3.162 \(\int \frac{c+\frac{d}{x}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ -\frac{(5 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b c-2 a d}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{5 b c-2 a d}{3 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c x}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.198709, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{(5 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{5 b c-2 a d}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{5 b c-2 a d}{3 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c x}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d/x)/(a + b/x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.3615, size = 90, normalized size = 0.87 \[ \frac{c x}{a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{2 \left (a d - \frac{5 b c}{2}\right )}{3 a^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{2 \left (a d - \frac{5 b c}{2}\right )}{a^{3} \sqrt{a + \frac{b}{x}}} + \frac{2 \left (a d - \frac{5 b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d/x)/(a+b/x)**(5/2),x)

[Out]

c*x/(a*(a + b/x)**(3/2)) - 2*(a*d - 5*b*c/2)/(3*a**2*(a + b/x)**(3/2)) - 2*(a*d
- 5*b*c/2)/(a**3*sqrt(a + b/x)) + 2*(a*d - 5*b*c/2)*atanh(sqrt(a + b/x)/sqrt(a))
/a**(7/2)

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Mathematica [A]  time = 0.140561, size = 102, normalized size = 0.99 \[ \frac{(2 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{7/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (a^2 x (3 c x-8 d)+a b (20 c x-6 d)+15 b^2 c\right )}{3 a^3 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d/x)/(a + b/x)^(5/2),x]

[Out]

(Sqrt[a + b/x]*x*(15*b^2*c + a^2*x*(-8*d + 3*c*x) + a*b*(-6*d + 20*c*x)))/(3*a^3
*(b + a*x)^2) + ((-5*b*c + 2*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2
*a^(7/2))

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Maple [B]  time = 0.019, size = 548, normalized size = 5.3 \[ -{\frac{x}{6\,b \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 12\,{a}^{15/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}d-30\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}bc-12\,{a}^{13/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xd+36\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}bd+24\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xbc-90\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}{b}^{2}c-8\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}bd+36\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}d+20\,{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}{b}^{2}c-90\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }x{b}^{3}c+12\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}d-30\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{4}c-6\,{a}^{7}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}bd+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{6}{b}^{2}c-18\,{a}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{b}^{2}d+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{5}{b}^{3}c-18\,{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{3}d+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{4}c-6\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{4}d+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{5}c \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)/(a+b/x)^(5/2),x)

[Out]

-1/6*((a*x+b)/x)^(1/2)*x*(12*a^(15/2)*(x*(a*x+b))^(1/2)*x^3*d-30*a^(13/2)*(x*(a*
x+b))^(1/2)*x^3*b*c-12*a^(13/2)*(x*(a*x+b))^(3/2)*x*d+36*a^(13/2)*(x*(a*x+b))^(1
/2)*x^2*b*d+24*a^(11/2)*(x*(a*x+b))^(3/2)*x*b*c-90*a^(11/2)*(x*(a*x+b))^(1/2)*x^
2*b^2*c-8*a^(11/2)*(x*(a*x+b))^(3/2)*b*d+36*a^(11/2)*(x*(a*x+b))^(1/2)*x*b^2*d+2
0*a^(9/2)*(x*(a*x+b))^(3/2)*b^2*c-90*a^(9/2)*(x*(a*x+b))^(1/2)*x*b^3*c+12*a^(9/2
)*(x*(a*x+b))^(1/2)*b^3*d-30*a^(7/2)*(x*(a*x+b))^(1/2)*b^4*c-6*a^7*ln(1/2*(2*(x*
(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*b*d+15*ln(1/2*(2*(x*(a*x+b))^(1/2)*
a^(1/2)+2*a*x+b)/a^(1/2))*x^3*a^6*b^2*c-18*a^6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/
2)+2*a*x+b)/a^(1/2))*x^2*b^2*d+45*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*x^2*a^5*b^3*c-18*a^5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2
))*x*b^3*d+45*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^4*b^4*c-
6*a^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^4*d+15*ln(1/2*(2*(
x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*b^5*c)/a^(13/2)/(x*(a*x+b))^(1/2)
/b/(a*x+b)^3

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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((c + d/x)/(a + b/x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241657, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{2} c - 2 \, a b d +{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (3 \, a^{2} c x^{2} + 15 \, b^{2} c - 6 \, a b d + 4 \,{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{a}}{6 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{3 \,{\left (5 \, b^{2} c - 2 \, a b d +{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} c x^{2} + 15 \, b^{2} c - 6 \, a b d + 4 \,{\left (5 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{-a}}{3 \,{\left (a^{4} x + a^{3} b\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)/(a + b/x)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(5*b^2*c - 2*a*b*d + (5*a*b*c - 2*a^2*d)*x)*sqrt((a*x + b)/x)*log(2*a*x
*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) - 2*(3*a^2*c*x^2 + 15*b^2*c - 6*a*b*d
+ 4*(5*a*b*c - 2*a^2*d)*x)*sqrt(a))/((a^4*x + a^3*b)*sqrt(a)*sqrt((a*x + b)/x)),
1/3*(3*(5*b^2*c - 2*a*b*d + (5*a*b*c - 2*a^2*d)*x)*sqrt((a*x + b)/x)*arctan(a/(
sqrt(-a)*sqrt((a*x + b)/x))) + (3*a^2*c*x^2 + 15*b^2*c - 6*a*b*d + 4*(5*a*b*c -
2*a^2*d)*x)*sqrt(-a))/((a^4*x + a^3*b)*sqrt(-a)*sqrt((a*x + b)/x))]

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Sympy [A]  time = 39.1088, size = 1479, normalized size = 14.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d/x)/(a+b/x)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.255601, size = 185, normalized size = 1.8 \[ -\frac{1}{3} \, b{\left (\frac{3 \, c \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3}} - \frac{3 \,{\left (5 \, b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b} - \frac{2 \,{\left (a b c - a^{2} d + \frac{6 \,{\left (a x + b\right )} b c}{x} - \frac{3 \,{\left (a x + b\right )} a d}{x}\right )} x}{{\left (a x + b\right )} a^{3} b \sqrt{\frac{a x + b}{x}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)/(a + b/x)^(5/2),x, algorithm="giac")

[Out]

-1/3*b*(3*c*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a^3) - 3*(5*b*c - 2*a*d)*arctan
(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3*b) - 2*(a*b*c - a^2*d + 6*(a*x + b)*b
*c/x - 3*(a*x + b)*a*d/x)*x/((a*x + b)*a^3*b*sqrt((a*x + b)/x)))