3.146 \(\int \frac{\left (c+\frac{d}{x}\right )^3}{\sqrt{a+\frac{b}{x}}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (-2 a^2 d^2+9 a b c d+3 b^2 c^2\right )+\frac{b d (2 a d+3 b c)}{x}\right )}{3 a b^2}+\frac{c x \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2}{a} \]

[Out]

-(d*Sqrt[a + b/x]*(2*(3*b^2*c^2 + 9*a*b*c*d - 2*a^2*d^2) + (b*d*(3*b*c + 2*a*d))
/x))/(3*a*b^2) + (c*Sqrt[a + b/x]*(c + d/x)^2*x)/a - (c^2*(b*c - 6*a*d)*ArcTanh[
Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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Rubi [A]  time = 0.283658, antiderivative size = 126, 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 \[ -\frac{c^2 (b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (-2 a^2 d^2+9 a b c d+3 b^2 c^2\right )+\frac{b d (2 a d+3 b c)}{x}\right )}{3 a b^2}+\frac{c x \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2}{a} \]

Antiderivative was successfully verified.

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

[Out]

-(d*Sqrt[a + b/x]*(2*(3*b^2*c^2 + 9*a*b*c*d - 2*a^2*d^2) + (b*d*(3*b*c + 2*a*d))
/x))/(3*a*b^2) + (c*Sqrt[a + b/x]*(c + d/x)^2*x)/a - (c^2*(b*c - 6*a*d)*ArcTanh[
Sqrt[a + b/x]/Sqrt[a]])/a^(3/2)

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Rubi in Sympy [A]  time = 25.5374, size = 114, normalized size = 0.9 \[ \frac{c x \sqrt{a + \frac{b}{x}} \left (c + \frac{d}{x}\right )^{2}}{a} + \frac{4 d \sqrt{a + \frac{b}{x}} \left (a^{2} d^{2} - \frac{9 a b c d}{2} - \frac{3 b^{2} c^{2}}{2} - \frac{b d \left (2 a d + 3 b c\right )}{4 x}\right )}{3 a b^{2}} + \frac{c^{2} \left (6 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d/x)**3/(a+b/x)**(1/2),x)

[Out]

c*x*sqrt(a + b/x)*(c + d/x)**2/a + 4*d*sqrt(a + b/x)*(a**2*d**2 - 9*a*b*c*d/2 -
3*b**2*c**2/2 - b*d*(2*a*d + 3*b*c)/(4*x))/(3*a*b**2) + c**2*(6*a*d - b*c)*atanh
(sqrt(a + b/x)/sqrt(a))/a**(3/2)

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Mathematica [A]  time = 0.288788, size = 99, normalized size = 0.79 \[ \frac{c^2 (6 a d-b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{3/2}}+\sqrt{a+\frac{b}{x}} \left (\frac{4 a d^3}{3 b^2}+\frac{c^3 x}{a}-\frac{2 d^2 (9 c x+d)}{3 b x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b/x]*((4*a*d^3)/(3*b^2) + (c^3*x)/a - (2*d^2*(d + 9*c*x))/(3*b*x)) + (c
^2*(-(b*c) + 6*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2*a^(3/2))

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Maple [B]  time = 0.021, size = 540, normalized size = 4.3 \[{\frac{1}{6\,{b}^{3}{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 3\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){d}^{3}b{x}^{3}-3\,{d}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{4}b{x}^{3}-9\,{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) c{d}^{2}{b}^{2}{x}^{3}+9\,{d}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}c{b}^{2}{x}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){c}^{2}d{b}^{3}{x}^{3}{a}^{2}+9\,d\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){c}^{2}{b}^{3}{x}^{3}{a}^{2}-6\,{d}^{3}\sqrt{a{x}^{2}+bx}{a}^{9/2}{x}^{3}+18\,{d}^{2}\sqrt{a{x}^{2}+bx}c{a}^{7/2}b{x}^{3}+18\,d\sqrt{a{x}^{2}+bx}{c}^{2}{b}^{2}{x}^{3}{a}^{5/2}-6\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{d}^{3}{x}^{3}+18\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }c{d}^{2}b{x}^{3}-18\,\sqrt{x \left ( ax+b \right ) }{c}^{2}d{b}^{2}{x}^{3}{a}^{5/2}+6\,\sqrt{x \left ( ax+b \right ) }{c}^{3}{b}^{3}{x}^{3}{a}^{3/2}-3\,{b}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){c}^{3}{x}^{3}a+12\,{d}^{3} \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}x-36\,{d}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}cbx{a}^{5/2}-4\,{d}^{3} \left ( a{x}^{2}+bx \right ) ^{3/2}b{a}^{5/2} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^3/(a+b/x)^(1/2),x)

[Out]

1/6*((a*x+b)/x)^(1/2)/x^2*(3*a^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*d^3*b*x^3-3*d^3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4
*b*x^3-9*a^3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*c*d^2*b^2*x^3
+9*d^2*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*c*b^2*x^3+9*ln(
1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*c^2*d*b^3*x^3*a^2+9*d*ln(1/2*
(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*c^2*b^3*x^3*a^2-6*d^3*(a*x^2+b*x)
^(1/2)*a^(9/2)*x^3+18*d^2*(a*x^2+b*x)^(1/2)*c*a^(7/2)*b*x^3+18*d*(a*x^2+b*x)^(1/
2)*c^2*b^2*x^3*a^(5/2)-6*a^(9/2)*(x*(a*x+b))^(1/2)*d^3*x^3+18*a^(7/2)*(x*(a*x+b)
)^(1/2)*c*d^2*b*x^3-18*(x*(a*x+b))^(1/2)*c^2*d*b^2*x^3*a^(5/2)+6*(x*(a*x+b))^(1/
2)*c^3*b^3*x^3*a^(3/2)-3*b^4*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2
))*c^3*x^3*a+12*d^3*(a*x^2+b*x)^(3/2)*a^(7/2)*x-36*d^2*(a*x^2+b*x)^(3/2)*c*b*x*a
^(5/2)-4*d^3*(a*x^2+b*x)^(3/2)*b*a^(5/2))/(x*(a*x+b))^(1/2)/b^3/a^(5/2)

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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)^3/sqrt(a + b/x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253138, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d\right )} x \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (3 \, b^{2} c^{3} x^{2} - 2 \, a b d^{3} - 2 \,{\left (9 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{6 \, a^{\frac{3}{2}} b^{2} x}, \frac{3 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d\right )} x \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, b^{2} c^{3} x^{2} - 2 \, a b d^{3} - 2 \,{\left (9 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{3 \, \sqrt{-a} a b^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*(b^3*c^3 - 6*a*b^2*c^2*d)*x*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*s
qrt(a)) - 2*(3*b^2*c^3*x^2 - 2*a*b*d^3 - 2*(9*a*b*c*d^2 - 2*a^2*d^3)*x)*sqrt(a)*
sqrt((a*x + b)/x))/(a^(3/2)*b^2*x), 1/3*(3*(b^3*c^3 - 6*a*b^2*c^2*d)*x*arctan(a/
(sqrt(-a)*sqrt((a*x + b)/x))) + (3*b^2*c^3*x^2 - 2*a*b*d^3 - 2*(9*a*b*c*d^2 - 2*
a^2*d^3)*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*a*b^2*x)]

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Sympy [A]  time = 15.5631, size = 377, normalized size = 2.99 \[ \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} d^{3} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} d^{3} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} d^{3} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{4} b d^{3} x^{\frac{5}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{3} b^{2} d^{3} x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + 3 c d^{2} \left (\begin{cases} - \frac{1}{\sqrt{a} x} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{\sqrt{b} c^{3} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} + \frac{6 c^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} - \frac{b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d/x)**3/(a+b/x)**(1/2),x)

[Out]

4*a**(7/2)*b**(3/2)*d**3*x**2*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(
3/2)*b**4*x**(3/2)) + 2*a**(5/2)*b**(5/2)*d**3*x*sqrt(a*x/b + 1)/(3*a**(5/2)*b**
3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 2*a**(3/2)*b**(7/2)*d**3*sqrt(a*x/b + 1
)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 4*a**4*b*d**3*x**(5/2)
/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 4*a**3*b**2*d**3*x**(3/
2)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) + 3*c*d**2*Piecewise((-
1/(sqrt(a)*x), Eq(b, 0)), (-2*sqrt(a + b/x)/b, True)) + sqrt(b)*c**3*sqrt(x)*sqr
t(a*x/b + 1)/a + 6*c**2*d*asinh(sqrt(a)*sqrt(x)/sqrt(b))/sqrt(a) - b*c**3*asinh(
sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

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GIAC/XCAS [A]  time = 0.256988, size = 207, normalized size = 1.64 \[ -\frac{1}{3} \,{\left (\frac{3 \, c^{3} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} - \frac{3 \,{\left (b c^{3} - 6 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b} + \frac{2 \,{\left (9 \, b^{7} c d^{2} \sqrt{\frac{a x + b}{x}} - 3 \, a b^{6} d^{3} \sqrt{\frac{a x + b}{x}} + \frac{{\left (a x + b\right )} b^{6} d^{3} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{b^{9}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*c^3*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a) - 3*(b*c^3 - 6*a*c^2*d)*arct
an(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b) + 2*(9*b^7*c*d^2*sqrt((a*x + b)/x)
- 3*a*b^6*d^3*sqrt((a*x + b)/x) + (a*x + b)*b^6*d^3*sqrt((a*x + b)/x)/x)/b^9)*b