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

Optimal. Leaf size=213 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac{(b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^4}+\frac{d \sqrt{a+\frac{b}{x}} (11 b c-12 a d)}{4 c^3 \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{3 d \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )^2} \]

[Out]

(3*d*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) + (d*(11*b*c - 12*a*d)*Sqrt[a + b/x])/(4
*c^3*(b*c - a*d)*(c + d/x)) + (Sqrt[a + b/x]*x)/(c*(c + d/x)^2) + (Sqrt[d]*(15*b
^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]
])/(4*c^4*(b*c - a*d)^(3/2)) + ((b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(S
qrt[a]*c^4)

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Rubi [A]  time = 0.968426, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{3/2}}+\frac{(b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{\sqrt{a} c^4}+\frac{d \sqrt{a+\frac{b}{x}} (11 b c-12 a d)}{4 c^3 \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{3 d \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*d*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) + (d*(11*b*c - 12*a*d)*Sqrt[a + b/x])/(4
*c^3*(b*c - a*d)*(c + d/x)) + (Sqrt[a + b/x]*x)/(c*(c + d/x)^2) + (Sqrt[d]*(15*b
^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]
])/(4*c^4*(b*c - a*d)^(3/2)) + ((b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(S
qrt[a]*c^4)

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Rubi in Sympy [A]  time = 111.143, size = 182, normalized size = 0.85 \[ \frac{x \sqrt{a + \frac{b}{x}}}{c \left (c + \frac{d}{x}\right )^{2}} + \frac{3 d \sqrt{a + \frac{b}{x}}}{2 c^{2} \left (c + \frac{d}{x}\right )^{2}} + \frac{d \sqrt{a + \frac{b}{x}} \left (12 a d - 11 b c\right )}{4 c^{3} \left (c + \frac{d}{x}\right ) \left (a d - b c\right )} + \frac{\sqrt{d} \left (24 a^{2} d^{2} - 40 a b c d + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{4 c^{4} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{\left (6 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{\sqrt{a} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b/x)/(c*(c + d/x)**2) + 3*d*sqrt(a + b/x)/(2*c**2*(c + d/x)**2) + d*s
qrt(a + b/x)*(12*a*d - 11*b*c)/(4*c**3*(c + d/x)*(a*d - b*c)) + sqrt(d)*(24*a**2
*d**2 - 40*a*b*c*d + 15*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/
(4*c**4*(a*d - b*c)**(3/2)) - (6*a*d - b*c)*atanh(sqrt(a + b/x)/sqrt(a))/(sqrt(a
)*c**4)

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Mathematica [C]  time = 0.78896, size = 275, normalized size = 1.29 \[ \frac{\frac{i \sqrt{d} \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right ) \log \left (-\frac{8 i c^5 \sqrt{b c-a d} \left (-2 i \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 a d x+b c x-b d\right )}{d^{3/2} (c x+d) \left (24 a^2 d^2-40 a b c d+15 b^2 c^2\right )}\right )}{(b c-a d)^{3/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (b c \left (4 c^2 x^2+17 c d x+11 d^2\right )-2 a d \left (2 c^2 x^2+9 c d x+6 d^2\right )\right )}{(c x+d)^2 (b c-a d)}+\frac{4 (b c-6 a d) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{\sqrt{a}}}{8 c^4} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*Sqrt[a + b/x]*x*(-2*a*d*(6*d^2 + 9*c*d*x + 2*c^2*x^2) + b*c*(11*d^2 + 17*c
*d*x + 4*c^2*x^2)))/((b*c - a*d)*(d + c*x)^2) + (4*(b*c - 6*a*d)*Log[b + 2*a*x +
2*Sqrt[a]*Sqrt[a + b/x]*x])/Sqrt[a] + (I*Sqrt[d]*(15*b^2*c^2 - 40*a*b*c*d + 24*
a^2*d^2)*Log[((-8*I)*c^5*Sqrt[b*c - a*d]*(-(b*d) + b*c*x - 2*a*d*x - (2*I)*Sqrt[
d]*Sqrt[b*c - a*d]*Sqrt[a + b/x]*x))/(d^(3/2)*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*
d^2)*(d + c*x))])/(b*c - a*d)^(3/2))/(8*c^4)

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Maple [B]  time = 0.028, size = 1963, normalized size = 9.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/(c+d/x)^3,x)

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(104*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2
))*a^2*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^3*d^3-24*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)
*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(7/2)*d^6+128*ln((2*(x*(a*x+b))^(1
/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(5/2)*x*b*c^2*d^4-11
0*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*
a^(3/2)*x*b^2*c^3*d^3-46*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*b*c^3
*d^3-12*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*c^5*d+14*(x*(a*x+b
))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*b*c^6+12*(x*(a*x+b))^(3/2)*a^(3/2)*
((a*d-b*c)*d/c^2)^(1/2)*x*c^5*d+64*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/
2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(5/2)*x^2*b*c^3*d^3-55*ln((2*(x*(a*x+b))^(1/2
)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(3/2)*x^2*b^2*c^4*d^2+
36*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*x*c^3*d^3-48*ln(1/2*(2*(x*(
a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*x*c^2*d^4-14
*(x*(a*x+b))^(3/2)*a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^6+22*(x*(a*x+b))^(1/2)*
a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^6+52*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/
2)+2*a*x+b)/a^(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*b*c^2*d^4+15*ln((2*(x*(a*x+b))^
(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(1/2)*x^2*b^3*c^5*
d-10*(x*(a*x+b))^(3/2)*a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*b*c^5*d-32*ln(1/2*(2*(x*(
a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^3*d^3+8*
ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*
b^3*c^5*d+30*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d
)/(c*x+d))*a^(1/2)*x*b^3*c^4*d^2+22*(x*(a*x+b))^(1/2)*a^(1/2)*((a*d-b*c)*d/c^2)^
(1/2)*b^2*c^4*d^2-24*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b
*c*x-b*d)/(c*x+d))*a^(7/2)*x^2*c^2*d^4+8*(x*(a*x+b))^(3/2)*a^(3/2)*((a*d-b*c)*d/
c^2)^(1/2)*c^4*d^2-48*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+
b*c*x-b*d)/(c*x+d))*a^(7/2)*x*c*d^5+24*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^
2)^(1/2)*c^2*d^4+64*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*
c*x-b*d)/(c*x+d))*a^(5/2)*b*c*d^5-55*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(
1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*a^(3/2)*b^2*c^2*d^4-24*ln(1/2*(2*(x*(a*x+b))^
(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*c*d^5+4*ln(1/2*(2*(x
*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^3*c^6+4*
ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*b^
3*c^4*d^2+15*ln((2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*c-2*a*d*x+b*c*x-b*d
)/(c*x+d))*a^(1/2)*b^3*c^3*d^3+52*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^4*d^2-24*ln(1/2*(2*(x*(a*x+b))^(1/2)
*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*x^2*c^3*d^3-32*ln(1/2*(2*
(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^
5*d-64*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*((a*d-b*c)*d/c^2)
^(1/2)*x*b^2*c^4*d^2+44*(x*(a*x+b))^(1/2)*a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*
c^5*d-78*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^4*d^2-18*(x*(a*
x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^5*d)/(x*(a*x+b))^(1/2)/c^5/(
a*d-b*c)^2/(c*x+d)^2/((a*d-b*c)*d/c^2)^(1/2)/a^(1/2)

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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 = 0.402124, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*((15*b^2*c^2*d^2 - 40*a*b*c*d^3 + 24*a^2*d^4 + (15*b^2*c^4 - 40*a*b*c^3*d
+ 24*a^2*c^2*d^2)*x^2 + 2*(15*b^2*c^3*d - 40*a*b*c^2*d^2 + 24*a^2*c*d^3)*x)*sqrt
(a)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x +
b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) - 2*(4*(b*c^4 - a*c^3*d)*x^3 + (17*b*c
^3*d - 18*a*c^2*d^2)*x^2 + (11*b*c^2*d^2 - 12*a*c*d^3)*x)*sqrt(a)*sqrt((a*x + b)
/x) + 4*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*
c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*log(-2*a*x*sqrt((a
*x + b)/x) + (2*a*x + b)*sqrt(a)))/((b*c^5*d^2 - a*c^4*d^3 + (b*c^7 - a*c^6*d)*x
^2 + 2*(b*c^6*d - a*c^5*d^2)*x)*sqrt(a)), 1/4*((15*b^2*c^2*d^2 - 40*a*b*c*d^3 +
24*a^2*d^4 + (15*b^2*c^4 - 40*a*b*c^3*d + 24*a^2*c^2*d^2)*x^2 + 2*(15*b^2*c^3*d
- 40*a*b*c^2*d^2 + 24*a^2*c*d^3)*x)*sqrt(a)*sqrt(d/(b*c - a*d))*arctan(-(b*c - a
*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) + (4*(b*c^4 - a*c^3*d)*x^3 + (17*
b*c^3*d - 18*a*c^2*d^2)*x^2 + (11*b*c^2*d^2 - 12*a*c*d^3)*x)*sqrt(a)*sqrt((a*x +
b)/x) - 2*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a
^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d^2 + 6*a^2*c*d^3)*x)*log(-2*a*x*sqrt
((a*x + b)/x) + (2*a*x + b)*sqrt(a)))/((b*c^5*d^2 - a*c^4*d^3 + (b*c^7 - a*c^6*d
)*x^2 + 2*(b*c^6*d - a*c^5*d^2)*x)*sqrt(a)), -1/8*((15*b^2*c^2*d^2 - 40*a*b*c*d^
3 + 24*a^2*d^4 + (15*b^2*c^4 - 40*a*b*c^3*d + 24*a^2*c^2*d^2)*x^2 + 2*(15*b^2*c^
3*d - 40*a*b*c^2*d^2 + 24*a^2*c*d^3)*x)*sqrt(-a)*sqrt(-d/(b*c - a*d))*log(-(2*(b
*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x
+ d)) - 2*(4*(b*c^4 - a*c^3*d)*x^3 + (17*b*c^3*d - 18*a*c^2*d^2)*x^2 + (11*b*c^
2*d^2 - 12*a*c*d^3)*x)*sqrt(-a)*sqrt((a*x + b)/x) + 8*(b^2*c^2*d^2 - 7*a*b*c*d^3
+ 6*a^2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*
b*c^2*d^2 + 6*a^2*c*d^3)*x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))))/((b*c^5*d^2
- a*c^4*d^3 + (b*c^7 - a*c^6*d)*x^2 + 2*(b*c^6*d - a*c^5*d^2)*x)*sqrt(-a)), 1/4*
((15*b^2*c^2*d^2 - 40*a*b*c*d^3 + 24*a^2*d^4 + (15*b^2*c^4 - 40*a*b*c^3*d + 24*a
^2*c^2*d^2)*x^2 + 2*(15*b^2*c^3*d - 40*a*b*c^2*d^2 + 24*a^2*c*d^3)*x)*sqrt(-a)*s
qrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))
) + (4*(b*c^4 - a*c^3*d)*x^3 + (17*b*c^3*d - 18*a*c^2*d^2)*x^2 + (11*b*c^2*d^2 -
12*a*c*d^3)*x)*sqrt(-a)*sqrt((a*x + b)/x) - 4*(b^2*c^2*d^2 - 7*a*b*c*d^3 + 6*a^
2*d^4 + (b^2*c^4 - 7*a*b*c^3*d + 6*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 7*a*b*c^2*d
^2 + 6*a^2*c*d^3)*x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))))/((b*c^5*d^2 - a*c^4
*d^3 + (b*c^7 - a*c^6*d)*x^2 + 2*(b*c^6*d - a*c^5*d^2)*x)*sqrt(-a))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.561121, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x