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

Optimal. Leaf size=98 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{5 A b-2 a B}{a^3 \sqrt{a+b x}}-\frac{5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}-\frac{A}{a x (a+b x)^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.143555, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{5 A b-2 a B}{a^3 \sqrt{a+b x}}-\frac{5 A b-2 a B}{3 a^2 (a+b x)^{3/2}}-\frac{A}{a x (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.106, size = 90, normalized size = 0.92 \[ - \frac{A}{a x \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \left (\frac{5 A b}{2} - B a\right )}{3 a^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \left (\frac{5 A b}{2} - B a\right )}{a^{3} \sqrt{a + b x}} + \frac{2 \left (\frac{5 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.133798, size = 86, normalized size = 0.88 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{a^2 (8 B x-3 A)+2 a b x (3 B x-10 A)-15 A b^2 x^2}{3 a^3 x (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-15*A*b^2*x^2 + 2*a*b*x*(-10*A + 3*B*x) + a^2*(-3*A + 8*B*x))/(3*a^3*x*(a + b*x
)^(3/2)) + ((5*A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(7/2)

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 88, normalized size = 0.9 \[ -{\frac{2\,Ab-2\,Ba}{3\,{a}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}}-2\,{\frac{2\,Ab-Ba}{{a}^{3}\sqrt{bx+a}}}-2\,{\frac{1}{{a}^{3}} \left ( 1/2\,{\frac{A\sqrt{bx+a}}{x}}-1/2\,{\frac{5\,Ab-2\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.230916, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (3 \, A a^{2} - 3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{2} - 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{a}}{6 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{b x + a} \sqrt{a}}, \frac{3 \,{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{2} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (3 \, A a^{2} - 3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{2} - 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{-a}}{3 \,{\left (a^{3} b x^{2} + a^{4} x\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 39.6129, size = 1520, normalized size = 15.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-6*a**17*sqrt(1 + b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*
b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 46*a**16*b*x*sqrt(1 + b*x/a)/(6*a**(39/2)*x
+ 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 15*a*
*16*b*x*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3
+ 6*a**(33/2)*b**3*x**4) + 30*a**16*b*x*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x
+ 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 70*a*
*15*b**2*x**2*sqrt(1 + b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2
)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 45*a**15*b**2*x**2*log(b*x/a)/(6*a**(39/2
)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90
*a**15*b**2*x**2*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 +
18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) - 30*a**14*b**3*x**3*sqrt(1 + b
*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2
)*b**3*x**4) - 45*a**14*b**3*x**3*log(b*x/a)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**
2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 90*a**14*b**3*x**3*log(sqr
t(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3
+ 6*a**(33/2)*b**3*x**4) - 15*a**13*b**4*x**4*log(b*x/a)/(6*a**(39/2)*x + 18*a**
(37/2)*b*x**2 + 18*a**(35/2)*b**2*x**3 + 6*a**(33/2)*b**3*x**4) + 30*a**13*b**4*
x**4*log(sqrt(1 + b*x/a) + 1)/(6*a**(39/2)*x + 18*a**(37/2)*b*x**2 + 18*a**(35/2
)*b**2*x**3 + 6*a**(33/2)*b**3*x**4)) + B*(8*a**7*sqrt(1 + b*x/a)/(3*a**(19/2) +
9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 3*a**7*log(b
*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*
x**3) - 6*a**7*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(1
5/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 14*a**6*b*x*sqrt(1 + b*x/a)/(3*a**(19/
2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 9*a**6*b
*x*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/
2)*b**3*x**3) - 18*a**6*b*x*log(sqrt(1 + b*x/a) + 1)/(3*a**(19/2) + 9*a**(17/2)*
b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) + 6*a**5*b**2*x**2*sqrt(1 +
b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**
3*x**3) + 9*a**5*b**2*x**2*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/
2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3) - 18*a**5*b**2*x**2*log(sqrt(1 + b*x/a) +
1)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**
3) + 3*a**4*b**3*x**3*log(b*x/a)/(3*a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b*
*2*x**2 + 3*a**(13/2)*b**3*x**3) - 6*a**4*b**3*x**3*log(sqrt(1 + b*x/a) + 1)/(3*
a**(19/2) + 9*a**(17/2)*b*x + 9*a**(15/2)*b**2*x**2 + 3*a**(13/2)*b**3*x**3))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216973, size = 122, normalized size = 1.24 \[ \frac{{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{\sqrt{b x + a} A}{a^{3} x} + \frac{2 \,{\left (3 \,{\left (b x + a\right )} B a + B a^{2} - 6 \,{\left (b x + a\right )} A b - A a b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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