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

Optimal. Leaf size=47 \[ -\frac{a^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0290423, antiderivative size = 47, 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 \[ -\frac{a^2}{\sqrt{c x^2}}+\frac{2 a b x \log (x)}{\sqrt{c x^2}}+\frac{b^2 x^2}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c*x**2)/(c*x**2) + 2*a*b*sqrt(c*x**2)*log(x)/(c*x) + sqrt(c*x**2)*Int
egral(b**2, x)/(c*x)

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Mathematica [A]  time = 0.0169981, size = 34, normalized size = 0.72 \[ \frac{c x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 29, normalized size = 0.6 \[{(2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2}){\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34546, size = 47, normalized size = 1. \[ \frac{2 \, a b \log \left (x\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{a^{2}}{\sqrt{c} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206527, size = 46, normalized size = 0.98 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{2}}}{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211898, size = 88, normalized size = 1.87 \[ \frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{2 \,{\left (a b{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{a^{2} \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}\right )}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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