3.155 \(\int \frac{c+\frac{d}{x}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.154825, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{(3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{3 b c-2 a d}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{c x}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d/x)/(a + b/x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.5111, size = 66, normalized size = 0.87 \[ \frac{c x}{a \sqrt{a + \frac{b}{x}}} - \frac{2 \left (a d - \frac{3 b c}{2}\right )}{a^{2} \sqrt{a + \frac{b}{x}}} + \frac{2 \left (a d - \frac{3 b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d/x)/(a+b/x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.116594, size = 81, normalized size = 1.07 \[ \frac{(2 a d-3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{5/2}}+\frac{x \sqrt{a+\frac{b}{x}} (a c x-2 a d+3 b c)}{a^2 (a x+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d/x)/(a + b/x)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.018, size = 394, normalized size = 5.2 \[ -{\frac{x}{2\,b \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 4\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}d-6\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}bc-4\,{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}d+8\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }xbd+4\,{a}^{7/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}cb-12\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}c+4\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{2}d-6\,{a}^{5/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}c-2\,{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}bd+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{4}{b}^{2}c-4\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{2}d+6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{3}{b}^{3}c-2\,{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{3}d+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{2}{b}^{4}c \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)/(a+b/x)^(3/2),x)

[Out]

-1/2*((a*x+b)/x)^(1/2)*x*(4*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*d-6*a^(9/2)*(x*(a*x+b
))^(1/2)*x^2*b*c-4*a^(9/2)*(x*(a*x+b))^(3/2)*d+8*a^(9/2)*(x*(a*x+b))^(1/2)*x*b*d
+4*a^(7/2)*(x*(a*x+b))^(3/2)*c*b-12*a^(7/2)*(x*(a*x+b))^(1/2)*x*b^2*c+4*a^(7/2)*
(x*(a*x+b))^(1/2)*b^2*d-6*a^(5/2)*(x*(a*x+b))^(1/2)*b^3*c-2*a^5*ln(1/2*(2*(x*(a*
x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*b*d+3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1
/2)+2*a*x+b)/a^(1/2))*x^2*a^4*b^2*c-4*a^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*
a*x+b)/a^(1/2))*x*b^2*d+6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*
x*a^3*b^3*c-2*a^3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^3*d+3*
ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^4*c)/a^(9/2)/(x*(a*x
+b))^(1/2)/b/(a*x+b)^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((c + d/x)/(a + b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24653, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b c - 2 \, a d\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (a c x + 3 \, b c - 2 \, a d\right )} \sqrt{a}}{2 \, a^{\frac{5}{2}} \sqrt{\frac{a x + b}{x}}}, \frac{{\left (3 \, b c - 2 \, a d\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (a c x + 3 \, b c - 2 \, a d\right )} \sqrt{-a}}{\sqrt{-a} a^{2} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)/(a + b/x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 21.4323, size = 224, normalized size = 2.95 \[ c \left (\frac{x^{\frac{3}{2}}}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} \sqrt{x}}{a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}}\right ) + d \left (- \frac{2 a^{3} x \sqrt{1 + \frac{b}{a x}}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{3} x \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{3} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d/x)/(a+b/x)**(3/2),x)

[Out]

c*(x**(3/2)/(a*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*sqrt(x)/(a**2*sqrt(a*x/b + 1
)) - 3*b*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(5/2)) + d*(-2*a**3*x*sqrt(1 + b/(a*x
))/(a**(9/2)*x + a**(7/2)*b) - a**3*x*log(b/(a*x))/(a**(9/2)*x + a**(7/2)*b) + 2
*a**3*x*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b) - a**2*b*log(b/(a*x
))/(a**(9/2)*x + a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x +
a**(7/2)*b))

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GIAC/XCAS [A]  time = 0.254238, size = 165, normalized size = 2.17 \[ b{\left (\frac{{\left (3 \, b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b} + \frac{2 \, a b c - 2 \, a^{2} d - \frac{3 \,{\left (a x + b\right )} b c}{x} + \frac{2 \,{\left (a x + b\right )} a d}{x}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )} a^{2} b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)/(a + b/x)^(3/2),x, algorithm="giac")

[Out]

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