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

Optimal. Leaf size=58 \[ \frac{a^2 \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a \sqrt{c x^2}}{b^2}+\frac{x \sqrt{c x^2}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0458353, antiderivative size = 58, 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 \[ \frac{a^2 \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a \sqrt{c x^2}}{b^2}+\frac{x \sqrt{c x^2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(c*x**2)/(a + b*x), x)

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Mathematica [A]  time = 0.0178285, size = 40, normalized size = 0.69 \[ \frac{c x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 0.7 \[{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx}{2\,{b}^{3}x}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2)^(1/2)/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.20929, size = 53, normalized size = 0.91 \[ \frac{{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \, b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(c*x**2)/(a + b*x), x)

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GIAC/XCAS [A]  time = 0.205359, size = 73, normalized size = 1.26 \[ \frac{1}{2} \, \sqrt{c}{\left (\frac{2 \, a^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} - \frac{2 \, a^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{3}} + \frac{b x^{2}{\rm sign}\left (x\right ) - 2 \, a x{\rm sign}\left (x\right )}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(c)*(2*a^2*ln(abs(b*x + a))*sign(x)/b^3 - 2*a^2*ln(abs(a))*sign(x)/b^3 +
(b*x^2*sign(x) - 2*a*x*sign(x))/b^2)