3.810 \(\int \frac{\sqrt{c x^2} (a+b x)^2}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0318719, 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 \[ -\frac{a^2 \sqrt{c x^2}}{x^2}+\frac{2 a b \sqrt{c x^2} \log (x)}{x}+b^2 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(c*x**2)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0166452, size = 31, normalized size = 0.63 \[ \frac{c \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c*x^2]*(a + b*x)^2)/x^3,x]

[Out]

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

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Maple [A]  time = 0.007, size = 32, normalized size = 0.7 \[{\frac{2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2}}{{x}^{2}}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(c*x^2)^(1/2)/x^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^2/x^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.22565, size = 42, normalized size = 0.86 \[ \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}} \left (a + b x\right )^{2}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(c*x**2)**(1/2)/x**3,x)

[Out]

Integral(sqrt(c*x**2)*(a + b*x)**2/x**3, x)

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GIAC/XCAS [A]  time = 0.20874, size = 42, normalized size = 0.86 \[{\left (b^{2} x{\rm sign}\left (x\right ) + 2 \, a b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right ) - \frac{a^{2}{\rm sign}\left (x\right )}{x}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^2/x^3,x, algorithm="giac")

[Out]

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