3.2123 \(\int \frac{\left (a+b \sqrt{x}\right )^2}{x} \, dx\)

Optimal. Leaf size=21 \[ a^2 \log (x)+4 a b \sqrt{x}+b^2 x \]

[Out]

4*a*b*Sqrt[x] + b^2*x + a^2*Log[x]

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Rubi [A]  time = 0.0380703, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^2 \log (x)+4 a b \sqrt{x}+b^2 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^2/x,x]

[Out]

4*a*b*Sqrt[x] + b^2*x + a^2*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*log(sqrt(x)) + 4*a*b*sqrt(x) + 2*b**2*Integral(x, (x, sqrt(x)))

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Mathematica [A]  time = 0.0102791, size = 21, normalized size = 1. \[ a^2 \log (x)+4 a b \sqrt{x}+b^2 x \]

Antiderivative was successfully verified.

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

[Out]

4*a*b*Sqrt[x] + b^2*x + a^2*Log[x]

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Maple [A]  time = 0.002, size = 20, normalized size = 1. \[{b}^{2}x+{a}^{2}\ln \left ( x \right ) +4\,ab\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^2/x,x)

[Out]

b^2*x+a^2*ln(x)+4*a*b*x^(1/2)

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Maxima [A]  time = 1.45189, size = 26, normalized size = 1.24 \[ b^{2} x + a^{2} \log \left (x\right ) + 4 \, a b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x + a^2*log(x) + 4*a*b*sqrt(x)

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Fricas [A]  time = 0.235815, size = 30, normalized size = 1.43 \[ b^{2} x + 2 \, a^{2} \log \left (\sqrt{x}\right ) + 4 \, a b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x + 2*a^2*log(sqrt(x)) + 4*a*b*sqrt(x)

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Sympy [A]  time = 0.473696, size = 20, normalized size = 0.95 \[ a^{2} \log{\left (x \right )} + 4 a b \sqrt{x} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*log(x) + 4*a*b*sqrt(x) + b**2*x

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GIAC/XCAS [A]  time = 0.218265, size = 27, normalized size = 1.29 \[ b^{2} x + a^{2}{\rm ln}\left ({\left | x \right |}\right ) + 4 \, a b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x + a^2*ln(abs(x)) + 4*a*b*sqrt(x)