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

Optimal. Leaf size=32 \[ \frac{b \sqrt{c x^2} \log (x)}{x}-\frac{a \sqrt{c x^2}}{x^2} \]

[Out]

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

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Rubi [A]  time = 0.019719, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{b \sqrt{c x^2} \log (x)}{x}-\frac{a \sqrt{c x^2}}{x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.17114, size = 27, normalized size = 0.84 \[ - \frac{a \sqrt{c x^{2}}}{x^{2}} + \frac{b \sqrt{c x^{2}} \log{\left (x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00830452, size = 20, normalized size = 0.62 \[ \frac{c (b x \log (x)-a)}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*(-a + b*x*Log[x]))/Sqrt[c*x^2]

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Maple [A]  time = 0.006, size = 21, normalized size = 0.7 \[{\frac{b\ln \left ( x \right ) x-a}{{x}^{2}}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2)^(1/2)*(b*ln(x)*x-a)/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)/x^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.215828, size = 27, normalized size = 0.84 \[ \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*(b*x*log(x) - a)/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207842, size = 27, normalized size = 0.84 \[{\left (b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right ) - \frac{a{\rm sign}\left (x\right )}{x}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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