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

Optimal. Leaf size=143 \[ -\frac{c^2 (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{(b c-a d) \left (-4 a^3 d^2 x-2 a^2 b d (5 c x+3 d)+a b^2 c (20 c x-3 d)+15 b^3 c^2\right )}{3 a^3 b^2 x \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c x \left (c+\frac{d}{x}\right )^2}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(c*(c + d/x)^2*x)/(a*(a + b/x)^(3/2)) + ((b*c - a*d)*(15*b^3*c^2 - 4*a^3*d^2*x -
2*a^2*b*d*(3*d + 5*c*x) + a*b^2*c*(-3*d + 20*c*x)))/(3*a^3*b^2*(a + b/x)^(3/2)*
x) - (c^2*(5*b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)

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Rubi [A]  time = 0.428991, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{c^2 (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{(b c-a d) \left (-4 a^3 d^2 x-2 a^2 b d (5 c x+3 d)-a b^2 c (3 d-20 c x)+15 b^3 c^2\right )}{3 a^3 b^2 x \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c x \left (c+\frac{d}{x}\right )^2}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(c + d/x)^2*x)/(a*(a + b/x)^(3/2)) + ((b*c - a*d)*(15*b^3*c^2 - 4*a^3*d^2*x -
a*b^2*c*(3*d - 20*c*x) - 2*a^2*b*d*(3*d + 5*c*x)))/(3*a^3*b^2*(a + b/x)^(3/2)*x
) - (c^2*(5*b*c - 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/a^(7/2)

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Rubi in Sympy [A]  time = 27.8366, size = 138, normalized size = 0.97 \[ \frac{c x \left (c + \frac{d}{x}\right )^{2}}{a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{4 \left (a d - b c\right ) \left (\frac{a \left (2 a^{2} d^{2} + 5 a b c d - 10 b^{2} c^{2}\right )}{2} + \frac{3 b \left (2 a^{2} d^{2} + a b c d - 5 b^{2} c^{2}\right )}{4 x}\right )}{3 a^{3} b^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{c^{2} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x*(c + d/x)**2/(a*(a + b/x)**(3/2)) + 4*(a*d - b*c)*(a*(2*a**2*d**2 + 5*a*b*c*
d - 10*b**2*c**2)/2 + 3*b*(2*a**2*d**2 + a*b*c*d - 5*b**2*c**2)/(4*x))/(3*a**3*b
**2*(a + b/x)**(3/2)) + c**2*(6*a*d - 5*b*c)*atanh(sqrt(a + b/x)/sqrt(a))/a**(7/
2)

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Mathematica [A]  time = 0.242433, size = 145, normalized size = 1.01 \[ \frac{c^2 (6 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{7/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (4 a^4 d^3 x+6 a^3 b d^2 (c x+d)+3 a^2 b^2 c^2 x (c x-8 d)+2 a b^3 c^2 (10 c x-9 d)+15 b^4 c^3\right )}{3 a^3 b^2 (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*x*(15*b^4*c^3 + 4*a^4*d^3*x + 3*a^2*b^2*c^2*x*(-8*d + c*x) + 6*a^
3*b*d^2*(d + c*x) + 2*a*b^3*c^2*(-9*d + 10*c*x)))/(3*a^3*b^2*(b + a*x)^2) + (c^2
*(-5*b*c + 6*a*d)*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[a + b/x]*x])/(2*a^(7/2))

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Maple [B]  time = 0.023, size = 1157, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^3/(a+b/x)^(5/2),x)

[Out]

1/6*((a*x+b)/x)^(1/2)*x/a^(13/2)*(90*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*b^4*c^3+9*ln
(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^7*x*b^3*d^3-9*ln(1/2*(2*(x
*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^7*x*b^3*d^3+18*a^(15/2)*(a*x^2+b*x)^
(1/2)*x*b^2*d^3+12*a^(13/2)*(x*(a*x+b))^(3/2)*b^2*c*d^2+24*a^(11/2)*(x*(a*x+b))^
(3/2)*b^3*c^2*d+18*a^(15/2)*(x*(a*x+b))^(1/2)*x*b^2*d^3+90*a^(9/2)*(x*(a*x+b))^(
1/2)*x*b^5*c^3-36*a^(9/2)*(x*(a*x+b))^(1/2)*b^5*c^2*d+18*ln(1/2*(2*(x*(a*x+b))^(
1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*b^6*c^2*d-15*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1
/2)+2*a*x+b)/a^(1/2))*x^3*a^6*b^4*c^3-45*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a
*x+b)/a^(1/2))*x^2*a^5*b^5*c^3-45*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a
^(1/2))*x*a^4*b^6*c^3+3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^
9*x^3*b*d^3-3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^9*x^3*b*d^
3+30*a^(13/2)*(x*(a*x+b))^(1/2)*x^3*b^3*c^3+9*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*a^8*x^2*b^2*d^3-9*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+
b)/a^(1/2))*a^8*x^2*b^2*d^3+18*a^(17/2)*(a*x^2+b*x)^(1/2)*x^2*b*d^3-24*a^(11/2)*
(x*(a*x+b))^(3/2)*x*b^3*c^3+18*a^(17/2)*(x*(a*x+b))^(1/2)*x^2*b*d^3+30*a^(7/2)*(
x*(a*x+b))^(1/2)*b^6*c^3-15*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2)
)*a^3*b^7*c^3+6*a^(19/2)*(a*x^2+b*x)^(1/2)*x^3*d^3+6*a^(19/2)*(x*(a*x+b))^(1/2)*
x^3*d^3-12*a^(17/2)*(x*(a*x+b))^(3/2)*x*d^3-16*a^(15/2)*(x*(a*x+b))^(3/2)*b*d^3-
20*a^(9/2)*(x*(a*x+b))^(3/2)*b^4*c^3+3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x
+b)/a^(1/2))*a^6*b^4*d^3-3*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))
*a^6*b^4*d^3+6*a^(13/2)*(a*x^2+b*x)^(1/2)*b^3*d^3+6*a^(13/2)*(x*(a*x+b))^(1/2)*b
^3*d^3+54*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^5*x*b^5*c^2*d-
36*a^(15/2)*(x*(a*x+b))^(1/2)*x^3*b^2*c^2*d+36*a^(13/2)*(x*(a*x+b))^(3/2)*x*b^2*
c^2*d-108*a^(13/2)*(x*(a*x+b))^(1/2)*x^2*b^3*c^2*d+18*ln(1/2*(2*(x*(a*x+b))^(1/2
)*a^(1/2)+2*a*x+b)/a^(1/2))*a^7*x^3*b^3*c^2*d+54*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(
1/2)+2*a*x+b)/a^(1/2))*a^6*x^2*b^4*c^2*d-108*a^(11/2)*(x*(a*x+b))^(1/2)*x*b^4*c^
2*d)/(x*(a*x+b))^(1/2)/b^3/(a*x+b)^3

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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((c + d/x)^3/(a + b/x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254988, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (3 \, a^{2} b^{2} c^{3} x^{2} + 15 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 6 \, a^{3} b d^{3} + 2 \,{\left (10 \, a b^{3} c^{3} - 12 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3}\right )} x\right )} \sqrt{a}}{6 \,{\left (a^{4} b^{2} x + a^{3} b^{3}\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{3 \,{\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d +{\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} b^{2} c^{3} x^{2} + 15 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 6 \, a^{3} b d^{3} + 2 \,{\left (10 \, a b^{3} c^{3} - 12 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3}\right )} x\right )} \sqrt{-a}}{3 \,{\left (a^{4} b^{2} x + a^{3} b^{3}\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)^3/(a + b/x)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*(5*b^4*c^3 - 6*a*b^3*c^2*d + (5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x)*sqrt((a
*x + b)/x)*log(2*a*x*sqrt((a*x + b)/x) + (2*a*x + b)*sqrt(a)) - 2*(3*a^2*b^2*c^3
*x^2 + 15*b^4*c^3 - 18*a*b^3*c^2*d + 6*a^3*b*d^3 + 2*(10*a*b^3*c^3 - 12*a^2*b^2*
c^2*d + 3*a^3*b*c*d^2 + 2*a^4*d^3)*x)*sqrt(a))/((a^4*b^2*x + a^3*b^3)*sqrt(a)*sq
rt((a*x + b)/x)), 1/3*(3*(5*b^4*c^3 - 6*a*b^3*c^2*d + (5*a*b^3*c^3 - 6*a^2*b^2*c
^2*d)*x)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x + b)/x))) + (3*a^2*b^2*c
^3*x^2 + 15*b^4*c^3 - 18*a*b^3*c^2*d + 6*a^3*b*d^3 + 2*(10*a*b^3*c^3 - 12*a^2*b^
2*c^2*d + 3*a^3*b*c*d^2 + 2*a^4*d^3)*x)*sqrt(-a))/((a^4*b^2*x + a^3*b^3)*sqrt(-a
)*sqrt((a*x + b)/x))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.259834, size = 267, normalized size = 1.87 \[ -\frac{1}{3} \, b{\left (\frac{3 \, c^{3} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3}} - \frac{3 \,{\left (5 \, b c^{3} - 6 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b} - \frac{2 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} + \frac{6 \,{\left (a x + b\right )} b^{3} c^{3}}{x} - \frac{9 \,{\left (a x + b\right )} a b^{2} c^{2} d}{x} + \frac{3 \,{\left (a x + b\right )} a^{3} d^{3}}{x}\right )} x}{{\left (a x + b\right )} a^{3} b^{3} \sqrt{\frac{a x + b}{x}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c + d/x)^3/(a + b/x)^(5/2),x, algorithm="giac")

[Out]

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