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

Optimal. Leaf size=122 \[ -\frac{c (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{c (5 b c-4 a d)}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{2 a^2 d^2+b c (5 b c-4 a d)}{3 a^2 b \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c^2 x}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.296017, antiderivative size = 118, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{c (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2}}+\frac{c (5 b c-4 a d)}{a^3 \sqrt{a+\frac{b}{x}}}+\frac{\frac{c (5 b c-4 a d)}{a^2}+\frac{2 d^2}{b}}{3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{c^2 x}{a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.7247, size = 107, normalized size = 0.88 \[ \frac{c^{2} x}{a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{2 \left (a^{2} d^{2} - \frac{b c \left (4 a d - 5 b c\right )}{2}\right )}{3 a^{2} b \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{c \left (4 a d - 5 b c\right )}{a^{3} \sqrt{a + \frac{b}{x}}} + \frac{c \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*x/(a*(a + b/x)**(3/2)) + 2*(a**2*d**2 - b*c*(4*a*d - 5*b*c)/2)/(3*a**2*b*(a
+ b/x)**(3/2)) - c*(4*a*d - 5*b*c)/(a**3*sqrt(a + b/x)) + c*(4*a*d - 5*b*c)*ata
nh(sqrt(a + b/x)/sqrt(a))/a**(7/2)

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Mathematica [A]  time = 0.189269, size = 123, normalized size = 1.01 \[ \frac{c (4 a d-5 b c) \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{7/2}}+\frac{x \sqrt{a+\frac{b}{x}} \left (2 a^3 d^2 x+a^2 b c x (3 c x-16 d)+4 a b^2 c (5 c x-3 d)+15 b^3 c^2\right )}{3 a^3 b (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x]*x*(15*b^3*c^2 + 2*a^3*d^2*x + a^2*b*c*x*(-16*d + 3*c*x) + 4*a*b^2
*c*(-3*d + 5*c*x)))/(3*a^3*b*(b + a*x)^2) + (c*(-5*b*c + 4*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 = 0.02, size = 595, normalized size = 4.9 \[ -{\frac{x}{6\,b \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 24\,{a}^{15/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}cd-30\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{3}b{c}^{2}-24\,{a}^{13/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xcd+72\,{a}^{13/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}bcd-4\,{a}^{13/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}{d}^{2}+24\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}xb{c}^{2}-90\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }{x}^{2}{b}^{2}{c}^{2}-16\,{a}^{11/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}bcd+72\,{a}^{11/2}\sqrt{x \left ( ax+b \right ) }x{b}^{2}cd+20\,{a}^{9/2} \left ( x \left ( ax+b \right ) \right ) ^{3/2}{b}^{2}{c}^{2}-90\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }x{b}^{3}{c}^{2}+24\,{a}^{9/2}\sqrt{x \left ( ax+b \right ) }{b}^{3}cd-30\,{a}^{7/2}\sqrt{x \left ( ax+b \right ) }{b}^{4}{c}^{2}-12\,{a}^{7}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}bcd+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{6}{b}^{2}{c}^{2}-36\,{a}^{6}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{b}^{2}cd+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{5}{b}^{3}{c}^{2}-36\,{a}^{5}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{3}cd+45\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{4}{b}^{4}{c}^{2}-12\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{4}cd+15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{b}^{5}{c}^{2} \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( ax+b \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d/x)^2/(a+b/x)^(5/2),x)

[Out]

-1/6*((a*x+b)/x)^(1/2)*x*(24*a^(15/2)*(x*(a*x+b))^(1/2)*x^3*c*d-30*a^(13/2)*(x*(
a*x+b))^(1/2)*x^3*b*c^2-24*a^(13/2)*(x*(a*x+b))^(3/2)*x*c*d+72*a^(13/2)*(x*(a*x+
b))^(1/2)*x^2*b*c*d-4*a^(13/2)*(x*(a*x+b))^(3/2)*d^2+24*a^(11/2)*(x*(a*x+b))^(3/
2)*x*b*c^2-90*a^(11/2)*(x*(a*x+b))^(1/2)*x^2*b^2*c^2-16*a^(11/2)*(x*(a*x+b))^(3/
2)*b*c*d+72*a^(11/2)*(x*(a*x+b))^(1/2)*x*b^2*c*d+20*a^(9/2)*(x*(a*x+b))^(3/2)*b^
2*c^2-90*a^(9/2)*(x*(a*x+b))^(1/2)*x*b^3*c^2+24*a^(9/2)*(x*(a*x+b))^(1/2)*b^3*c*
d-30*a^(7/2)*(x*(a*x+b))^(1/2)*b^4*c^2-12*a^7*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2
)+2*a*x+b)/a^(1/2))*x^3*b*c*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^2*c^2-36*a^6*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/
2))*x^2*b^2*c*d+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^3*c^2-36*a^5*ln(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*b^3*c*d+
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^4*c^2-12*a^4*ln
(1/2*(2*(x*(a*x+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^4*c*d+15*ln(1/2*(2*(x*(a*x
+b))^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*b^5*c^2)/a^(13/2)/(x*(a*x+b))^(1/2)/b/(
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)^2/(a + b/x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254442, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d +{\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \log \left (2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right ) - 2 \,{\left (3 \, a^{2} b c^{2} x^{2} + 15 \, b^{3} c^{2} - 12 \, a b^{2} c d + 2 \,{\left (10 \, a b^{2} c^{2} - 8 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt{a}}{6 \,{\left (a^{4} b x + a^{3} b^{2}\right )} \sqrt{a} \sqrt{\frac{a x + b}{x}}}, \frac{3 \,{\left (5 \, b^{3} c^{2} - 4 \, a b^{2} c d +{\left (5 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a^{2} b c^{2} x^{2} + 15 \, b^{3} c^{2} - 12 \, a b^{2} c d + 2 \,{\left (10 \, a b^{2} c^{2} - 8 \, a^{2} b c d + a^{3} d^{2}\right )} x\right )} \sqrt{-a}}{3 \,{\left (a^{4} b x + a^{3} b^{2}\right )} \sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(3*(5*b^3*c^2 - 4*a*b^2*c*d + (5*a*b^2*c^2 - 4*a^2*b*c*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*c^2*x^2 + 1
5*b^3*c^2 - 12*a*b^2*c*d + 2*(10*a*b^2*c^2 - 8*a^2*b*c*d + a^3*d^2)*x)*sqrt(a))/
((a^4*b*x + a^3*b^2)*sqrt(a)*sqrt((a*x + b)/x)), 1/3*(3*(5*b^3*c^2 - 4*a*b^2*c*d
+ (5*a*b^2*c^2 - 4*a^2*b*c*d)*x)*sqrt((a*x + b)/x)*arctan(a/(sqrt(-a)*sqrt((a*x
+ b)/x))) + (3*a^2*b*c^2*x^2 + 15*b^3*c^2 - 12*a*b^2*c*d + 2*(10*a*b^2*c^2 - 8*
a^2*b*c*d + a^3*d^2)*x)*sqrt(-a))/((a^4*b*x + a^3*b^2)*sqrt(-a)*sqrt((a*x + b)/x
))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257682, size = 217, normalized size = 1.78 \[ -\frac{1}{3} \, b{\left (\frac{3 \, c^{2} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3}} - \frac{3 \,{\left (5 \, b c^{2} - 4 \, a c d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b} - \frac{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + \frac{6 \,{\left (a x + b\right )} b^{2} c^{2}}{x} - \frac{6 \,{\left (a x + b\right )} a b c d}{x}\right )} x}{{\left (a x + b\right )} a^{3} b^{2} \sqrt{\frac{a x + b}{x}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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