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

Optimal. Leaf size=54 \[ -\frac{\left (a+b \sqrt{c+d x}\right )^2}{x}-\frac{2 a b d \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+b^2 d \log (x) \]

[Out]

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

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Rubi [A]  time = 0.153833, antiderivative size = 54, 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 \[ -\frac{\left (a+b \sqrt{c+d x}\right )^2}{x}-\frac{2 a b d \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+b^2 d \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.8642, size = 68, normalized size = 1.26 \[ - \frac{2 a b d \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} + b^{2} d \log{\left (- d x \right )} - \frac{\left (a + b \sqrt{c + d x}\right ) \left (2 a + 2 b \sqrt{c + d x}\right )}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.121485, size = 63, normalized size = 1.17 \[ -\frac{a^2+2 a b \sqrt{c+d x}+\frac{2 a b d x \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{\sqrt{c}}+b^2 c-b^2 d x \log (x)}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 60, normalized size = 1.1 \[{b}^{2}d\ln \left ( x \right ) -{\frac{{b}^{2}c}{x}}-2\,{\frac{ab\sqrt{dx+c}}{x}}-2\,{\frac{abd}{\sqrt{c}}{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) }-{\frac{{a}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*d*ln(x)-b^2*c/x-2*a*b/x*(d*x+c)^(1/2)-2*a*b*d*arctanh((d*x+c)^(1/2)/c^(1/2))
/c^(1/2)-a^2/x

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28203, size = 1, normalized size = 0.02 \[ \left [\frac{a b d x \log \left (\frac{{\left (d x + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x + c} c}{x}\right ) - 2 \, \sqrt{d x + c} a b \sqrt{c} +{\left (b^{2} d x \log \left (x\right ) - b^{2} c - a^{2}\right )} \sqrt{c}}{\sqrt{c} x}, \frac{2 \, a b d x \arctan \left (\frac{c}{\sqrt{d x + c} \sqrt{-c}}\right ) - 2 \, \sqrt{d x + c} a b \sqrt{-c} +{\left (b^{2} d x \log \left (x\right ) - b^{2} c - a^{2}\right )} \sqrt{-c}}{\sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(a*b*d*x*log(((d*x + 2*c)*sqrt(c) - 2*sqrt(d*x + c)*c)/x) - 2*sqrt(d*x + c)*a*b
*sqrt(c) + (b^2*d*x*log(x) - b^2*c - a^2)*sqrt(c))/(sqrt(c)*x), (2*a*b*d*x*arcta
n(c/(sqrt(d*x + c)*sqrt(-c))) - 2*sqrt(d*x + c)*a*b*sqrt(-c) + (b^2*d*x*log(x) -
b^2*c - a^2)*sqrt(-c))/(sqrt(-c)*x)]

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Sympy [A]  time = 19.2259, size = 196, normalized size = 3.63 \[ - \frac{a^{2}}{x} - a b c d \sqrt{\frac{1}{c^{3}}} \log{\left (- c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )} + a b c d \sqrt{\frac{1}{c^{3}}} \log{\left (c^{2} \sqrt{\frac{1}{c^{3}}} + \sqrt{c + d x} \right )} - 4 a b d \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right ) - \frac{2 a b \sqrt{c + d x}}{x} - \frac{b^{2} c}{x} + b^{2} d \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/x - a*b*c*d*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(c + d*x)) + a*b*c
*d*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(c + d*x)) - 4*a*b*d*Piecewise((-a
tan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt(c + d*x)/sqrt(c))/sqr
t(c), (-c < 0) & (c < c + d*x)), (atanh(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0)
& (c > c + d*x))) - 2*a*b*sqrt(c + d*x)/x - b**2*c/x + b**2*d*log(x)

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GIAC/XCAS [A]  time = 0.2943, size = 153, normalized size = 2.83 \[ \frac{b^{2} d^{2}{\rm ln}\left (d x\right ) + \frac{2 \, a b d^{2} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} - \frac{b^{2} c d^{2}{\rm ln}\left (-c\right ) + b^{2} c d^{2} + a^{2} d^{2}}{c} - \frac{b^{2} c d^{2} + 2 \, \sqrt{d x + c} a b d^{2} + a^{2} d^{2}}{d x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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