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

Optimal. Leaf size=99 \[ \frac{c^2 x \left (a+\frac{b}{x}\right )^{3/2}}{a}-\frac{c \sqrt{a+\frac{b}{x}} (4 a d+b c)}{a}+\frac{c (4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.218174, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{c^2 x \left (a+\frac{b}{x}\right )^{3/2}}{a}-\frac{c \sqrt{a+\frac{b}{x}} (4 a d+b c)}{a}+\frac{c (4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{2 d^2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.3938, size = 82, normalized size = 0.83 \[ - \frac{2 d^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} + \frac{c^{2} x \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{a} - \frac{c \sqrt{a + \frac{b}{x}} \left (4 a d + b c\right )}{a} + \frac{c \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

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

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Mathematica [A]  time = 0.133863, size = 85, normalized size = 0.86 \[ \sqrt{a+\frac{b}{x}} \left (-\frac{2 d^2 (a x+b)}{3 b x}+c^2 x-4 c d\right )+\frac{c (4 a d+b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b/x]*(-4*c*d + c^2*x - (2*d^2*(b + a*x))/(3*b*x)) + (c*(b*c + 4*a*d)*Lo
g[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2*Sqrt[a])

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Maple [B]  time = 0.017, size = 191, normalized size = 1.9 \[{\frac{1}{6\,b{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 12\,cda\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}b+24\,cd{a}^{3/2}\sqrt{a{x}^{2}+bx}{x}^{3}+6\,{c}^{2}\sqrt{a{x}^{2}+bx}\sqrt{a}{x}^{3}b+3\,{c}^{2}{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}-24\,cd \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}x-4\,{d}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^2*(a+b/x)^(1/2),x)

[Out]

1/6*((a*x+b)/x)^(1/2)/x^2*(12*c*d*a*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)
/a^(1/2))*x^3*b+24*c*d*a^(3/2)*(a*x^2+b*x)^(1/2)*x^3+6*c^2*(a*x^2+b*x)^(1/2)*a^(
1/2)*x^3*b+3*c^2*b^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3-2
4*c*d*(a*x^2+b*x)^(3/2)*a^(1/2)*x-4*d^2*(a*x^2+b*x)^(3/2)*a^(1/2))/(x*(a*x+b))^(
1/2)/a^(1/2)/b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25669, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{2} c^{2} + 4 \, a b c d\right )} x \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (3 \, b c^{2} x^{2} - 2 \, b d^{2} - 2 \,{\left (6 \, b c d + a d^{2}\right )} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{6 \, \sqrt{a} b x}, -\frac{3 \,{\left (b^{2} c^{2} + 4 \, a b c d\right )} x \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (3 \, b c^{2} x^{2} - 2 \, b d^{2} - 2 \,{\left (6 \, b c d + a d^{2}\right )} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{3 \, \sqrt{-a} b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.0423, size = 156, normalized size = 1.58 \[ 4 \sqrt{a} c d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{4 a c d \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \sqrt{b} c^{2} \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{4 \sqrt{b} c d}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} + d^{2} \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + \frac{b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError