3.157 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )} \, dx\)
Optimal. Leaf size=147 \[ -\frac{(2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^2}+\frac{b (3 b c-a d)}{a^2 c \sqrt{a+\frac{b}{x}} (b c-a d)}+\frac{2 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 (b c-a d)^{3/2}}+\frac{x}{a c \sqrt{a+\frac{b}{x}}} \]
[Out]
(b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b/x]) + x/(a*c*Sqrt[a + b/x]) + (2
*d^(5/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c^2*(b*c - a*d)^(3/2)
) - ((3*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(5/2)*c^2)
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Rubi [A] time = 0.618832, antiderivative size = 147, 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
\[ -\frac{(2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^2}+\frac{b (3 b c-a d)}{a^2 c \sqrt{a+\frac{b}{x}} (b c-a d)}+\frac{2 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^2 (b c-a d)^{3/2}}+\frac{x}{a c \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(3/2)*(c + d/x)),x]
[Out]
(b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*Sqrt[a + b/x]) + x/(a*c*Sqrt[a + b/x]) + (2
*d^(5/2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c^2*(b*c - a*d)^(3/2)
) - ((3*b*c + 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(5/2)*c^2)
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Rubi in Sympy [A] time = 67.6655, size = 122, normalized size = 0.83 \[ \frac{2 d^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{2} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{x}{a c \sqrt{a + \frac{b}{x}}} + \frac{b \left (a d - 3 b c\right )}{a^{2} c \sqrt{a + \frac{b}{x}} \left (a d - b c\right )} - \frac{\left (2 a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2)/(c+d/x),x)
[Out]
2*d**(5/2)*atanh(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**2*(a*d - b*c)**(3/2)
) + x/(a*c*sqrt(a + b/x)) + b*(a*d - 3*b*c)/(a**2*c*sqrt(a + b/x)*(a*d - b*c)) -
(2*a*d + 3*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/(a**(5/2)*c**2)
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Mathematica [A] time = 1.22763, size = 197, normalized size = 1.34 \[ \frac{-\frac{(2 a d+3 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{5/2}}+\frac{2 c x \sqrt{a+\frac{b}{x}} \left (a^2 d x+a b (d-c x)-3 b^2 c\right )}{a^2 (a x+b) (a d-b c)}+\frac{2 d^{5/2} \log (c x+d)}{(a d-b c)^{3/2}}-\frac{2 d^{5/2} \log \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{a d-b c}-2 a d x+b c x-b d\right )}{(a d-b c)^{3/2}}}{2 c^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(3/2)*(c + d/x)),x]
[Out]
((2*c*Sqrt[a + b/x]*x*(-3*b^2*c + a^2*d*x + a*b*(d - c*x)))/(a^2*(-(b*c) + a*d)*
(b + a*x)) + (2*d^(5/2)*Log[d + c*x])/(-(b*c) + a*d)^(3/2) - ((3*b*c + 2*a*d)*Lo
g[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(5/2) - (2*d^(5/2)*Log[-(b*d) + b*c*
x - 2*a*d*x + 2*Sqrt[d]*Sqrt[-(b*c) + a*d]*Sqrt[a + b/x]*x])/(-(b*c) + a*d)^(3/2
))/(2*c^2)
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Maple [B] time = 0.026, size = 1480, normalized size = 10.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2)/(c+d/x),x)
[Out]
1/2*((a*x+b)/x)^(1/2)*x/a^(9/2)*(-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)*a^2*b^5*c^4-4*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c
^2)^(1/2)*a^(7/2)*b^2*c^4-4*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^(13/2)*x*b*d^4+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))*a^(9/2)*b^3*c*d^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))*a^(15/2)*x^2*d^4-2*
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))*a^
(11/2)*b^2*d^4+4*(x*(a*x+b))^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*b*c^3*d+12*(x
*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(7/2)*x*b^3*c^4-2*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)*a^5*b^2*c*d^3+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^3*b^4*
c^3*d+4*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*b^2*c*d^3+2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*b
^2*c^2*d^2-8*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(7/2)*b^3*c^3*d+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))*a^(13/2
)*x^2*b*c*d^3+6*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(5/2)*b^4*c^4+4*(x*(
a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(11/2)*x*b*c^2*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)*a^6*x^2*b*c^2*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^6*x
*b*c*d^3-8*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(11/2)*x^2*b*c^3*d-16*(x*
(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*x*b^2*c^3*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)*a^5*x*b^2*c^2*d^2+4*l
n(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^5
*x^2*b^2*c^3*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)*a^4*x*b^3*c^3*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^4*b^3*c^2*d^2-3*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^2*a^4*b^3*c^4-6*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^3*b^4*c^4-2*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)*a^7*x
^2*c*d^3+2*(x*(a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*x^2*c^2*d^2+6*(x*(
a*x+b))^(1/2)*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*x^2*b^2*c^4)/(x*(a*x+b))^(1/2)/(a*
d-b*c)^2/c^3/((a*d-b*c)*d/c^2)^(1/2)/(a*x+b)^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/((a + b/x)^(3/2)*(c + d/x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.38584, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*(c + d/x)),x, algorithm="fricas")
[Out]
[-1/2*(2*a^(5/2)*d^2*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x)*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)) - (
3*b^2*c^2 - a*b*c*d - 2*a^2*d^2)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x)
+ (2*a*x + b)*sqrt(a)) - 2*(3*b^2*c^2 - a*b*c*d + (a*b*c^2 - a^2*c*d)*x)*sqrt(a)
)/((a^2*b*c^3 - a^3*c^2*d)*sqrt(a)*sqrt((a*x + b)/x)), -(sqrt(-a)*a^2*d^2*sqrt(-
d/(b*c - a*d))*sqrt((a*x + b)/x)*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)) - (3*b^2*c^2 - a*b*c*d - 2*a^2
*d^2)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (3*b^2*c^2 - a*
b*c*d + (a*b*c^2 - a^2*c*d)*x)*sqrt(-a))/((a^2*b*c^3 - a^3*c^2*d)*sqrt(-a)*sqrt(
(a*x + b)/x)), 1/2*(4*a^(5/2)*d^2*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)*arctan(-
(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) + (3*b^2*c^2 - a*b*c*d -
2*a^2*d^2)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a))
+ 2*(3*b^2*c^2 - a*b*c*d + (a*b*c^2 - a^2*c*d)*x)*sqrt(a))/((a^2*b*c^3 - a^3*c^
2*d)*sqrt(a)*sqrt((a*x + b)/x)), (2*sqrt(-a)*a^2*d^2*sqrt(d/(b*c - a*d))*sqrt((a
*x + b)/x)*arctan(-(b*c - a*d)*sqrt(d/(b*c - a*d))/(d*sqrt((a*x + b)/x))) + (3*b
^2*c^2 - a*b*c*d - 2*a^2*d^2)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b
)/x))) + (3*b^2*c^2 - a*b*c*d + (a*b*c^2 - a^2*c*d)*x)*sqrt(-a))/((a^2*b*c^3 - a
^3*c^2*d)*sqrt(-a)*sqrt((a*x + b)/x))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (c x + d\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(3/2)/(c+d/x),x)
[Out]
Integral(x/((a + b/x)**(3/2)*(c*x + d)), x)
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GIAC/XCAS [A] time = 0.254063, size = 261, normalized size = 1.78 \[{\left (\frac{2 \, d^{3} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{2} c^{3} - a b c^{2} d\right )} \sqrt{b c d - a d^{2}}} + \frac{2 \, a b c - \frac{3 \,{\left (a x + b\right )} b c}{x} + \frac{{\left (a x + b\right )} a d}{x}}{{\left (a^{2} b c^{2} - a^{3} c d\right )}{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}} + \frac{{\left (3 \, b c + 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b c^{2}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*(c + d/x)),x, algorithm="giac")
[Out]
(2*d^3*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^2*c^3 - a*b*c^2*d)*sq
rt(b*c*d - a*d^2)) + (2*a*b*c - 3*(a*x + b)*b*c/x + (a*x + b)*a*d/x)/((a^2*b*c^2
- a^3*c*d)*(a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x + b)/x)/x)) + (3*b*c + 2*
a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2*b*c^2))*b