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

Optimal. Leaf size=94 \[ -\frac{c (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 a^2 d^2+b c (3 b c-4 a d)}{a^2 b \sqrt{a+\frac{b}{x}}}+\frac{c^2 x}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

(2*a^2*d^2 + b*c*(3*b*c - 4*a*d))/(a^2*b*Sqrt[a + b/x]) + (c^2*x)/(a*Sqrt[a + b/
x]) - (c*(3*b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)

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Rubi [A]  time = 0.247795, antiderivative size = 90, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{c (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{\frac{c (3 b c-4 a d)}{a^2}+\frac{2 d^2}{b}}{\sqrt{a+\frac{b}{x}}}+\frac{c^2 x}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

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

[Out]

((2*d^2)/b + (c*(3*b*c - 4*a*d))/a^2)/Sqrt[a + b/x] + (c^2*x)/(a*Sqrt[a + b/x])
- (c*(3*b*c - 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 18.8213, size = 82, normalized size = 0.87 \[ \frac{c^{2} x}{a \sqrt{a + \frac{b}{x}}} + \frac{2 \left (a^{2} d^{2} - \frac{b c \left (4 a d - 3 b c\right )}{2}\right )}{a^{2} b \sqrt{a + \frac{b}{x}}} + \frac{c \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*x/(a*sqrt(a + b/x)) + 2*(a**2*d**2 - b*c*(4*a*d - 3*b*c)/2)/(a**2*b*sqrt(a
+ b/x)) + c*(4*a*d - 3*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/a**(5/2)

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Mathematica [A]  time = 0.187898, size = 100, normalized size = 1.06 \[ \frac{c (4 a d-3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{5/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (2 a^2 d^2+a b c (c x-4 d)+3 b^2 c^2\right )}{a^2 b (a x+b)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*x*(3*b^2*c^2 + 2*a^2*d^2 + a*b*c*(-4*d + c*x)))/(a^2*b*(b + a*x))
+ (c*(-3*b*c + 4*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2*a^(5/2))

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Maple [B]  time = 0.022, size = 796, normalized size = 8.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^2/(a+b/x)^(3/2),x)

[Out]

1/2*((a*x+b)/x)^(1/2)*x/a^(9/2)*(6*a^(9/2)*(x*(a*x+b))^(1/2)*x^2*b^2*c^2+a^6*ln(
1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*b*d^2-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*d^2+4*a^(11/2)*(a*x^2+b*x)^(1/2)*x*
b*d^2+8*a^(9/2)*(x*(a*x+b))^(3/2)*c*d*b+4*a^(11/2)*(x*(a*x+b))^(1/2)*x*b*d^2-3*l
n(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^4*b^3*c^2-6*ln(1/2*(2
*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^3*b^4*c^2+6*a^(5/2)*(x*(a*x+b))
^(1/2)*b^4*c^2-3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*b^5*c
^2+2*a^(13/2)*(a*x^2+b*x)^(1/2)*x^2*d^2+2*a^(13/2)*(x*(a*x+b))^(1/2)*x^2*d^2-4*a
^(7/2)*(x*(a*x+b))^(3/2)*c^2*b^2+2*a^(9/2)*(a*x^2+b*x)^(1/2)*b^2*d^2+2*a^(9/2)*(
x*(a*x+b))^(1/2)*b^2*d^2+a^4*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2
))*b^3*d^2-a^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^3*d^2-4*a
^(11/2)*(x*(a*x+b))^(3/2)*d^2+12*a^(7/2)*(x*(a*x+b))^(1/2)*x*b^3*c^2+2*a^5*ln(1/
2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*b^2*d^2-2*a^5*ln(1/2*(2*(x*(a
*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*b^2*d^2-8*a^(7/2)*(x*(a*x+b))^(1/2)*b^3
*c*d+4*a^3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^4*c*d-16*a^(9
/2)*(x*(a*x+b))^(1/2)*x*b^2*c*d+4*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^2*c*d+8*a^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/
2))*x*b^3*c*d-8*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*b*c*d)/(x*(a*x+b))^(1/2)/b^2/(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)^2/(a + b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (a b c^{2} x + 3 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2}\right )} \sqrt{a}}{2 \, a^{\frac{5}{2}} b \sqrt{\frac{a x + b}{x}}}, \frac{{\left (3 \, b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (a b c^{2} x + 3 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2}\right )} \sqrt{-a}}{\sqrt{-a} a^{2} b \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x + d\right )^{2}}{x^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((c*x + d)**2/(x**2*(a + b/x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.251661, size = 217, normalized size = 2.31 \[ b{\left (\frac{{\left (3 \, b c^{2} - 4 \, a c d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b} + \frac{2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} - \frac{3 \,{\left (a x + b\right )} b^{2} c^{2}}{x} + \frac{4 \,{\left (a x + b\right )} a b c d}{x} - \frac{2 \,{\left (a x + b\right )} a^{2} d^{2}}{x}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )} a^{2} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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