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

Optimal. Leaf size=209 \[ -\frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 \sqrt{d} \sqrt{b c-a d}}+\frac{3 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}-\frac{3 \sqrt{a+\frac{b}{x}} (b c-4 a d)}{4 c^3 \left (c+\frac{d}{x}\right )}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )^2} \]

[Out]

-((b*c - 3*a*d)*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) - (3*(b*c - 4*a*d)*Sqrt[a + b
/x])/(4*c^3*(c + d/x)) + (a*Sqrt[a + b/x]*x)/(c*(c + d/x)^2) - (3*(b^2*c^2 - 8*a
*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*Sqrt
[d]*Sqrt[b*c - a*d]) + (3*Sqrt[a]*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/
c^4

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Rubi [A]  time = 0.983858, antiderivative size = 209, 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 \[ -\frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 \sqrt{d} \sqrt{b c-a d}}+\frac{3 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}-\frac{3 \sqrt{a+\frac{b}{x}} (b c-4 a d)}{4 c^3 \left (c+\frac{d}{x}\right )}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - 3*a*d)*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) - (3*(b*c - 4*a*d)*Sqrt[a + b
/x])/(4*c^3*(c + d/x)) + (a*Sqrt[a + b/x]*x)/(c*(c + d/x)^2) - (3*(b^2*c^2 - 8*a
*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*Sqrt
[d]*Sqrt[b*c - a*d]) + (3*Sqrt[a]*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/
c^4

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Rubi in Sympy [A]  time = 106.166, size = 182, normalized size = 0.87 \[ - \frac{6 \sqrt{a} \left (a d - \frac{b c}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{c^{4}} + \frac{a x \sqrt{a + \frac{b}{x}}}{c \left (c + \frac{d}{x}\right )^{2}} + \frac{\sqrt{a + \frac{b}{x}} \left (3 a d - b c\right )}{2 c^{2} \left (c + \frac{d}{x}\right )^{2}} + \frac{3 \sqrt{a + \frac{b}{x}} \left (4 a d - b c\right )}{4 c^{3} \left (c + \frac{d}{x}\right )} + \frac{3 \left (8 a^{2} d^{2} - 8 a b c d + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{4 c^{4} \sqrt{d} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*sqrt(a)*(a*d - b*c/2)*atanh(sqrt(a + b/x)/sqrt(a))/c**4 + a*x*sqrt(a + b/x)/(
c*(c + d/x)**2) + sqrt(a + b/x)*(3*a*d - b*c)/(2*c**2*(c + d/x)**2) + 3*sqrt(a +
b/x)*(4*a*d - b*c)/(4*c**3*(c + d/x)) + 3*(8*a**2*d**2 - 8*a*b*c*d + b**2*c**2)
*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(4*c**4*sqrt(d)*sqrt(a*d - b*c))

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Mathematica [C]  time = 0.512074, size = 256, normalized size = 1.22 \[ \frac{\frac{3 i \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \log \left (\frac{8 c^5 \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}+2 i a d x+i b (d-c x)\right )}{3 \sqrt{d} (c x+d) \sqrt{b c-a d} \left (8 a^2 d^2-8 a b c d+b^2 c^2\right )}\right )}{\sqrt{d} \sqrt{b c-a d}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (2 a \left (2 c^2 x^2+9 c d x+6 d^2\right )-b c (5 c x+3 d)\right )}{(c x+d)^2}-12 \sqrt{a} (2 a d-b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{8 c^4} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*Sqrt[a + b/x]*x*(-(b*c*(3*d + 5*c*x)) + 2*a*(6*d^2 + 9*c*d*x + 2*c^2*x^2))
)/(d + c*x)^2 - 12*Sqrt[a]*(-(b*c) + 2*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b
/x]*x] + ((3*I)*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*Log[(8*c^5*((2*I)*a*d*x + 2*Sq
rt[d]*Sqrt[b*c - a*d]*Sqrt[a + b/x]*x + I*b*(d - c*x)))/(3*Sqrt[d]*Sqrt[b*c - a*
d]*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*(d + c*x))])/(Sqrt[d]*Sqrt[b*c - a*d]))/(8*
c^4)

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Maple [B]  time = 0.021, size = 2373, normalized size = 11.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/(c+d/x)^3,x)

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(-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^4*d^7+60*a^(5/2)*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*c^4*d^3+120*a^(5/2)*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*c^3*d^4-
48*a^(3/2)*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^2*c^5*d^2+18*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*a^2*b*
c^6*d-96*a^(3/2)*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^2*c^4*d^3+12*a^(1/2)*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*d-18*(x*(a*x+b))^(3/2)*((a*d-b
*c)*d/c^2)^(1/2)*x*a*b*c^6*d-18*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*a^
2*b*c^5*d^2+24*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*a*b^2*c^6*d+24*a^(1
/2)*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^2-90*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a^2*b*c^4*d^3+66*
(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a*b^2*c^5*d^2-48*a^(3/2)*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^2*c^3*d^4-3
*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))*b
^4*c^4*d^3-24*a^(7/2)*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)*c*d^6+6*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^7-
6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^3*c^7-24*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))*x^2*a^4*c^2*d^5-3*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))*x^2*b^4
*c^6*d+8*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*a^2*c^4*d^3+2*(x*(a*x+b))^(3/
2)*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^6*d-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))*x*a^4*c*d^6-6*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))*x*b^4*c^5*d^2+24*(x*(a*x+b))^(
1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^3*c^2*d^5-6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(
1/2)*b^3*c^5*d^2+72*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*b*c*d^6-75*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^2*b^2*c^2*d^5+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*b^3*c^3*d^4+12*(x*(a*x+b))^(3/
2)*((a*d-b*c)*d/c^2)^(1/2)*x*a^2*c^5*d^2+36*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^
(1/2)*x*a^3*c^3*d^4-12*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^3*c^6*d+12*
a^(1/2)*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^3+144*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))*x*a^3*b*c^2*d^5-150*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))*x*a^2*b^2*c^3*d^4+60*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))*x*a*b^3*c^4*d^3-54*(x*(
a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^2*b*c^3*d^4+36*(x*(a*x+b))^(1/2)*((a*d-b
*c)*d/c^2)^(1/2)*a*b^2*c^4*d^3-6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*a
*b^2*c^7+60*a^(5/2)*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*c^2*d^5-48*a^(7/2)*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*c^2*d^5-12*(x*(a*x+b))^(1/2)*((a*d-b*c)*d
/c^2)^(1/2)*x^3*a^3*c^5*d^2-24*a^(7/2)*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*c^3*d^4+72*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))*x^2*a^3*b*c^3*d^4-75*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))*x^2*a^2*b^2
*c^4*d^3+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))*x^2*a*b^3*c^5*d^2-10*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*a*b*c^5
*d^2)/(x*(a*x+b))^(1/2)/c^5/(a*d-b*c)^2/((a*d-b*c)*d/c^2)^(1/2)/(c*x+d)^2/d

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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((a + b/x)^(3/2)/(c + d/x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.328166, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.550936, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x