3.165 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^2} \, dx\)

Optimal. Leaf size=287 \[ -\frac{(4 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^3}+\frac{b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a^2 c^2 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^2}+\frac{b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a^3 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^3}-\frac{d^{7/2} (9 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{7/2}}+\frac{d (b c-2 a d)}{a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )} \]

[Out]

(b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(a + b/x)^(3/2)
) + (b*(b*c - 2*a*d)*(5*b^2*c^2 - a*b*c*d + a^2*d^2))/(a^3*c^2*(b*c - a*d)^3*Sqr
t[a + b/x]) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)) +
x/(a*c*(a + b/x)^(3/2)*(c + d/x)) - (d^(7/2)*(9*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqr
t[a + b/x])/Sqrt[b*c - a*d]])/(c^3*(b*c - a*d)^(7/2)) - ((5*b*c + 4*a*d)*ArcTanh
[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^3)

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Rubi [A]  time = 1.40881, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{(4 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^3}+\frac{b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a^2 c^2 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^2}+\frac{b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a^3 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^3}-\frac{d^{7/2} (9 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{7/2}}+\frac{d (b c-2 a d)}{a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^(5/2)*(c + d/x)^2),x]

[Out]

(b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(a + b/x)^(3/2)
) + (b*(b*c - 2*a*d)*(5*b^2*c^2 - a*b*c*d + a^2*d^2))/(a^3*c^2*(b*c - a*d)^3*Sqr
t[a + b/x]) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)) +
x/(a*c*(a + b/x)^(3/2)*(c + d/x)) - (d^(7/2)*(9*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqr
t[a + b/x])/Sqrt[b*c - a*d]])/(c^3*(b*c - a*d)^(7/2)) - ((5*b*c + 4*a*d)*ArcTanh
[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^3)

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Rubi in Sympy [A]  time = 160.074, size = 252, normalized size = 0.88 \[ - \frac{d x}{c \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (c + \frac{d}{x}\right ) \left (a d - b c\right )} + \frac{d^{\frac{7}{2}} \left (4 a d - 9 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + \frac{b}{x}}}{\sqrt{a d - b c}} \right )}}{c^{3} \left (a d - b c\right )^{\frac{7}{2}}} + \frac{x \left (2 a d - b c\right )}{a c^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{b \left (6 a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}\right )}{3 a^{2} c^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{b \left (2 a d - b c\right ) \left (a^{2} d^{2} - a b c d + 5 b^{2} c^{2}\right )}{a^{3} c^{2} \sqrt{a + \frac{b}{x}} \left (a d - b c\right )^{3}} - \frac{\left (4 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(5/2)/(c+d/x)**2,x)

[Out]

-d*x/(c*(a + b/x)**(3/2)*(c + d/x)*(a*d - b*c)) + d**(7/2)*(4*a*d - 9*b*c)*atanh
(sqrt(d)*sqrt(a + b/x)/sqrt(a*d - b*c))/(c**3*(a*d - b*c)**(7/2)) + x*(2*a*d - b
*c)/(a*c**2*(a + b/x)**(3/2)*(a*d - b*c)) + b*(6*a**2*d**2 - 6*a*b*c*d + 5*b**2*
c**2)/(3*a**2*c**2*(a + b/x)**(3/2)*(a*d - b*c)**2) + b*(2*a*d - b*c)*(a**2*d**2
- a*b*c*d + 5*b**2*c**2)/(a**3*c**2*sqrt(a + b/x)*(a*d - b*c)**3) - (4*a*d + 5*
b*c)*atanh(sqrt(a + b/x)/sqrt(a))/(a**(7/2)*c**3)

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Mathematica [C]  time = 1.84408, size = 364, normalized size = 1.27 \[ \frac{-\frac{3 (4 a d+5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{a^{7/2}}+\frac{2 \sqrt{a+\frac{b}{x}} \left (-3 a^4 d^4 (a x+b)^2 (c x+d)+3 a^4 d^5 (a x+b)^2+2 b^5 c^3 (c x+d) (b c-a d)+14 b^4 c^4 (a x+b)^2 (c x+d)-4 b^4 c^3 (a x+b) (c x+d) (4 b c-7 a d)-26 a b^3 c^3 d (a x+b)^2 (c x+d)+3 a c x (a x+b)^2 (c x+d) (b c-a d)^3\right )}{a^4 (a x+b)^2 (c x+d) (b c-a d)^3}+\frac{3 i d^{7/2} (4 a d-9 b c) \log \left (\frac{2 c^4 (b c-a d)^{5/2} \left (2 \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{b c-a d}-2 i a d x-i b (d-c x)\right )}{d^{9/2} (c x+d) (9 b c-4 a d)}\right )}{(b c-a d)^{7/2}}}{6 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^(5/2)*(c + d/x)^2),x]

[Out]

((2*Sqrt[a + b/x]*(3*a^4*d^5*(b + a*x)^2 + 2*b^5*c^3*(b*c - a*d)*(d + c*x) - 4*b
^4*c^3*(4*b*c - 7*a*d)*(b + a*x)*(d + c*x) + 14*b^4*c^4*(b + a*x)^2*(d + c*x) -
26*a*b^3*c^3*d*(b + a*x)^2*(d + c*x) - 3*a^4*d^4*(b + a*x)^2*(d + c*x) + 3*a*c*(
b*c - a*d)^3*x*(b + a*x)^2*(d + c*x)))/(a^4*(b*c - a*d)^3*(b + a*x)^2*(d + c*x))
- (3*(5*b*c + 4*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/a^(7/2) + ((3*
I)*d^(7/2)*(-9*b*c + 4*a*d)*Log[(2*c^4*(b*c - a*d)^(5/2)*((-2*I)*a*d*x + 2*Sqrt[
d]*Sqrt[b*c - a*d]*Sqrt[a + b/x]*x - I*b*(d - c*x)))/(d^(9/2)*(9*b*c - 4*a*d)*(d
+ c*x))])/(b*c - a*d)^(7/2))/(6*c^3)

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Maple [B]  time = 0.028, size = 4648, normalized size = 16.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^(5/2)/(c+d/x)^2,x)

[Out]

-1/6*((a*x+b)/x)^(1/2)*x/a^(13/2)*(27*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^(19/2)*x^4*b^2*c^3*d^4-6*((a*d-b*c)*d/c^2)
^(1/2)*a^(19/2)*(x*(a*x+b))^(3/2)*x^3*c^4*d^3-6*((a*d-b*c)*d/c^2)^(1/2)*a^(21/2)
*(x*(a*x+b))^(1/2)*x^4*c^3*d^4-30*((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*(x*(a*x+b))^(
1/2)*x^4*b^4*c^7-3*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))*a^(21/2)*x^3*b*c*d^6-90*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))*a^(19/2)*x^3*b^2*c^2*d^5+81*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^(17/2)*x^3*b^
3*c^3*d^4+24*((a*d-b*c)*d/c^2)^(1/2)*a^(11/2)*(x*(a*x+b))^(3/2)*x^2*b^4*c^7-12*(
(a*d-b*c)*d/c^2)^(1/2)*a^(21/2)*(x*(a*x+b))^(1/2)*x^3*c^2*d^5-90*((a*d-b*c)*d/c^
2)^(1/2)*a^(11/2)*(x*(a*x+b))^(1/2)*x^3*b^5*c^7+12*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^8*b^3*c*d^6-81*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^(19/2)*x^2*b^2
*c*d^6-36*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^(17/2)*x^2*b^3*c^2*d^5+81*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*b^4*c^3*d^4+20*((a*d-b*c)*d/c^2)^
(1/2)*a^(9/2)*(x*(a*x+b))^(3/2)*x*b^5*c^7-90*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*(x*
(a*x+b))^(1/2)*x^2*b^6*c^7-48*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^7*c^5*d^2+15*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^4*a^6*b^5*c^7-39*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)*b^4*c*d^6
+27*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)*b^5*c^2*d^5+12*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^(23/2)*x^4*c*d^6+36*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^(21/2)*x^2*b*d^7+36*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^(19/2)*x*b^2*d
^7-30*((a*d-b*c)*d/c^2)^(1/2)*a^(7/2)*(x*(a*x+b))^(1/2)*x*b^7*c^7-84*((a*d-b*c)*
d/c^2)^(1/2)*a^(11/2)*(x*(a*x+b))^(1/2)*b^5*c^4*d^3+96*((a*d-b*c)*d/c^2)^(1/2)*a
^(9/2)*(x*(a*x+b))^(1/2)*b^6*c^5*d^2+42*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^6*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*d-b*c)*d/c^2)^(1/2)*a^11*x^4*c^2*d^5+12*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^11*x^
3*c*d^6+6*((a*d-b*c)*d/c^2)^(1/2)*a^(21/2)*(x*(a*x+b))^(1/2)*x^5*c^4*d^3-39*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^(21/
2)*x^4*b*c^2*d^5-156*((a*d-b*c)*d/c^2)^(1/2)*a^(15/2)*(x*(a*x+b))^(1/2)*x^3*b^3*
c^5*d^2+12*((a*d-b*c)*d/c^2)^(1/2)*a^(19/2)*(x*(a*x+b))^(1/2)*x^3*b*c^3*d^4+258*
((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*(x*(a*x+b))^(1/2)*x^3*b^4*c^6*d+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^7*x*b^4*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*d-b*c)*d/c^2)^(1
/2)*a^7*x^4*b^4*c^6*d+30*((a*d-b*c)*d/c^2)^(1/2)*a^(15/2)*(x*(a*x+b))^(3/2)*x*b^
2*c^4*d^3-28*((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*(x*(a*x+b))^(3/2)*x*b^3*c^5*d^2-40
*((a*d-b*c)*d/c^2)^(1/2)*a^(11/2)*(x*(a*x+b))^(3/2)*x*b^4*c^6*d+96*((a*d-b*c)*d/
c^2)^(1/2)*a^(15/2)*(x*(a*x+b))^(1/2)*x^4*b^3*c^6*d-84*((a*d-b*c)*d/c^2)^(1/2)*a
^(17/2)*(x*(a*x+b))^(1/2)*x^4*b^2*c^5*d^2+42*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^8*x^4*b^3*c^5*d^2-63*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^8*x^2*b^3*c^3*
d^4+78*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^8*x^3*b^3*c^4*d^3+36*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^9*x*b^2*c*d^6-87*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^8*x*b^3*c^2*d^5+36*((a*d-b*c)*d/c^2
)^(1/2)*a^(13/2)*(x*(a*x+b))^(1/2)*x^2*b^4*c^5*d^2+12*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^9*x^4*b^2*c^4*d^3+48*((a*d
-b*c)*d/c^2)^(1/2)*a^(19/2)*(x*(a*x+b))^(1/2)*x^4*b*c^4*d^3+12*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^(23/2)*x^3*d^7+12
*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
^(17/2)*b^3*d^7+204*((a*d-b*c)*d/c^2)^(1/2)*a^(11/2)*(x*(a*x+b))^(1/2)*x*b^5*c^5
*d^2+6*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*(x*(a*x+b))^(1/2)*x*b^6*c^6*d+45*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^3*a^5*b^
6*c^7-30*((a*d-b*c)*d/c^2)^(1/2)*a^(7/2)*(x*(a*x+b))^(1/2)*b^7*c^6*d+45*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^
7*c^7+15*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^8*c^7+15*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^8*c^6*d-33*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*b^4*c^2*d^5-105*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^(17/2)*x*b^3*c*d^6+42*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))*a^(
15/2)*x*b^4*c^2*d^5+27*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^5*c^3*d^4+38*((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*
(x*(a*x+b))^(3/2)*b^3*c^4*d^3-64*((a*d-b*c)*d/c^2)^(1/2)*a^(11/2)*(x*(a*x+b))^(3
/2)*b^4*c^5*d^2+20*((a*d-b*c)*d/c^2)^(1/2)*a^(9/2)*(x*(a*x+b))^(3/2)*b^5*c^6*d+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)*
a^6*b^5*c^3*d^4-12*((a*d-b*c)*d/c^2)^(1/2)*a^(15/2)*(x*(a*x+b))^(1/2)*b^3*c^2*d^
5+30*((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*(x*(a*x+b))^(1/2)*b^4*c^3*d^4-36*((a*d-b*c
)*d/c^2)^(1/2)*a^(17/2)*(x*(a*x+b))^(1/2)*x*b^2*c^2*d^5+84*((a*d-b*c)*d/c^2)^(1/
2)*a^(15/2)*(x*(a*x+b))^(1/2)*x*b^3*c^3*d^4-18*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^5*c^5*d^2-129*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^3*b^5*c
^6*d-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^4*x*b^7*c^6*d+78*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^3*b^4*c^5*d^2-72*((a*d-b*c)*d/c^2)^(1/2)*a^(13/2)*(x
*(a*x+b))^(3/2)*x^2*b^3*c^6*d-87*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^9*x^3*b^2*c^3*d^4-99*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^2*b^6*c^6*d+198*((a*d-
b*c)*d/c^2)^(1/2)*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*b^5*c^6*d+138*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^5*c^4*d^3-1
56*((a*d-b*c)*d/c^2)^(1/2)*a^(15/2)*(x*(a*x+b))^(1/2)*x^2*b^3*c^4*d^3-102*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^
6*c^5*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)*a^10*x^3*b*c^2*d^5-33*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^10*x^4*b*c^3*d^4-36*((a*d-b*c)*d/c^2)^(1/2)*a^(1
9/2)*(x*(a*x+b))^(1/2)*x^2*b*c^2*d^5+72*((a*d-b*c)*d/c^2)^(1/2)*a^(17/2)*(x*(a*x
+b))^(1/2)*x^2*b^2*c^3*d^4+36*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^10*x^2*b*c*d^6-63*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^9*x^2*b^2*c^2*d^5-222*((a*d-b*c)
*d/c^2)^(1/2)*a^(13/2)*(x*(a*x+b))^(1/2)*x*b^4*c^4*d^3-18*((a*d-b*c)*d/c^2)^(1/2
)*a^(17/2)*(x*(a*x+b))^(3/2)*x^2*b*c^4*d^3+48*((a*d-b*c)*d/c^2)^(1/2)*a^(15/2)*(
x*(a*x+b))^(3/2)*x^2*b^2*c^5*d^2+162*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*b^4*c^4*d^3+24*((a*d-b*c)*d/c^2)^(1/2
)*a^(17/2)*(x*(a*x+b))^(1/2)*x^3*b^2*c^4*d^3)/(x*(a*x+b))^(1/2)/(a*d-b*c)^4/(a*x
+b)^3/c^4/((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(1/((a + b/x)^(5/2)*(c + d/x)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.54242, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*(c + d/x)^2),x, algorithm="fricas")

[Out]

[1/6*(3*(9*a^3*b^2*c*d^4 - 4*a^4*b*d^5 + (9*a^4*b*c^2*d^3 - 4*a^5*c*d^4)*x^2 + (
9*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - 4*a^5*d^5)*x)*sqrt(a)*sqrt(-d/(b*c - a*d))*s
qrt((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*(5*b^5*c^4*d - 11*a*b^4*c^3*d^2 + 3*a^2*b^
3*c^2*d^3 + 7*a^3*b^2*c*d^4 - 4*a^4*b*d^5 + (5*a*b^4*c^5 - 11*a^2*b^3*c^4*d + 3*
a^3*b^2*c^3*d^2 + 7*a^4*b*c^2*d^3 - 4*a^5*c*d^4)*x^2 + (5*b^5*c^5 - 6*a*b^4*c^4*
d - 8*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 3*a^4*b*c*d^4 - 4*a^5*d^5)*x)*sqrt(
(a*x + b)/x)*log(-2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) + 2*(15*b^5*c^4
*d - 33*a*b^4*c^3*d^2 + 9*a^2*b^3*c^2*d^3 - 6*a^3*b^2*c*d^4 + 3*(a^2*b^3*c^5 - 3
*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*x^3 + (20*a*b^4*c^5 - 41*a^2*b^3
*c^4*d + 9*a^3*b^2*c^3*d^2 + 3*a^4*b*c^2*d^3 - 6*a^5*c*d^4)*x^2 + (15*b^5*c^5 -
13*a*b^4*c^4*d - 35*a^2*b^3*c^3*d^2 + 15*a^3*b^2*c^2*d^3 - 12*a^4*b*c*d^4)*x)*sq
rt(a))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4 +
(a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^2 + (a^3*b^4*
c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x)*sqrt(a)*sqrt((a*x + b)
/x)), -1/6*(6*(9*a^3*b^2*c*d^4 - 4*a^4*b*d^5 + (9*a^4*b*c^2*d^3 - 4*a^5*c*d^4)*x
^2 + (9*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - 4*a^5*d^5)*x)*sqrt(a)*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*(5*b^5*c^4*d - 11*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 + 7*a^3*b^2*c*d^4 -
4*a^4*b*d^5 + (5*a*b^4*c^5 - 11*a^2*b^3*c^4*d + 3*a^3*b^2*c^3*d^2 + 7*a^4*b*c^2
*d^3 - 4*a^5*c*d^4)*x^2 + (5*b^5*c^5 - 6*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 + 10*a^
3*b^2*c^2*d^3 + 3*a^4*b*c*d^4 - 4*a^5*d^5)*x)*sqrt((a*x + b)/x)*log(-2*a*x*sqrt(
(a*x + b)/x) + (2*a*x + b)*sqrt(a)) - 2*(15*b^5*c^4*d - 33*a*b^4*c^3*d^2 + 9*a^2
*b^3*c^2*d^3 - 6*a^3*b^2*c*d^4 + 3*(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*
d^2 - a^5*c^2*d^3)*x^3 + (20*a*b^4*c^5 - 41*a^2*b^3*c^4*d + 9*a^3*b^2*c^3*d^2 +
3*a^4*b*c^2*d^3 - 6*a^5*c*d^4)*x^2 + (15*b^5*c^5 - 13*a*b^4*c^4*d - 35*a^2*b^3*c
^3*d^2 + 15*a^3*b^2*c^2*d^3 - 12*a^4*b*c*d^4)*x)*sqrt(a))/((a^3*b^4*c^6*d - 3*a^
4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6
*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^2 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6
*b*c^4*d^3 - a^7*c^3*d^4)*x)*sqrt(a)*sqrt((a*x + b)/x)), 1/6*(3*(9*a^3*b^2*c*d^4
- 4*a^4*b*d^5 + (9*a^4*b*c^2*d^3 - 4*a^5*c*d^4)*x^2 + (9*a^3*b^2*c^2*d^3 + 5*a^
4*b*c*d^4 - 4*a^5*d^5)*x)*sqrt(-a)*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)) + 6*(5*b^5*c^4*d - 11*a*b^4*c^3*d^2 + 3*a^2*b^3*c^2*d^3 + 7*a^3*b^2*c
*d^4 - 4*a^4*b*d^5 + (5*a*b^4*c^5 - 11*a^2*b^3*c^4*d + 3*a^3*b^2*c^3*d^2 + 7*a^4
*b*c^2*d^3 - 4*a^5*c*d^4)*x^2 + (5*b^5*c^5 - 6*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 +
10*a^3*b^2*c^2*d^3 + 3*a^4*b*c*d^4 - 4*a^5*d^5)*x)*sqrt((a*x + b)/x)*arctan(a/(
sqrt(-a)*sqrt((a*x + b)/x))) + 2*(15*b^5*c^4*d - 33*a*b^4*c^3*d^2 + 9*a^2*b^3*c^
2*d^3 - 6*a^3*b^2*c*d^4 + 3*(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a
^5*c^2*d^3)*x^3 + (20*a*b^4*c^5 - 41*a^2*b^3*c^4*d + 9*a^3*b^2*c^3*d^2 + 3*a^4*b
*c^2*d^3 - 6*a^5*c*d^4)*x^2 + (15*b^5*c^5 - 13*a*b^4*c^4*d - 35*a^2*b^3*c^3*d^2
+ 15*a^3*b^2*c^2*d^3 - 12*a^4*b*c*d^4)*x)*sqrt(-a))/((a^3*b^4*c^6*d - 3*a^4*b^3*
c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d^4 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3
*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^2 + (a^3*b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4
*d^3 - a^7*c^3*d^4)*x)*sqrt(-a)*sqrt((a*x + b)/x)), -1/3*(3*(9*a^3*b^2*c*d^4 - 4
*a^4*b*d^5 + (9*a^4*b*c^2*d^3 - 4*a^5*c*d^4)*x^2 + (9*a^3*b^2*c^2*d^3 + 5*a^4*b*
c*d^4 - 4*a^5*d^5)*x)*sqrt(-a)*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*(5*b^5*c^4*d - 11*a*b^4*
c^3*d^2 + 3*a^2*b^3*c^2*d^3 + 7*a^3*b^2*c*d^4 - 4*a^4*b*d^5 + (5*a*b^4*c^5 - 11*
a^2*b^3*c^4*d + 3*a^3*b^2*c^3*d^2 + 7*a^4*b*c^2*d^3 - 4*a^5*c*d^4)*x^2 + (5*b^5*
c^5 - 6*a*b^4*c^4*d - 8*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 3*a^4*b*c*d^4 - 4
*a^5*d^5)*x)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) - (15*b^5*
c^4*d - 33*a*b^4*c^3*d^2 + 9*a^2*b^3*c^2*d^3 - 6*a^3*b^2*c*d^4 + 3*(a^2*b^3*c^5
- 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3)*x^3 + (20*a*b^4*c^5 - 41*a^2*
b^3*c^4*d + 9*a^3*b^2*c^3*d^2 + 3*a^4*b*c^2*d^3 - 6*a^5*c*d^4)*x^2 + (15*b^5*c^5
- 13*a*b^4*c^4*d - 35*a^2*b^3*c^3*d^2 + 15*a^3*b^2*c^2*d^3 - 12*a^4*b*c*d^4)*x)
*sqrt(-a))/((a^3*b^4*c^6*d - 3*a^4*b^3*c^5*d^2 + 3*a^5*b^2*c^4*d^3 - a^6*b*c^3*d
^4 + (a^4*b^3*c^7 - 3*a^5*b^2*c^6*d + 3*a^6*b*c^5*d^2 - a^7*c^4*d^3)*x^2 + (a^3*
b^4*c^7 - 2*a^4*b^3*c^6*d + 2*a^6*b*c^4*d^3 - a^7*c^3*d^4)*x)*sqrt(-a)*sqrt((a*x
+ b)/x))]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.280938, size = 779, normalized size = 2.71 \[ -\frac{1}{3} \, b{\left (\frac{3 \,{\left (9 \, b c d^{4} - 4 \, a d^{5}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{4} c^{6} - 3 \, a b^{3} c^{5} d + 3 \, a^{2} b^{2} c^{4} d^{2} - a^{3} b c^{3} d^{3}\right )} \sqrt{b c d - a d^{2}}} - \frac{2 \,{\left (a b^{3} c - a^{2} b^{2} d + \frac{6 \,{\left (a x + b\right )} b^{3} c}{x} - \frac{12 \,{\left (a x + b\right )} a b^{2} d}{x}\right )} x}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )}{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}} + \frac{3 \,{\left (b^{4} c^{4} \sqrt{\frac{a x + b}{x}} - 4 \, a b^{3} c^{3} d \sqrt{\frac{a x + b}{x}} + 6 \, a^{2} b^{2} c^{2} d^{2} \sqrt{\frac{a x + b}{x}} - 4 \, a^{3} b c d^{3} \sqrt{\frac{a x + b}{x}} + 2 \, a^{4} d^{4} \sqrt{\frac{a x + b}{x}} + \frac{{\left (a x + b\right )} b^{3} c^{3} d \sqrt{\frac{a x + b}{x}}}{x} - \frac{3 \,{\left (a x + b\right )} a b^{2} c^{2} d^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{3 \,{\left (a x + b\right )} a^{2} b c d^{3} \sqrt{\frac{a x + b}{x}}}{x} - \frac{2 \,{\left (a x + b\right )} a^{3} d^{4} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (a^{3} b^{3} c^{5} - 3 \, a^{4} b^{2} c^{4} d + 3 \, a^{5} b c^{3} d^{2} - a^{6} c^{2} d^{3}\right )}{\left (a b c - a^{2} d - \frac{{\left (a x + b\right )} b c}{x} + \frac{2 \,{\left (a x + b\right )} a d}{x} - \frac{{\left (a x + b\right )}^{2} d}{x^{2}}\right )}} - \frac{3 \,{\left (5 \, b c + 4 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x)^(5/2)*(c + d/x)^2),x, algorithm="giac")

[Out]

-1/3*b*(3*(9*b*c*d^4 - 4*a*d^5)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/
((b^4*c^6 - 3*a*b^3*c^5*d + 3*a^2*b^2*c^4*d^2 - a^3*b*c^3*d^3)*sqrt(b*c*d - a*d^
2)) - 2*(a*b^3*c - a^2*b^2*d + 6*(a*x + b)*b^3*c/x - 12*(a*x + b)*a*b^2*d/x)*x/(
(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*(a*x + b)*sqrt((a*x +
b)/x)) + 3*(b^4*c^4*sqrt((a*x + b)/x) - 4*a*b^3*c^3*d*sqrt((a*x + b)/x) + 6*a^2*
b^2*c^2*d^2*sqrt((a*x + b)/x) - 4*a^3*b*c*d^3*sqrt((a*x + b)/x) + 2*a^4*d^4*sqrt
((a*x + b)/x) + (a*x + b)*b^3*c^3*d*sqrt((a*x + b)/x)/x - 3*(a*x + b)*a*b^2*c^2*
d^2*sqrt((a*x + b)/x)/x + 3*(a*x + b)*a^2*b*c*d^3*sqrt((a*x + b)/x)/x - 2*(a*x +
b)*a^3*d^4*sqrt((a*x + b)/x)/x)/((a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d
^2 - a^6*c^2*d^3)*(a*b*c - a^2*d - (a*x + b)*b*c/x + 2*(a*x + b)*a*d/x - (a*x +
b)^2*d/x^2)) - 3*(5*b*c + 4*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^
3*b*c^3))