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

Optimal. Leaf size=143 \[ -\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (-2 a^2 d^2+15 a b c d+57 b^2 c^2\right )+\frac{b d (2 a d+33 b c)}{x}\right )}{15 b^2}+\frac{c^2 (6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}}+x \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^3-\frac{7}{5} d \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 \]

[Out]

(-7*d*Sqrt[a + b/x]*(c + d/x)^2)/5 - (d*Sqrt[a + b/x]*(2*(57*b^2*c^2 + 15*a*b*c*
d - 2*a^2*d^2) + (b*d*(33*b*c + 2*a*d))/x))/(15*b^2) + Sqrt[a + b/x]*(c + d/x)^3
*x + (c^2*(b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]

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Rubi [A]  time = 0.417175, antiderivative size = 143, 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 \[ -\frac{d \sqrt{a+\frac{b}{x}} \left (2 \left (-2 a^2 d^2+15 a b c d+57 b^2 c^2\right )+\frac{b d (2 a d+33 b c)}{x}\right )}{15 b^2}+\frac{c^2 (6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a}}+x \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^3-\frac{7}{5} d \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 \]

Antiderivative was successfully verified.

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

[Out]

(-7*d*Sqrt[a + b/x]*(c + d/x)^2)/5 - (d*Sqrt[a + b/x]*(2*(57*b^2*c^2 + 15*a*b*c*
d - 2*a^2*d^2) + (b*d*(33*b*c + 2*a*d))/x))/(15*b^2) + Sqrt[a + b/x]*(c + d/x)^3
*x + (c^2*(b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/Sqrt[a]

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Rubi in Sympy [A]  time = 50.01, size = 131, normalized size = 0.92 \[ - \frac{7 d \sqrt{a + \frac{b}{x}} \left (c + \frac{d}{x}\right )^{2}}{5} + x \sqrt{a + \frac{b}{x}} \left (c + \frac{d}{x}\right )^{3} + \frac{8 d \sqrt{a + \frac{b}{x}} \left (\frac{a^{2} d^{2}}{2} - \frac{15 a b c d}{4} - \frac{57 b^{2} c^{2}}{4} - \frac{b d \left (2 a d + 33 b c\right )}{8 x}\right )}{15 b^{2}} + \frac{c^{2} \left (6 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*d*sqrt(a + b/x)*(c + d/x)**2/5 + x*sqrt(a + b/x)*(c + d/x)**3 + 8*d*sqrt(a +
b/x)*(a**2*d**2/2 - 15*a*b*c*d/4 - 57*b**2*c**2/4 - b*d*(2*a*d + 33*b*c)/(8*x))/
(15*b**2) + c**2*(6*a*d + b*c)*atanh(sqrt(a + b/x)/sqrt(a))/sqrt(a)

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Mathematica [A]  time = 0.20535, size = 131, normalized size = 0.92 \[ \sqrt{\frac{a x+b}{x}} \left (-\frac{2 d \left (-2 a^2 d^2+15 a b c d+45 b^2 c^2\right )}{15 b^2}-\frac{2 d^2 (a d+15 b c)}{15 b x}+c^3 x-\frac{2 d^3}{5 x^2}\right )+\frac{c^2 (6 a d+b c) \log \left (2 \sqrt{a} x \sqrt{\frac{a x+b}{x}}+2 a x+b\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[(b + a*x)/x]*((-2*d*(45*b^2*c^2 + 15*a*b*c*d - 2*a^2*d^2))/(15*b^2) - (2*d^
3)/(5*x^2) - (2*d^2*(15*b*c + a*d))/(15*b*x) + c^3*x) + (c^2*(b*c + 6*a*d)*Log[b
+ 2*a*x + 2*Sqrt[a]*x*Sqrt[(b + a*x)/x]])/(2*Sqrt[a])

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Maple [A]  time = 0.02, size = 248, normalized size = 1.7 \[{\frac{1}{30\,{x}^{3}{b}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 90\,d{c}^{2}a\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{2}{x}^{4}+180\,d{c}^{2}{a}^{3/2}\sqrt{a{x}^{2}+bx}b{x}^{4}+30\,{c}^{3}\sqrt{a{x}^{2}+bx}\sqrt{a}{b}^{2}{x}^{4}+15\,{c}^{3}{b}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{4}-180\,d{c}^{2} \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}b{x}^{2}+8\,{a}^{3/2} \left ( a{x}^{2}+bx \right ) ^{3/2}x{d}^{3}-60\,{d}^{2}c \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}bx-12\,\sqrt{a} \left ( a{x}^{2}+bx \right ) ^{3/2}b{d}^{3} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^3*(a+b/x)^(1/2),x)

[Out]

1/30*((a*x+b)/x)^(1/2)/x^3*(90*d*c^2*a*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x
+b)/a^(1/2))*b^2*x^4+180*d*c^2*a^(3/2)*(a*x^2+b*x)^(1/2)*b*x^4+30*c^3*(a*x^2+b*x
)^(1/2)*a^(1/2)*b^2*x^4+15*c^3*b^3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/
a^(1/2))*x^4-180*d*c^2*(a*x^2+b*x)^(3/2)*a^(1/2)*b*x^2+8*a^(3/2)*(a*x^2+b*x)^(3/
2)*x*d^3-60*d^2*c*(a*x^2+b*x)^(3/2)*a^(1/2)*b*x-12*a^(1/2)*(a*x^2+b*x)^(3/2)*b*d
^3)/(x*(a*x+b))^(1/2)/a^(1/2)/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(sqrt(a + b/x)*(c + d/x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260376, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{3} c^{3} + 6 \, a b^{2} c^{2} d\right )} x^{2} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) + 2 \,{\left (15 \, b^{2} c^{3} x^{3} - 6 \, b^{2} d^{3} - 2 \,{\left (45 \, b^{2} c^{2} d + 15 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} x^{2} - 2 \,{\left (15 \, b^{2} c d^{2} + a b d^{3}\right )} x\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}{30 \, \sqrt{a} b^{2} x^{2}}, -\frac{15 \,{\left (b^{3} c^{3} + 6 \, a b^{2} c^{2} d\right )} x^{2} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) -{\left (15 \, b^{2} c^{3} x^{3} - 6 \, b^{2} d^{3} - 2 \,{\left (45 \, b^{2} c^{2} d + 15 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} x^{2} - 2 \,{\left (15 \, b^{2} c d^{2} + a b d^{3}\right )} x\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}{15 \, \sqrt{-a} b^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*(b^3*c^3 + 6*a*b^2*c^2*d)*x^2*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b
)*sqrt(a)) + 2*(15*b^2*c^3*x^3 - 6*b^2*d^3 - 2*(45*b^2*c^2*d + 15*a*b*c*d^2 - 2*
a^2*d^3)*x^2 - 2*(15*b^2*c*d^2 + a*b*d^3)*x)*sqrt(a)*sqrt((a*x + b)/x))/(sqrt(a)
*b^2*x^2), -1/15*(15*(b^3*c^3 + 6*a*b^2*c^2*d)*x^2*arctan(a/(sqrt(-a)*sqrt((a*x
+ b)/x))) - (15*b^2*c^3*x^3 - 6*b^2*d^3 - 2*(45*b^2*c^2*d + 15*a*b*c*d^2 - 2*a^2
*d^3)*x^2 - 2*(15*b^2*c*d^2 + a*b*d^3)*x)*sqrt(-a)*sqrt((a*x + b)/x))/(sqrt(-a)*
b^2*x^2)]

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Sympy [A]  time = 15.4931, size = 491, normalized size = 3.43 \[ \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} d^{3} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} d^{3} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} d^{3} x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} d^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + 6 \sqrt{a} c^{2} d \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{4 a^{6} b d^{3} x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{2} d^{3} x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a c^{2} d \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \sqrt{b} c^{3} \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{6 \sqrt{b} c^{2} d}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} + 3 c d^{2} \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + \frac{b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(11/2)*b**(3/2)*d**3*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a
**(5/2)*b**4*x**(5/2)) + 2*a**(9/2)*b**(5/2)*d**3*x**2*sqrt(a*x/b + 1)/(15*a**(7
/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**(7/2)*b**(7/2)*d**3*x*sqrt
(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**(5/2)
*b**(9/2)*d**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**
(5/2)) + 6*sqrt(a)*c**2*d*asinh(sqrt(a)*sqrt(x)/sqrt(b)) - 4*a**6*b*d**3*x**(7/2
)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**5*b**2*d**3*x**
(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a*c**2*d*sqrt(
x)/(sqrt(b)*sqrt(a*x/b + 1)) + sqrt(b)*c**3*sqrt(x)*sqrt(a*x/b + 1) - 6*sqrt(b)*
c**2*d/(sqrt(x)*sqrt(a*x/b + 1)) + 3*c*d**2*Piecewise((-sqrt(a)/x, Eq(b, 0)), (-
2*(a + b/x)**(3/2)/(3*b), True)) + b*c**3*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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError