}} - \frac{5 A b - 4 B a}{4 a^{2} x^{2} \sqrt{a + b x^{2}}} + \frac{3 \sqrt{a + b x^{2}} \left (5 A b - 4 B a\right )}{8 a^{3} x^{2}} - \frac{3 b \left (5 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**5/(b*x**2+a)**(3/2),x)

[Out]

-A/(4*a*x**4*sqrt(a + b*x**2)) - (5*A*b - 4*B*a)/(4*a**2*x**2*sqrt(a + b*x**2))
+ 3*sqrt(a + b*x**2)*(5*A*b - 4*B*a)/(8*a**3*x**2) - 3*b*(5*A*b - 4*B*a)*atanh(s
qrt(a + b*x**2)/sqrt(a))/(8*a**(7/2))

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Mathematica [A]  time = 0.275984, size = 115, normalized size = 0.96 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 \left (A+2 B x^2\right )+a b x^2 \left (5 A-12 B x^2\right )+15 A b^2 x^4\right )}{x^4 \sqrt{a+b x^2}}+3 b (4 a B-5 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+3 b \log (x) (5 A b-4 a B)}{8 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)),x]

[Out]

((Sqrt[a]*(15*A*b^2*x^4 + a*b*x^2*(5*A - 12*B*x^2) - 2*a^2*(A + 2*B*x^2)))/(x^4*
Sqrt[a + b*x^2]) + 3*b*(5*A*b - 4*a*B)*Log[x] + 3*b*(-5*A*b + 4*a*B)*Log[a + Sqr
t[a]*Sqrt[a + b*x^2]])/(8*a^(7/2))

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Maple [A]  time = 0.015, size = 153, normalized size = 1.3 \[ -{\frac{A}{4\,a{x}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Ab}{8\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,{b}^{2}A}{8\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{B}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Bb}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^5/(b*x^2+a)^(3/2),x)

[Out]

-1/4*A/a/x^4/(b*x^2+a)^(1/2)+5/8*A*b/a^2/x^2/(b*x^2+a)^(1/2)+15/8*A*b^2/a^3/(b*x
^2+a)^(1/2)-15/8*A*b^2/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/2*B/a/x^2
/(b*x^2+a)^(1/2)-3/2*B*b/a^2/(b*x^2+a)^(1/2)+3/2*B*b/a^(5/2)*ln((2*a+2*a^(1/2)*(
b*x^2+a)^(1/2))/x)

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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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245352, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} + 3 \,{\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} +{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{16 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )} \sqrt{a}}, -\frac{{\left (3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} +{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{8 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^5),x, algorithm="fricas")

[Out]

[-1/16*(2*(3*(4*B*a*b - 5*A*b^2)*x^4 + 2*A*a^2 + (4*B*a^2 - 5*A*a*b)*x^2)*sqrt(b
*x^2 + a)*sqrt(a) + 3*((4*B*a*b^2 - 5*A*b^3)*x^6 + (4*B*a^2*b - 5*A*a*b^2)*x^4)*
log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a)*a)/x^2))/((a^3*b*x^6 + a^4*x^4)*
sqrt(a)), -1/8*((3*(4*B*a*b - 5*A*b^2)*x^4 + 2*A*a^2 + (4*B*a^2 - 5*A*a*b)*x^2)*
sqrt(b*x^2 + a)*sqrt(-a) - 3*((4*B*a*b^2 - 5*A*b^3)*x^6 + (4*B*a^2*b - 5*A*a*b^2
)*x^4)*arctan(sqrt(-a)/sqrt(b*x^2 + a)))/((a^3*b*x^6 + a^4*x^4)*sqrt(-a))]

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Sympy [A]  time = 56.0351, size = 180, normalized size = 1.5 \[ A \left (- \frac{1}{4 a \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 \sqrt{b}}{8 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{15 b^{\frac{3}{2}}}{8 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{7}{2}}}\right ) + B \left (- \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**5/(b*x**2+a)**(3/2),x)

[Out]

A*(-1/(4*a*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + 5*sqrt(b)/(8*a**2*x**3*sqrt(a/(b
*x**2) + 1)) + 15*b**(3/2)/(8*a**3*x*sqrt(a/(b*x**2) + 1)) - 15*b**2*asinh(sqrt(
a)/(sqrt(b)*x))/(8*a**(7/2))) + B*(-1/(2*a*sqrt(b)*x**3*sqrt(a/(b*x**2) + 1)) -
3*sqrt(b)/(2*a**2*x*sqrt(a/(b*x**2) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*x))/(2*a*
*(5/2)))

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GIAC/XCAS [A]  time = 0.233742, size = 185, normalized size = 1.54 \[ -\frac{3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{3}} - \frac{B a b - A b^{2}}{\sqrt{b x^{2} + a} a^{3}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x^{2} + a} B a^{2} b - 7 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{2} + 9 \, \sqrt{b x^{2} + a} A a b^{2}}{8 \, a^{3} b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^5),x, algorithm="giac")

[Out]

-3/8*(4*B*a*b - 5*A*b^2)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^3) - (B*a*
b - A*b^2)/(sqrt(b*x^2 + a)*a^3) - 1/8*(4*(b*x^2 + a)^(3/2)*B*a*b - 4*sqrt(b*x^2
+ a)*B*a^2*b - 7*(b*x^2 + a)^(3/2)*A*b^2 + 9*sqrt(b*x^2 + a)*A*a*b^2)/(a^3*b^2*
x^4)