3.811 \(\int \frac{\sqrt{c x^2} (a+b x)^2}{x^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0316508, antiderivative size = 54, 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 \[ -\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.3947, size = 49, normalized size = 0.91 \[ - \frac{a^{2} \sqrt{c x^{2}}}{2 x^{3}} - \frac{2 a b \sqrt{c x^{2}}}{x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(c*x**2)**(1/2)/x**4,x)

[Out]

-a**2*sqrt(c*x**2)/(2*x**3) - 2*a*b*sqrt(c*x**2)/x**2 + b**2*sqrt(c*x**2)*log(x)
/x

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Mathematica [A]  time = 0.0150725, size = 36, normalized size = 0.67 \[ \frac{\sqrt{c x^2} \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[c*x^2]*(a + b*x)^2)/x^4,x]

[Out]

(Sqrt[c*x^2]*(-(a*(a + 4*b*x)) + 2*b^2*x^2*Log[x]))/(2*x^3)

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Maple [A]  time = 0.007, size = 34, normalized size = 0.6 \[{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2}}{2\,{x}^{3}}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(c*x^2)^(1/2)/x^4,x)

[Out]

1/2*(c*x^2)^(1/2)*(2*b^2*ln(x)*x^2-4*a*b*x-a^2)/x^3

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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^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.216448, size = 45, normalized size = 0.83 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^2/x^4,x, algorithm="fricas")

[Out]

1/2*(2*b^2*x^2*log(x) - 4*a*b*x - a^2)*sqrt(c*x^2)/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(c*x**2)**(1/2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.205849, size = 47, normalized size = 0.87 \[ \frac{1}{2} \,{\left (2 \, b^{2}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right ) - \frac{4 \, a b x{\rm sign}\left (x\right ) + a^{2}{\rm sign}\left (x\right )}{x^{2}}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2)*(b*x + a)^2/x^4,x, algorithm="giac")

[Out]

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