3.145 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{\left (c+\frac{d}{x}\right )^3} \, dx\)

Optimal. Leaf size=237 \[ \frac{a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}-\frac{\sqrt{b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 d^{3/2}}-\frac{\sqrt{a+\frac{b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{4 c^3 d \left (c+\frac{d}{x}\right )}+\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left (c+\frac{d}{x}\right )^2}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c \left (c+\frac{d}{x}\right )^2} \]

[Out]

((b*c - 3*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(2*c^2*d*(c + d/x)^2) - ((b^2*c^2 + 7*
a*b*c*d - 12*a^2*d^2)*Sqrt[a + b/x])/(4*c^3*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)
/(c*(c + d/x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 24*a^2*d^2)*ArcTan[(S
qrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*d^(3/2)) + (a^(3/2)*(5*b*c - 6*a*
d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^4

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Rubi [A]  time = 1.0225, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}-\frac{\sqrt{b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 d^{3/2}}-\frac{\sqrt{a+\frac{b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{4 c^3 d \left (c+\frac{d}{x}\right )}+\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left (c+\frac{d}{x}\right )^2}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(5/2)/(c + d/x)^3,x]

[Out]

((b*c - 3*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(2*c^2*d*(c + d/x)^2) - ((b^2*c^2 + 7*
a*b*c*d - 12*a^2*d^2)*Sqrt[a + b/x])/(4*c^3*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)
/(c*(c + d/x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 24*a^2*d^2)*ArcTan[(S
qrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*d^(3/2)) + (a^(3/2)*(5*b*c - 6*a*
d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^4

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Rubi in Sympy [A]  time = 106.304, size = 204, normalized size = 0.86 \[ - \frac{a^{\frac{3}{2}} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{c^{4}} + \frac{a x \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{c \left (c + \frac{d}{x}\right )^{2}} + \frac{\sqrt{a + \frac{b}{x}} \left (a d - b c\right ) \left (3 a d - b c\right )}{2 c^{2} d \left (c + \frac{d}{x}\right )^{2}} + \frac{\sqrt{a + \frac{b}{x}} \left (12 a^{2} d^{2} - 7 a b c d - b^{2} c^{2}\right )}{4 c^{3} d \left (c + \frac{d}{x}\right )} + \frac{\sqrt{a d - b c} \left (24 a^{2} d^{2} - 8 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{4 c^{4} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**(5/2)/(c+d/x)**3,x)

[Out]

-a**(3/2)*(6*a*d - 5*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/c**4 + a*x*(a + b/x)**(3/
2)/(c*(c + d/x)**2) + sqrt(a + b/x)*(a*d - b*c)*(3*a*d - b*c)/(2*c**2*d*(c + d/x
)**2) + sqrt(a + b/x)*(12*a**2*d**2 - 7*a*b*c*d - b**2*c**2)/(4*c**3*d*(c + d/x)
) + sqrt(a*d - b*c)*(24*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*d**(3/2))

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Mathematica [C]  time = 0.651414, size = 304, normalized size = 1.28 \[ \frac{-4 a^{3/2} (6 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (2 a^2 d \left (2 c^2 x^2+9 c d x+6 d^2\right )-a b c d (11 c x+7 d)+b^2 c^2 (c x-d)\right )}{d (c x+d)^2}-\frac{i \left (24 a^3 d^3-32 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \log \left (\frac{8 c^5 \left (2 d x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d^{3/2} x-i b \sqrt{d} (d-c x)\right )}{(c x+d) \sqrt{b c-a d} \left (24 a^3 d^3-32 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right )}\right )}{d^{3/2} \sqrt{b c-a d}}}{8 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^(5/2)/(c + d/x)^3,x]

[Out]

((2*c*Sqrt[a + b/x]*x*(b^2*c^2*(-d + c*x) - a*b*c*d*(7*d + 11*c*x) + 2*a^2*d*(6*
d^2 + 9*c*d*x + 2*c^2*x^2)))/(d*(d + c*x)^2) - 4*a^(3/2)*(-5*b*c + 6*a*d)*Log[b
+ 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x] - (I*(b^3*c^3 + 7*a*b^2*c^2*d - 32*a^2*b*c*
d^2 + 24*a^3*d^3)*Log[(8*c^5*((-2*I)*a*d^(3/2)*x + 2*d*Sqrt[b*c - a*d]*Sqrt[a +
b/x]*x - I*b*Sqrt[d]*(d - c*x)))/(Sqrt[b*c - a*d]*(b^3*c^3 + 7*a*b^2*c^2*d - 32*
a^2*b*c*d^2 + 24*a^3*d^3)*(d + c*x))])/(d^(3/2)*Sqrt[b*c - a*d]))/(8*c^4)

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Maple [B]  time = 0.022, size = 2807, normalized size = 11.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(5/2)/(c+d/x)^3,x)

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(-22*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a^2*b*c
^6*d^2+8*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a*b^2*c^7*d-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^5*c^5*d^3+20*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^3*c^6*d^2+22*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*a^3*b*c^6*d
^2-128*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^2*c^4*d^4-24*a^(9/2)*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^7+2*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^
(1/2)*x*b^3*c^8-2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^4*c^8-24*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^5
*c^2*d^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^2*b^5*c^7*d+8*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*a^3*c^4*d^4+6*(
x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*b^3*c^7*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^5*c*d^7-2*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^5*c^6*d^2+24
*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^4*c^2*d^6-2*(x*(a*x+b))^(1/2)*((a*d
-b*c)*d/c^2)^(1/2)*b^4*c^6*d^2+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^4*b*c*d^7-95*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^2*c^2*d^6+45*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^3*c^3*d^5-5*
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^4*c^4*d^4+68*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*b*c^4*d^4+136*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*b*c^3*d^5-64*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^2*c^5*d^
3-102*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a^3*b*c^4*d^4+92*(x*(a*x+b))^(
1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a^2*b^2*c^5*d^3-22*(x*(a*x+b))^(1/2)*((a*d-b*c)*d
/c^2)^(1/2)*x*a*b^3*c^6*d^2-8*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*a^2*
b^2*c^7*d+40*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^3*c^5*d^3-190*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^2*c^3*d^5+90*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^3*c^4*d^4-10*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^4
*c^5*d^3-62*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^3*b*c^3*d^5+50*(x*(a*x+b
))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^2*b^2*c^4*d^4-10*(x*(a*x+b))^(1/2)*((a*d-b*c)
*d/c^2)^(1/2)*a*b^3*c^5*d^3-24*a^(9/2)*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^5-24*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^5*d^8-48*a^(9/2)*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^6+68*
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)*b*c^2*d^6-12*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*a^4*c^5*d^3-2*(
x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^3*a*b^3*c^8-64*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^2*c^3*d^5+12*(
x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a^3*c^5*d^3+20*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^3*c^4*d^4+80*l
n((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*b*c^3*d^5-95*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^2*c^4*d^4+45*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^3*c^5*d^3-5*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^4*c^6*d^2-10*(x
*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*a^2*b*c^5*d^3-4*(x*(a*x+b))^(3/2)*((a*d-
b*c)*d/c^2)^(1/2)*a*b^2*c^6*d^2+36*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*a
^4*c^3*d^5-4*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^4*c^7*d+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))*x*a^4*b*c^2
*d^6-18*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*a^3*b*c^5*d^3-14*(x*(a*x+b
))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*a*b^3*c^7*d+34*(x*(a*x+b))^(1/2)*((a*d-b*c)
*d/c^2)^(1/2)*x^2*a^2*b^2*c^6*d^2)/(x*(a*x+b))^(1/2)/c^5/(a*d-b*c)^2/d^2/((a*d-b
*c)*d/c^2)^(1/2)/(c*x+d)^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((a + b/x)^(5/2)/(c + d/x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.342362, 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)^(5/2)/(c + d/x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{\left (c x + d\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**(5/2)/(c+d/x)**3,x)

[Out]

Integral(x**3*(a + b/x)**(5/2)/(c*x + d)**3, x)

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GIAC/XCAS [A]  time = 0.549039, 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)^(5/2)/(c + d/x)^3,x, algorithm="giac")

[Out]

sage0*x