3.152 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^3} \, dx\)

Optimal. Leaf size=250 \[ -\frac{(6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^4}-\frac{d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}+\frac{d \sqrt{a+\frac{b}{x}} (b c-4 a d) (4 b c-3 a d)}{4 a c^3 \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{d \sqrt{a+\frac{b}{x}} (2 b c-3 a d)}{2 a c^2 \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left (c+\frac{d}{x}\right )^2} \]

[Out]

(d*(2*b*c - 3*a*d)*Sqrt[a + b/x])/(2*a*c^2*(b*c - a*d)*(c + d/x)^2) + (d*(b*c -
4*a*d)*(4*b*c - 3*a*d)*Sqrt[a + b/x])/(4*a*c^3*(b*c - a*d)^2*(c + d/x)) + (Sqrt[
a + b/x]*x)/(a*c*(c + d/x)^2) - (d^(3/2)*(35*b^2*c^2 - 56*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)^(5/2)) - ((b
*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(3/2)*c^4)

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Rubi [A]  time = 1.08312, antiderivative size = 250, 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 \[ -\frac{(6 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2} c^4}-\frac{d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{5/2}}+\frac{d \sqrt{a+\frac{b}{x}} (b c-4 a d) (4 b c-3 a d)}{4 a c^3 \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{d \sqrt{a+\frac{b}{x}} (2 b c-3 a d)}{2 a c^2 \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x \sqrt{a+\frac{b}{x}}}{a c \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*(c + d/x)^3),x]

[Out]

(d*(2*b*c - 3*a*d)*Sqrt[a + b/x])/(2*a*c^2*(b*c - a*d)*(c + d/x)^2) + (d*(b*c -
4*a*d)*(4*b*c - 3*a*d)*Sqrt[a + b/x])/(4*a*c^3*(b*c - a*d)^2*(c + d/x)) + (Sqrt[
a + b/x]*x)/(a*c*(c + d/x)^2) - (d^(3/2)*(35*b^2*c^2 - 56*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)^(5/2)) - ((b
*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(3/2)*c^4)

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Rubi in Sympy [A]  time = 117.385, size = 218, normalized size = 0.87 \[ - \frac{d x \sqrt{a + \frac{b}{x}}}{2 c \left (c + \frac{d}{x}\right )^{2} \left (a d - b c\right )} - \frac{3 d x \sqrt{a + \frac{b}{x}} \left (2 a d - 3 b c\right )}{4 c^{2} \left (c + \frac{d}{x}\right ) \left (a d - b c\right )^{2}} + \frac{d^{\frac{3}{2}} \left (24 a^{2} d^{2} - 56 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{4 c^{4} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{x \sqrt{a + \frac{b}{x}} \left (3 a d - 4 b c\right ) \left (4 a d - b c\right )}{4 a c^{3} \left (a d - b c\right )^{2}} - \frac{\left (6 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*x*sqrt(a + b/x)/(2*c*(c + d/x)**2*(a*d - b*c)) - 3*d*x*sqrt(a + b/x)*(2*a*d -
3*b*c)/(4*c**2*(c + d/x)*(a*d - b*c)**2) + d**(3/2)*(24*a**2*d**2 - 56*a*b*c*d
+ 35*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(4*c**4*(a*d - b*c)
**(5/2)) + x*sqrt(a + b/x)*(3*a*d - 4*b*c)*(4*a*d - b*c)/(4*a*c**3*(a*d - b*c)**
2) - (6*a*d + b*c)*atanh(sqrt(a + b/x)/sqrt(a))/(a**(3/2)*c**4)

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Mathematica [C]  time = 0.950155, size = 301, normalized size = 1.2 \[ \frac{-\frac{4 (6 a d+b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{3/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (2 a^2 d^2 \left (2 c^2 x^2+9 c d x+6 d^2\right )-a b c d \left (8 c^2 x^2+29 c d x+19 d^2\right )+4 b^2 c^2 (c x+d)^2\right )}{a (c x+d)^2 (b c-a d)^2}-\frac{i d^{3/2} \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right ) \log \left (\frac{8 c^5 (b c-a d)^{3/2} \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d x-i b (d-c x)\right )}{d^{5/2} (c x+d) \left (24 a^2 d^2-56 a b c d+35 b^2 c^2\right )}\right )}{(b c-a d)^{5/2}}}{8 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*(c + d/x)^3),x]

[Out]

((2*c*Sqrt[a + b/x]*x*(4*b^2*c^2*(d + c*x)^2 + 2*a^2*d^2*(6*d^2 + 9*c*d*x + 2*c^
2*x^2) - a*b*c*d*(19*d^2 + 29*c*d*x + 8*c^2*x^2)))/(a*(b*c - a*d)^2*(d + c*x)^2)
- (4*(b*c + 6*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(3/2) - (I*d^(
3/2)*(35*b^2*c^2 - 56*a*b*c*d + 24*a^2*d^2)*Log[(8*c^5*(b*c - a*d)^(3/2)*((-2*I)
*a*d*x + 2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[a + b/x]*x - I*b*(d - c*x)))/(d^(5/2)*(3
5*b^2*c^2 - 56*a*b*c*d + 24*a^2*d^2)*(d + c*x))])/(b*c - a*d)^(5/2))/(8*c^4)

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Maple [B]  time = 0.025, size = 2269, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(-12*(x*(a*x+b))^(1/2)*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*x
^3*c^5*d^2+12*(x*(a*x+b))^(3/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x*c^5*d^2+68*ln(
1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*b
*c^2*d^5+80*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^(9/2)*x^2*b*c^3*d^4-91*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*b^2*c^4*d^3+35*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^3*c^5*d^
2-18*(x*(a*x+b))^(3/2)*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*b*c^5*d^2+36*(x*(a*x+b))^
(1/2)*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*x*c^3*d^4-60*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)*b^2*c^3*d^4+160*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^(9/2)*x*b*c
^2*d^5-182*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*b^2*c^3*d^4+70*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^3*c^4*d^3-62*(x*(a*x+b))^(1/2)*a^(7/
2)*((a*d-b*c)*d/c^2)^(1/2)*b*c^3*d^4-8*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^
2)^(1/2)*x^2*b^3*c^7+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*a*b^4*c^7-8*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)
^(1/2)*b^3*c^5*d^2+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)*a*b^4*c^5*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^(11/2)*d^7+12*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^3*c^4*d^3-24*ln(1/2*(2*(x*
(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d/c^2)^(1/2)*x^2*c^3*d^4
-48*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d/c^2)^
(1/2)*x*c^2*d^5-18*(x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^5*d
^2+46*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^6*d+68*ln(1/2*
(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*x^2*b
*c^4*d^3+22*(x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*b*c^6*d+136*ln
(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*
x*b*c^3*d^4-60*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*b^2*c^5*d^2-22*(x*(a*x+b))^(3/2)*a^(5/2)*((a*d-b*c)*d/c^2)^
(1/2)*x*b*c^6*d+92*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^5*d
^2+24*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^3*c^5*d^2-120*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*b^2*c^4*d^3+12*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^3*c^6*d-102*(x*(a*x+b))^(1/
2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^4*d^3-16*(x*(a*x+b))^(1/2)*a^(3/2)*((a*
d-b*c)*d/c^2)^(1/2)*x*b^3*c^6*d+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*a*b^4*c^6*d-24*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(
1/2)+2*a*x+b)/a^(1/2))*a^5*((a*d-b*c)*d/c^2)^(1/2)*c*d^6-24*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^(11/2)*x^2*c^2*d^5+8
*(x*(a*x+b))^(3/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*c^4*d^3-48*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^(11/2)*x*c*d^6+24*
(x*(a*x+b))^(1/2)*a^(9/2)*((a*d-b*c)*d/c^2)^(1/2)*c^2*d^5+80*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^(9/2)*b*c*d^6-91*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)*b^2*c^2*d^5+35*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^3*c^3*d^4+46*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c
^2)^(1/2)*b^2*c^4*d^3)/(x*(a*x+b))^(1/2)/c^5/(a*d-b*c)^3/(c*x+d)^2/((a*d-b*c)*d/
c^2)^(1/2)/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(1/(sqrt(a + b/x)*(c + d/x)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.766872, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*(c + d/x)^3),x, algorithm="fricas")

[Out]

[1/8*((35*a*b^2*c^2*d^3 - 56*a^2*b*c*d^4 + 24*a^3*d^5 + (35*a*b^2*c^4*d - 56*a^2
*b*c^3*d^2 + 24*a^3*c^2*d^3)*x^2 + 2*(35*a*b^2*c^3*d^2 - 56*a^2*b*c^2*d^3 + 24*a
^3*c*d^4)*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^2*c^5 - 2*a
*b*c^4*d + a^2*c^3*d^2)*x^3 + (8*b^2*c^4*d - 29*a*b*c^3*d^2 + 18*a^2*c^2*d^3)*x^
2 + (4*b^2*c^3*d^2 - 19*a*b*c^2*d^3 + 12*a^2*c*d^4)*x)*sqrt(a)*sqrt((a*x + b)/x)
+ 4*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*
a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*d + 4*a*b^2*c^3
*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x
+ b)*sqrt(a)))/((a*b^2*c^6*d^2 - 2*a^2*b*c^5*d^3 + a^3*c^4*d^4 + (a*b^2*c^8 - 2*
a^2*b*c^7*d + a^3*c^6*d^2)*x^2 + 2*(a*b^2*c^7*d - 2*a^2*b*c^6*d^2 + a^3*c^5*d^3)
*x)*sqrt(a)), 1/8*((35*a*b^2*c^2*d^3 - 56*a^2*b*c*d^4 + 24*a^3*d^5 + (35*a*b^2*c
^4*d - 56*a^2*b*c^3*d^2 + 24*a^3*c^2*d^3)*x^2 + 2*(35*a*b^2*c^3*d^2 - 56*a^2*b*c
^2*d^3 + 24*a^3*c*d^4)*x)*sqrt(-a)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sq
rt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d)) + 2*(4*
(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*x^3 + (8*b^2*c^4*d - 29*a*b*c^3*d^2 + 18*a
^2*c^2*d^3)*x^2 + (4*b^2*c^3*d^2 - 19*a*b*c^2*d^3 + 12*a^2*c*d^4)*x)*sqrt(-a)*sq
rt((a*x + b)/x) + 8*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5
+ (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*
d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)*arctan(a/(sqrt(-a)*sqrt
((a*x + b)/x))))/((a*b^2*c^6*d^2 - 2*a^2*b*c^5*d^3 + a^3*c^4*d^4 + (a*b^2*c^8 -
2*a^2*b*c^7*d + a^3*c^6*d^2)*x^2 + 2*(a*b^2*c^7*d - 2*a^2*b*c^6*d^2 + a^3*c^5*d^
3)*x)*sqrt(-a)), -1/4*((35*a*b^2*c^2*d^3 - 56*a^2*b*c*d^4 + 24*a^3*d^5 + (35*a*b
^2*c^4*d - 56*a^2*b*c^3*d^2 + 24*a^3*c^2*d^3)*x^2 + 2*(35*a*b^2*c^3*d^2 - 56*a^2
*b*c^2*d^3 + 24*a^3*c*d^4)*x)*sqrt(a)*sqrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*sq
rt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) - (4*(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d
^2)*x^3 + (8*b^2*c^4*d - 29*a*b*c^3*d^2 + 18*a^2*c^2*d^3)*x^2 + (4*b^2*c^3*d^2 -
19*a*b*c^2*d^3 + 12*a^2*c*d^4)*x)*sqrt(a)*sqrt((a*x + b)/x) - 2*(b^3*c^3*d^2 +
4*a*b^2*c^2*d^3 - 11*a^2*b*c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2
*b*c^3*d^2 + 6*a^3*c^2*d^3)*x^2 + 2*(b^3*c^4*d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*
d^3 + 6*a^3*c*d^4)*x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)))/((a*b
^2*c^6*d^2 - 2*a^2*b*c^5*d^3 + a^3*c^4*d^4 + (a*b^2*c^8 - 2*a^2*b*c^7*d + a^3*c^
6*d^2)*x^2 + 2*(a*b^2*c^7*d - 2*a^2*b*c^6*d^2 + a^3*c^5*d^3)*x)*sqrt(a)), -1/4*(
(35*a*b^2*c^2*d^3 - 56*a^2*b*c*d^4 + 24*a^3*d^5 + (35*a*b^2*c^4*d - 56*a^2*b*c^3
*d^2 + 24*a^3*c^2*d^3)*x^2 + 2*(35*a*b^2*c^3*d^2 - 56*a^2*b*c^2*d^3 + 24*a^3*c*d
^4)*x)*sqrt(-a)*sqrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*s
qrt((a*x + b)/x))) - (4*(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*x^3 + (8*b^2*c^4*d
- 29*a*b*c^3*d^2 + 18*a^2*c^2*d^3)*x^2 + (4*b^2*c^3*d^2 - 19*a*b*c^2*d^3 + 12*a
^2*c*d^4)*x)*sqrt(-a)*sqrt((a*x + b)/x) - 4*(b^3*c^3*d^2 + 4*a*b^2*c^2*d^3 - 11*
a^2*b*c*d^4 + 6*a^3*d^5 + (b^3*c^5 + 4*a*b^2*c^4*d - 11*a^2*b*c^3*d^2 + 6*a^3*c^
2*d^3)*x^2 + 2*(b^3*c^4*d + 4*a*b^2*c^3*d^2 - 11*a^2*b*c^2*d^3 + 6*a^3*c*d^4)*x)
*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))))/((a*b^2*c^6*d^2 - 2*a^2*b*c^5*d^3 + a^3
*c^4*d^4 + (a*b^2*c^8 - 2*a^2*b*c^7*d + a^3*c^6*d^2)*x^2 + 2*(a*b^2*c^7*d - 2*a^
2*b*c^6*d^2 + a^3*c^5*d^3)*x)*sqrt(-a))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.261172, size = 459, normalized size = 1.84 \[ -\frac{1}{4} \, b{\left (\frac{{\left (35 \, b^{2} c^{2} d^{2} - 56 \, a b c d^{3} + 24 \, a^{2} d^{4}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{3} c^{6} - 2 \, a b^{2} c^{5} d + a^{2} b c^{4} d^{2}\right )} \sqrt{b c d - a d^{2}}} + \frac{13 \, b^{2} c^{2} d^{2} \sqrt{\frac{a x + b}{x}} - 21 \, a b c d^{3} \sqrt{\frac{a x + b}{x}} + 8 \, a^{2} d^{4} \sqrt{\frac{a x + b}{x}} + \frac{11 \,{\left (a x + b\right )} b c d^{3} \sqrt{\frac{a x + b}{x}}}{x} - \frac{8 \,{\left (a x + b\right )} a d^{4} \sqrt{\frac{a x + b}{x}}}{x}}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}{\left (b c - a d + \frac{{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac{4 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a c^{3}} - \frac{4 \,{\left (b c + 6 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*(c + d/x)^3),x, algorithm="giac")

[Out]

-1/4*b*((35*b^2*c^2*d^2 - 56*a*b*c*d^3 + 24*a^2*d^4)*arctan(d*sqrt((a*x + b)/x)/
sqrt(b*c*d - a*d^2))/((b^3*c^6 - 2*a*b^2*c^5*d + a^2*b*c^4*d^2)*sqrt(b*c*d - a*d
^2)) + (13*b^2*c^2*d^2*sqrt((a*x + b)/x) - 21*a*b*c*d^3*sqrt((a*x + b)/x) + 8*a^
2*d^4*sqrt((a*x + b)/x) + 11*(a*x + b)*b*c*d^3*sqrt((a*x + b)/x)/x - 8*(a*x + b)
*a*d^4*sqrt((a*x + b)/x)/x)/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*(b*c - a*d +
(a*x + b)*d/x)^2) + 4*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a*c^3) - 4*(b*c + 6*a
*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b*c^4))