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

Optimal. Leaf size=104 \[ -\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{8 A \sqrt{a+c x^2}}{3 a^3 x}+\frac{4 A+3 B x}{3 a^2 x \sqrt{a+c x^2}}+\frac{A+B x}{3 a x \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.265494, antiderivative size = 104, 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 \[ -\frac{B \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{8 A \sqrt{a+c x^2}}{3 a^3 x}+\frac{4 A+3 B x}{3 a^2 x \sqrt{a+c x^2}}+\frac{A+B x}{3 a x \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.7351, size = 88, normalized size = 0.85 \[ - \frac{8 A \sqrt{a + c x^{2}}}{3 a^{3} x} - \frac{B \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} + \frac{A + B x}{3 a x \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 A + 3 B x}{3 a^{2} x \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.205747, size = 95, normalized size = 0.91 \[ -\frac{B \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{5/2}}+\frac{B \log (x)}{a^{5/2}}+\frac{a^2 (4 B x-3 A)+3 a c x^2 (B x-4 A)-8 A c^2 x^4}{3 a^3 x \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 112, normalized size = 1.1 \[ -{\frac{A}{ax} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{4\,Acx}{3\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{8\,Acx}{3\,{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{B}{3\,a} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{B}{{a}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/a/x/(c*x^2+a)^(3/2)-4/3*A/a^2*c*x/(c*x^2+a)^(3/2)-8/3*A/a^3*c*x/(c*x^2+a)^(1/
2)+1/3*B/a/(c*x^2+a)^(3/2)+B/a^2/(c*x^2+a)^(1/2)-B/a^(5/2)*ln((2*a+2*a^(1/2)*(c*
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)/((c*x^2 + a)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.308557, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (8 \, A c^{2} x^{4} - 3 \, B a c x^{3} + 12 \, A a c x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a} - 3 \,{\left (B a c^{2} x^{5} + 2 \, B a^{2} c x^{3} + B a^{3} x\right )} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right )}{6 \,{\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt{a}}, -\frac{{\left (8 \, A c^{2} x^{4} - 3 \, B a c x^{3} + 12 \, A a c x^{2} - 4 \, B a^{2} x + 3 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a} + 3 \,{\left (B a c^{2} x^{5} + 2 \, B a^{2} c x^{3} + B a^{3} x\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right )}{3 \,{\left (a^{3} c^{2} x^{5} + 2 \, a^{4} c x^{3} + a^{5} x\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(2*(8*A*c^2*x^4 - 3*B*a*c*x^3 + 12*A*a*c*x^2 - 4*B*a^2*x + 3*A*a^2)*sqrt(c
*x^2 + a)*sqrt(a) - 3*(B*a*c^2*x^5 + 2*B*a^2*c*x^3 + B*a^3*x)*log(-((c*x^2 + 2*a
)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2))/((a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt
(a)), -1/3*((8*A*c^2*x^4 - 3*B*a*c*x^3 + 12*A*a*c*x^2 - 4*B*a^2*x + 3*A*a^2)*sqr
t(c*x^2 + a)*sqrt(-a) + 3*(B*a*c^2*x^5 + 2*B*a^2*c*x^3 + B*a^3*x)*arctan(sqrt(-a
)/sqrt(c*x^2 + a)))/((a^3*c^2*x^5 + 2*a^4*c*x^3 + a^5*x)*sqrt(-a))]

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Sympy [A]  time = 80.8726, size = 910, normalized size = 8.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-3*a**2*c**(9/2)*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c**5*x**2 + 3*a**
3*c**6*x**4) - 12*a*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4 + 6*a**4*c*
*5*x**2 + 3*a**3*c**6*x**4) - 8*c**(13/2)*x**4*sqrt(a/(c*x**2) + 1)/(3*a**5*c**4
+ 6*a**4*c**5*x**2 + 3*a**3*c**6*x**4)) + B*(8*a**7*sqrt(1 + c*x**2/a)/(6*a**(1
9/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 3
*a**7*log(c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4
+ 6*a**(13/2)*c**3*x**6) - 6*a**7*log(sqrt(1 + c*x**2/a) + 1)/(6*a**(19/2) + 18*
a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 14*a**6*c*x
**2*sqrt(1 + c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x*
*4 + 6*a**(13/2)*c**3*x**6) + 9*a**6*c*x**2*log(c*x**2/a)/(6*a**(19/2) + 18*a**(
17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) - 18*a**6*c*x**2*
log(sqrt(1 + c*x**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c*
*2*x**4 + 6*a**(13/2)*c**3*x**6) + 6*a**5*c**2*x**4*sqrt(1 + c*x**2/a)/(6*a**(19
/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) + 9*
a**5*c**2*x**4*log(c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c
**2*x**4 + 6*a**(13/2)*c**3*x**6) - 18*a**5*c**2*x**4*log(sqrt(1 + c*x**2/a) + 1
)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3
*x**6) + 3*a**4*c**3*x**6*log(c*x**2/a)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*
a**(15/2)*c**2*x**4 + 6*a**(13/2)*c**3*x**6) - 6*a**4*c**3*x**6*log(sqrt(1 + c*x
**2/a) + 1)/(6*a**(19/2) + 18*a**(17/2)*c*x**2 + 18*a**(15/2)*c**2*x**4 + 6*a**(
13/2)*c**3*x**6))

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GIAC/XCAS [A]  time = 0.277753, size = 161, normalized size = 1.55 \[ -\frac{{\left ({\left (\frac{5 \, A c^{2} x}{a^{3}} - \frac{3 \, B c}{a^{2}}\right )} x + \frac{6 \, A c}{a^{2}}\right )} x - \frac{4 \, B}{a}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} + \frac{2 \, B \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \, A \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(((5*A*c^2*x/a^3 - 3*B*c/a^2)*x + 6*A*c/a^2)*x - 4*B/a)/(c*x^2 + a)^(3/2) +
2*B*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^2) + 2*A*sqrt(c
)/(((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)*a^2)