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

Optimal. Leaf size=57 \[ \log (x) \left (a^2+b^2 c\right )+4 a b \sqrt{c+d x}-4 a b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+b^2 d x \]

[Out]

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

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Rubi [A]  time = 0.152317, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \log (x) \left (a^2+b^2 c\right )+4 a b \sqrt{c+d x}-4 a b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+b^2 d x \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 4 a b \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )} + 4 a b \sqrt{c + d x} + 2 b^{2} \int ^{\sqrt{c + d x}} x\, dx + \left (a^{2} + b^{2} c\right ) \log{\left (- d x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a*b*sqrt(c)*atanh(sqrt(c + d*x)/sqrt(c)) + 4*a*b*sqrt(c + d*x) + 2*b**2*Integ
ral(x, (x, sqrt(c + d*x))) + (a**2 + b**2*c)*log(-d*x)

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Mathematica [A]  time = 0.0592721, size = 62, normalized size = 1.09 \[ \left (a^2+b^2 c\right ) \log (d x)+b \left (4 a \sqrt{c+d x}+b c+b d x\right )-4 a b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 51, normalized size = 0.9 \[{b}^{2}c\ln \left ( x \right ) +{b}^{2}dx-4\,ab{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \sqrt{c}+4\,ab\sqrt{dx+c}+{a}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(d*x+c)^(1/2))^2/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31156, size = 1, normalized size = 0.02 \[ \left [b^{2} d x + 2 \, a b \sqrt{c} \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 4 \, \sqrt{d x + c} a b +{\left (b^{2} c + a^{2}\right )} \log \left (x\right ), b^{2} d x - 4 \, a b \sqrt{-c} \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) + 4 \, \sqrt{d x + c} a b +{\left (b^{2} c + a^{2}\right )} \log \left (x\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.05637, size = 129, normalized size = 2.26 \[ a^{2} \log{\left (- d x \right )} - 4 a b c \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + d x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + d x}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + d x \wedge - c < 0 \end{cases}\right ) + 4 a b \sqrt{c + d x} + b^{2} c \log{\left (- d x \right )} + b^{2} \left (c + d x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*log(-d*x) - 4*a*b*c*Piecewise((-atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), -c >
0), (acoth(sqrt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d*x)), (atanh(sq
rt(c + d*x)/sqrt(c))/sqrt(c), (-c < 0) & (c > c + d*x))) + 4*a*b*sqrt(c + d*x) +
b**2*c*log(-d*x) + b**2*(c + d*x)

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GIAC/XCAS [A]  time = 0.275968, size = 105, normalized size = 1.84 \[ -b^{2} c{\rm ln}\left (-c\right ) + \frac{4 \, a b c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} +{\left (d x + c\right )} b^{2} - a^{2}{\rm ln}\left (-c\right ) + 4 \, \sqrt{d x + c} a b +{\left (b^{2} c + a^{2}\right )}{\rm ln}\left (d x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^2*c*ln(-c) + 4*a*b*c*arctan(sqrt(d*x + c)/sqrt(-c))/sqrt(-c) + (d*x + c)*b^2
- a^2*ln(-c) + 4*sqrt(d*x + c)*a*b + (b^2*c + a^2)*ln(d*x)