3.147 \(\int \frac{\left (c+\frac{d}{x}\right )^2}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b} \]

[Out]

(-2*d^2*Sqrt[a + b/x])/b + (c^2*Sqrt[a + b/x]*x)/a - (c*(b*c - 4*a*d)*ArcTanh[Sq
rt[a + b/x]/Sqrt[a]])/a^(3/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.180524, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d^2*Sqrt[a + b/x])/b + (c^2*Sqrt[a + b/x]*x)/a - (c*(b*c - 4*a*d)*ArcTanh[Sq
rt[a + b/x]/Sqrt[a]])/a^(3/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.5146, size = 60, normalized size = 0.82 \[ - \frac{2 d^{2} \sqrt{a + \frac{b}{x}}}{b} + \frac{c^{2} x \sqrt{a + \frac{b}{x}}}{a} + \frac{c \left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d/x)**2/(a+b/x)**(1/2),x)

[Out]

-2*d**2*sqrt(a + b/x)/b + c**2*x*sqrt(a + b/x)/a + c*(4*a*d - b*c)*atanh(sqrt(a
+ b/x)/sqrt(a))/a**(3/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.161446, size = 75, normalized size = 1.03 \[ \frac{c (4 a d-b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{3/2}}+\sqrt{a+\frac{b}{x}} \left (\frac{c^2 x}{a}-\frac{2 d^2}{b}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b/x]*((-2*d^2)/b + (c^2*x)/a) + (c*(-(b*c) + 4*a*d)*Log[b + 2*a*x + 2*S
qrt[a]*Sqrt[a + b/x]*x])/(2*a^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 354, normalized size = 4.9 \[{\frac{1}{2\,x{b}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ({d}^{2}{a}^{3}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}b-{a}^{3}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){d}^{2}{x}^{2}b+2\,{d}^{2}{a}^{7/2}\sqrt{a{x}^{2}+bx}{x}^{2}+4\,d\sqrt{a{x}^{2}+bx}c{x}^{2}b{a}^{5/2}+2\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{d}^{2}{x}^{2}-4\,\sqrt{x \left ( ax+b \right ) }cd{x}^{2}b{a}^{5/2}+2\,\sqrt{x \left ( ax+b \right ) }{c}^{2}{x}^{2}{b}^{2}{a}^{3/2}+2\,d\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) c{x}^{2}{b}^{2}{a}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) cd{x}^{2}{b}^{2}{a}^{2}-4\,{d}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}-{b}^{3}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){c}^{2}{x}^{2}a \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^2/(a+b/x)^(1/2),x)

[Out]

1/2*((a*x+b)/x)^(1/2)/x*(d^2*a^3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*x^2*b-a^3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*d^2*x^2*b
+2*d^2*a^(7/2)*(a*x^2+b*x)^(1/2)*x^2+4*d*(a*x^2+b*x)^(1/2)*c*x^2*b*a^(5/2)+2*a^(
7/2)*(x*(a*x+b))^(1/2)*d^2*x^2-4*(x*(a*x+b))^(1/2)*c*d*x^2*b*a^(5/2)+2*(x*(a*x+b
))^(1/2)*c^2*x^2*b^2*a^(3/2)+2*d*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*c*x^2*b^2*a^2+2*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*c*d
*x^2*b^2*a^2-4*d^2*(a*x^2+b*x)^(3/2)*a^(5/2)-b^3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(
1/2)+2*a*x+b)/a^(1/2))*c^2*x^2*a)/(x*(a*x+b))^(1/2)/b^2/a^(5/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.248367, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b c^{2} x - 2 \, a d^{2}\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}} -{\left (b^{2} c^{2} - 4 \, a b c d\right )} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}} b}, \frac{{\left (b c^{2} x - 2 \, a d^{2}\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}} +{\left (b^{2} c^{2} - 4 \, a b c d\right )} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*(b*c^2*x - 2*a*d^2)*sqrt(a)*sqrt((a*x + b)/x) - (b^2*c^2 - 4*a*b*c*d)*lo
g(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)))/(a^(3/2)*b), ((b*c^2*x - 2*a*d
^2)*sqrt(-a)*sqrt((a*x + b)/x) + (b^2*c^2 - 4*a*b*c*d)*arctan(a/(sqrt(-a)*sqrt((
a*x + b)/x))))/(sqrt(-a)*a*b)]

_______________________________________________________________________________________

Sympy [A]  time = 12.382, size = 105, normalized size = 1.44 \[ d^{2} \left (\begin{cases} - \frac{1}{\sqrt{a} x} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{\sqrt{b} c^{2} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} + \frac{4 c d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} - \frac{b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d/x)**2/(a+b/x)**(1/2),x)

[Out]

d**2*Piecewise((-1/(sqrt(a)*x), Eq(b, 0)), (-2*sqrt(a + b/x)/b, True)) + sqrt(b)
*c**2*sqrt(x)*sqrt(a*x/b + 1)/a + 4*c*d*asinh(sqrt(a)*sqrt(x)/sqrt(b))/sqrt(a) -
b*c**2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.255677, size = 131, normalized size = 1.79 \[ -b{\left (\frac{c^{2} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} + \frac{2 \, d^{2} \sqrt{\frac{a x + b}{x}}}{b^{2}} - \frac{{\left (b c^{2} - 4 \, a c d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(c^2*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a) + 2*d^2*sqrt((a*x + b)/x)/b^2 -
(b*c^2 - 4*a*c*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b))