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

Optimal. Leaf size=52 \[ \frac{a^2 x \log (x)}{\sqrt{c x^2}}+\frac{2 a b x^2}{\sqrt{c x^2}}+\frac{b^2 x^3}{2 \sqrt{c x^2}} \]

[Out]

(2*a*b*x^2)/Sqrt[c*x^2] + (b^2*x^3)/(2*Sqrt[c*x^2]) + (a^2*x*Log[x])/Sqrt[c*x^2]

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Rubi [A]  time = 0.0264642, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^2 x \log (x)}{\sqrt{c x^2}}+\frac{2 a b x^2}{\sqrt{c x^2}}+\frac{b^2 x^3}{2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/Sqrt[c*x^2],x]

[Out]

(2*a*b*x^2)/Sqrt[c*x^2] + (b^2*x^3)/(2*Sqrt[c*x^2]) + (a^2*x*Log[x])/Sqrt[c*x^2]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0109473, size = 32, normalized size = 0.62 \[ \frac{x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(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 = 31, normalized size = 0.6 \[{\frac{x \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx \right ) }{2}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.34248, size = 47, normalized size = 0.9 \[ \frac{b^{2} x^{2}}{2 \, \sqrt{c}} + \frac{a^{2} \log \left (x\right )}{\sqrt{c}} + \frac{2 \, \sqrt{c x^{2}} a b}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211219, size = 47, normalized size = 0.9 \[ \frac{{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(c*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)/(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216858, size = 68, normalized size = 1.31 \[ -\frac{a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} + \frac{1}{2} \, \sqrt{c x^{2}}{\left (\frac{b^{2} x}{c} + \frac{4 \, a b}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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