3.144 \(\int \frac{\left (a+\frac{b}{x}\right )^{5/2}}{\left (c+\frac{d}{x}\right )^2} \, dx\)
Optimal. Leaf size=166 \[ \frac{a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^3}-\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 d^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} (b c-2 a d) (b c-a d)}{c^2 d \left (c+\frac{d}{x}\right )}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c \left (c+\frac{d}{x}\right )} \]
[Out]
((b*c - 2*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(c^2*d*(c + d/x)) + (a*(a + b/x)^(3/2)
*x)/(c*(c + d/x)) - ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/
x])/Sqrt[b*c - a*d]])/(c^3*d^(3/2)) + (a^(3/2)*(5*b*c - 4*a*d)*ArcTanh[Sqrt[a +
b/x]/Sqrt[a]])/c^3
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Rubi [A] time = 0.668661, antiderivative size = 166, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333
\[ \frac{a^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^3}-\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 d^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} (b c-2 a d) (b c-a d)}{c^2 d \left (c+\frac{d}{x}\right )}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c \left (c+\frac{d}{x}\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^(5/2)/(c + d/x)^2,x]
[Out]
((b*c - 2*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(c^2*d*(c + d/x)) + (a*(a + b/x)^(3/2)
*x)/(c*(c + d/x)) - ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/
x])/Sqrt[b*c - a*d]])/(c^3*d^(3/2)) + (a^(3/2)*(5*b*c - 4*a*d)*ArcTanh[Sqrt[a +
b/x]/Sqrt[a]])/c^3
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Rubi in Sympy [A] time = 63.2027, size = 138, normalized size = 0.83 \[ - \frac{a^{\frac{3}{2}} \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{c^{3}} + \frac{a x \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{c \left (c + \frac{d}{x}\right )} + \frac{\sqrt{a + \frac{b}{x}} \left (a d - b c\right ) \left (2 a d - b c\right )}{c^{2} d \left (c + \frac{d}{x}\right )} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{3} d^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(5/2)/(c+d/x)**2,x)
[Out]
-a**(3/2)*(4*a*d - 5*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/c**3 + a*x*(a + b/x)**(3/
2)/(c*(c + d/x)) + sqrt(a + b/x)*(a*d - b*c)*(2*a*d - b*c)/(c**2*d*(c + d/x)) +
(a*d - b*c)**(3/2)*(4*a*d + b*c)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c
**3*d**(3/2))
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Mathematica [C] time = 0.604074, size = 219, normalized size = 1.32 \[ -\frac{a^{3/2} (4 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )-\frac{2 c x \sqrt{a+\frac{b}{x}} \left (a^2 d (c x+2 d)-2 a b c d+b^2 c^2\right )}{d (c x+d)}+\frac{i (b c-a d)^{3/2} (4 a d+b c) \log \left (\frac{2 c^4 \left (2 d x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d^{3/2} x-i b \sqrt{d} (d-c x)\right )}{(c x+d) (b c-a d)^{5/2} (4 a d+b c)}\right )}{d^{3/2}}}{2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^(5/2)/(c + d/x)^2,x]
[Out]
-((-2*c*Sqrt[a + b/x]*x*(b^2*c^2 - 2*a*b*c*d + a^2*d*(2*d + c*x)))/(d*(d + c*x))
+ a^(3/2)*(-5*b*c + 4*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x] + (I*(b*c
- a*d)^(3/2)*(b*c + 4*a*d)*Log[(2*c^4*((-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)))/((b*c - a*d)^(5/2)*(b*c + 4*a*d)*(d +
c*x))])/d^(3/2))/(2*c^3)
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Maple [B] time = 0.024, size = 2014, normalized size = 12.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(5/2)/(c+d/x)^2,x)
[Out]
1/2*((a*x+b)/x)^(1/2)*x*(-4*a^(9/2)*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))*d^6+2*c^4*(x*(a*x+b))^(3/2)*a^(5/2)*d^2*((a*d-
b*c)*d/c^2)^(1/2)-4*a^(9/2)*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*c*d^5+4*(x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^
(1/2)*c^2*d^4+11*a^(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))*b*c*d^5-9*a^(5/2)*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))*b^2*c^2*d^4+a^(3/2)*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))*b^3*c^3*d^3-4*a^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)*c*d^5+2
*c^6*(x*(a*x+b))^(3/2)*b^2*a^(1/2)*((a*d-b*c)*d/c^2)^(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^4*c^6-ln(1/2*(2*(a*
x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*b^4*c^6+a^(1/
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))
*b^4*c^4*d^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^4*c^5*d-ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d
-b*c)*d/c^2)^(1/2)*b^4*c^5*d-5*a^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^2-5*a^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^3+a^(1/2)*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^4*c^5*d
-2*a^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*(a*x^2+b*x)^(1/2)*x*b^3*c^6-2*a^(1/2)*((a*d-b
*c)*d/c^2)^(1/2)*(a*x^2+b*x)^(1/2)*b^3*c^5*d-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^3*c^4*d^2+ln(1/2*(2*(a*x^2+b*x)^(
1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*((a*d-b*c)*d/c^2)^(1/2)*a*b^3*c^4*d^2+11*a^(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*c^2*d^4-9*a^(5/2)*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^2*c^3*d^3-2*(x*(a*x+b))^(1/2)*a^(7/2)*((a*d-b*c)*d/c^2)^(1
/2)*x^2*c^4*d^2-2*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^6-
4*c^5*(x*(a*x+b))^(3/2)*a^(3/2)*b*d*((a*d-b*c)*d/c^2)^(1/2)+2*(x*(a*x+b))^(1/2)*
a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*x*c^3*d^3-8*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)
*d/c^2)^(1/2)*b*c^3*d^3+4*(x*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*b^2*
c^4*d^2+2*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*(a*x^2+b*x)^(1/2)*b^2*c^4*d^2-4*a^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*c^
2*d^4+9*a^3*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^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*a*b^3*c^5*d+ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(
1/2))*((a*d-b*c)*d/c^2)^(1/2)*x*a*b^3*c^5*d-4*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*
c)*d/c^2)^(1/2)*x*b*c^4*d^2+4*(x*(a*x+b))^(1/2)*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*
x^2*b*c^5*d+2*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*(a*x^2+b*x)^(1/2)*x*b^2*c^5*d+2*(x
*(a*x+b))^(1/2)*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^5*d+9*a^3*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^3+a^(3
/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^3*c^4*d^2)/(x*(a*x+b))^(1/2)/c^4/d^2/a^(1/2)/(a*d-b*c)/((a*d-b*c)*d/c^2)^(
1/2)/(c*x+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)^(5/2)/(c + d/x)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.373037, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^(5/2)/(c + d/x)^2,x, algorithm="fricas")
[Out]
[-1/2*((5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*log(2*a
*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 +
(b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*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*(a^2*c^
2*d*x^2 + (b^2*c^3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(c^4*d*x +
c^3*d^2), 1/2*(2*(5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt
(-a)*arctan(sqrt((a*x + b)/x)/sqrt(-a)) - (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 +
(b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*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*(a^2*c^
2*d*x^2 + (b^2*c^3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(c^4*d*x +
c^3*d^2), -1/2*(2*(b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d
- 4*a^2*c*d^2)*x)*sqrt((b*c - a*d)/d)*arctan(sqrt((a*x + b)/x)/sqrt((b*c - a*d)
/d)) + (5*a*b*c*d^2 - 4*a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*log(2*a
*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(a^2*c^2*d*x^2 + (b^2*c^3 - 2*a*b*c^
2*d + 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(c^4*d*x + c^3*d^2), ((5*a*b*c*d^2 - 4*
a^2*d^3 + (5*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(-a)*arctan(sqrt((a*x + b)/x)/sqrt(
-a)) - (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d
^2)*x)*sqrt((b*c - a*d)/d)*arctan(sqrt((a*x + b)/x)/sqrt((b*c - a*d)/d)) + (a^2*
c^2*d*x^2 + (b^2*c^3 - 2*a*b*c^2*d + 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(c^4*d*x
+ c^3*d^2)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{\left (c x + d\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(5/2)/(c+d/x)**2,x)
[Out]
Integral(x**2*(a + b/x)**(5/2)/(c*x + d)**2, x)
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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((a + b/x)^(5/2)/(c + d/x)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError