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

Optimal. Leaf size=80 \[ \frac{a b d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2}}-\frac{\left (a+b \sqrt{c+d x}\right )^2}{2 x^2}-\frac{b d \left (a \sqrt{c+d x}+b c\right )}{2 c x} \]

[Out]

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

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Rubi [A]  time = 0.179639, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{a b d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2}}-\frac{\left (a+b \sqrt{c+d x}\right )^2}{2 x^2}-\frac{b d \left (a \sqrt{c+d x}+b c\right )}{2 c x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.6154, size = 68, normalized size = 0.85 \[ \frac{a b d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}}} - \frac{b d \left (a \sqrt{c + d x} + b c\right )}{2 c x} - \frac{\left (a + b \sqrt{c + d x}\right )^{2}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(d*x+c)**(1/2))**2/x**3,x)

[Out]

a*b*d**2*atanh(sqrt(c + d*x)/sqrt(c))/(2*c**(3/2)) - b*d*(a*sqrt(c + d*x) + b*c)
/(2*c*x) - (a + b*sqrt(c + d*x))**2/(2*x**2)

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Mathematica [A]  time = 0.129455, size = 77, normalized size = 0.96 \[ \frac{a b d^2 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{2 c^{3/2}}-\frac{a^2 c+a b \sqrt{c+d x} (2 c+d x)+b^2 c (c+2 d x)}{2 c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 81, normalized size = 1. \[{b}^{2} \left ( -{\frac{d}{x}}-{\frac{c}{2\,{x}^{2}}} \right ) +4\,ab{d}^{2} \left ({\frac{1}{{d}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ( dx+c \right ) ^{3/2}}{c}}-1/8\,\sqrt{dx+c} \right ) }+1/8\,{\frac{1}{{c}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) } \right ) -{\frac{{a}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(d*x+c)^(1/2))^2/x^3,x)

[Out]

b^2*(-d/x-1/2*c/x^2)+4*a*b*d^2*((-1/8/c*(d*x+c)^(3/2)-1/8*(d*x+c)^(1/2))/x^2/d^2
+1/8/c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2)))-1/2*a^2/x^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((sqrt(d*x + c)*b + a)^2/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278916, size = 1, normalized size = 0.01 \[ \left [\frac{a b d^{2} x^{2} \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x + c} c}{x}\right ) - 2 \,{\left (a b d x + 2 \, a b c\right )} \sqrt{d x + c} \sqrt{c} - 2 \,{\left (2 \, b^{2} c d x + b^{2} c^{2} + a^{2} c\right )} \sqrt{c}}{4 \, c^{\frac{3}{2}} x^{2}}, -\frac{a b d^{2} x^{2} \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) +{\left (a b d x + 2 \, a b c\right )} \sqrt{d x + c} \sqrt{-c} +{\left (2 \, b^{2} c d x + b^{2} c^{2} + a^{2} c\right )} \sqrt{-c}}{2 \, \sqrt{-c} c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(a*b*d^2*x^2*log(((d*x + 2*c)*sqrt(c) + 2*sqrt(d*x + c)*c)/x) - 2*(a*b*d*x
+ 2*a*b*c)*sqrt(d*x + c)*sqrt(c) - 2*(2*b^2*c*d*x + b^2*c^2 + a^2*c)*sqrt(c))/(c
^(3/2)*x^2), -1/2*(a*b*d^2*x^2*arctan(c/(sqrt(d*x + c)*sqrt(-c))) + (a*b*d*x + 2
*a*b*c)*sqrt(d*x + c)*sqrt(-c) + (2*b^2*c*d*x + b^2*c^2 + a^2*c)*sqrt(-c))/(sqrt
(-c)*c*x^2)]

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Sympy [A]  time = 38.4755, size = 292, normalized size = 3.65 \[ - \frac{a^{2}}{2 x^{2}} - \frac{20 a b c^{2} d^{2} \sqrt{c + d x}}{- 8 c^{4} - 16 c^{3} d x + 8 c^{2} \left (c + d x\right )^{2}} + \frac{12 a b c d^{2} \left (c + d x\right )^{\frac{3}{2}}}{- 8 c^{4} - 16 c^{3} d x + 8 c^{2} \left (c + d x\right )^{2}} + \frac{3 a b c d^{2} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{c + d x} \right )}}{4} - \frac{3 a b c d^{2} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{c + d x} \right )}}{4} - a b d^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )} + a b d^{2} \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )} - \frac{2 a b d \sqrt{c + d x}}{c x} - \frac{b^{2} c}{2 x^{2}} - \frac{b^{2} d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(d*x+c)**(1/2))**2/x**3,x)

[Out]

-a**2/(2*x**2) - 20*a*b*c**2*d**2*sqrt(c + d*x)/(-8*c**4 - 16*c**3*d*x + 8*c**2*
(c + d*x)**2) + 12*a*b*c*d**2*(c + d*x)**(3/2)/(-8*c**4 - 16*c**3*d*x + 8*c**2*(
c + d*x)**2) + 3*a*b*c*d**2*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(c + d*x
))/4 - 3*a*b*c*d**2*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(c + d*x))/4 - a*
b*d**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x)) + a*b*d**2*sqrt(c*
*(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x)) - 2*a*b*d*sqrt(c + d*x)/(c*x) - b
**2*c/(2*x**2) - b**2*d/x

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GIAC/XCAS [A]  time = 0.2944, size = 170, normalized size = 2.12 \[ -\frac{\frac{a b d^{3} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} + \frac{b^{2} c d^{3} - a^{2} d^{3}}{c^{2}} + \frac{2 \,{\left (d x + c\right )} b^{2} c d^{3} - b^{2} c^{2} d^{3} +{\left (d x + c\right )}^{\frac{3}{2}} a b d^{3} + \sqrt{d x + c} a b c d^{3} + a^{2} c d^{3}}{c d^{2} x^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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