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

Optimal. Leaf size=72 \[ -\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A}{a^2 \sqrt{a+b x^2}}+\frac{A b-a B}{3 a b \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.155064, antiderivative size = 72, 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 \[ -\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A}{a^2 \sqrt{a+b x^2}}+\frac{A b-a B}{3 a b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.2788, size = 60, normalized size = 0.83 \[ \frac{A}{a^{2} \sqrt{a + b x^{2}}} - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} + \frac{A b - B a}{3 a b \left (a + b x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.260301, size = 79, normalized size = 1.1 \[ \frac{\frac{\sqrt{a} \left (a^2 (-B)+4 a A b+3 A b^2 x^2\right )}{b \left (a+b x^2\right )^{3/2}}-3 A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+3 A \log (x)}{3 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 75, normalized size = 1. \[{\frac{A}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{A}{{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2275, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \, A b^{2} x^{2} - B a^{2} + 4 \, A a b\right )} \sqrt{b x^{2} + a} \sqrt{a} + 3 \,{\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{6 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{a}}, \frac{{\left (3 \, A b^{2} x^{2} - B a^{2} + 4 \, A a b\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 62.4018, size = 790, normalized size = 10.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(8*a**7*sqrt(1 + b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b
**2*x**4 + 6*a**(13/2)*b**3*x**6) + 3*a**7*log(b*x**2/a)/(6*a**(19/2) + 18*a**(1
7/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 6*a**7*log(sqrt(
1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 +
6*a**(13/2)*b**3*x**6) + 14*a**6*b*x**2*sqrt(1 + b*x**2/a)/(6*a**(19/2) + 18*a*
*(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 9*a**6*b*x**2
*log(b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a
**(13/2)*b**3*x**6) - 18*a**6*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) +
18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 6*a**5*b
**2*x**4*sqrt(1 + b*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b*
*2*x**4 + 6*a**(13/2)*b**3*x**6) + 9*a**5*b**2*x**4*log(b*x**2/a)/(6*a**(19/2) +
18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) - 18*a**5
*b**2*x**4*log(sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a
**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6) + 3*a**4*b**3*x**6*log(b*x**2/a)/(6*
a**(19/2) + 18*a**(17/2)*b*x**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6
) - 6*a**4*b**3*x**6*log(sqrt(1 + b*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*b*x
**2 + 18*a**(15/2)*b**2*x**4 + 6*a**(13/2)*b**3*x**6)) + B*Piecewise((-1/(3*a*b*
sqrt(a + b*x**2) + 3*b**2*x**2*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(5/2)),
True))

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GIAC/XCAS [A]  time = 0.236707, size = 89, normalized size = 1.24 \[ \frac{A \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{B a^{2} - 3 \,{\left (b x^{2} + a\right )} A b - A a b}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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