3.809 \(\int \frac{\sqrt{c x^2} (a+b x)^2}{x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{a^2 \sqrt{c x^2} \log (x)}{x}+2 a b \sqrt{c x^2}+\frac{1}{2} b^2 x \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0269627, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^2 \sqrt{c x^2} \log (x)}{x}+2 a b \sqrt{c x^2}+\frac{1}{2} b^2 x \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0150334, size = 33, normalized size = 0.67 \[ \frac{c x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 33, normalized size = 0.7 \[{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx}{2\,x}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.212906, size = 43, normalized size = 0.88 \[ \frac{{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*x^2 + 4*a*b*x + 2*a^2*log(x))*sqrt(c*x^2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.20721, size = 43, normalized size = 0.88 \[ \frac{1}{2} \,{\left (b^{2} x^{2}{\rm sign}\left (x\right ) + 4 \, a b x{\rm sign}\left (x\right ) + 2 \, a^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*x^2*sign(x) + 4*a*b*x*sign(x) + 2*a^2*ln(abs(x))*sign(x))*sqrt(c)