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

Optimal. Leaf size=76 \[ -\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}+\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.218707, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 A+2 B x}{3 a^2 \sqrt{a+b x^2}}+\frac{A+B x}{3 a \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.6381, size = 65, normalized size = 0.86 \[ - \frac{A \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} + \frac{A + B x}{3 a \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{3 A + 2 B x}{3 a^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.296461, size = 79, normalized size = 1.04 \[ \frac{\frac{\sqrt{a} \left (4 a A+3 a B x+3 A b x^2+2 b B x^3\right )}{\left (a+b x^2\right )^{3/2}}-3 A \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+3 A \log (x)}{3 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(4*a*A + 3*a*B*x + 3*A*b*x^2 + 2*b*B*x^3))/(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 = 0.008, size = 92, normalized size = 1.2 \[{\frac{Bx}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bx}{3\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{A}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{A}{{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{A\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*x/a/(b*x^2+a)^(3/2)+2/3*B/a^2*x/(b*x^2+a)^(1/2)+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)

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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 + A)/((b*x^2 + a)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256535, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x^{3} + 3 \, A b x^{2} + 3 \, B a x + 4 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{a} + 3 \,{\left (A b^{2} x^{4} + 2 \, A a b x^{2} + A a^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{6 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{a}}, \frac{{\left (2 \, B b x^{3} + 3 \, A b x^{2} + 3 \, B a x + 4 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left (A b^{2} x^{4} + 2 \, A a b x^{2} + A a^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 29.0716, size = 840, normalized size = 11.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+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*(3*a*x/(3*a**(7/2)*sq
rt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)) + 2*b*x**3/(3*a**(7/2)*
sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)))

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GIAC/XCAS [A]  time = 0.221261, size = 111, normalized size = 1.46 \[ \frac{{\left ({\left (\frac{2 \, B b x}{a^{2}} + \frac{3 \, A b}{a^{2}}\right )} x + \frac{3 \, B}{a}\right )} x + \frac{4 \, A}{a}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} + \frac{2 \, A \arctan \left (-\frac{\sqrt{b} x - \sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(((2*B*b*x/a^2 + 3*A*b/a^2)*x + 3*B/a)*x + 4*A/a)/(b*x^2 + a)^(3/2) + 2*A*ar
ctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2)