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

Optimal. Leaf size=43 \[ \frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0383983, antiderivative size = 43, 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 \[ \frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.1177, size = 42, normalized size = 0.98 \[ \frac{a \sqrt{c x^{2}}}{b^{2} c x \left (a + b x\right )} + \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{2} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0163927, size = 35, normalized size = 0.81 \[ \frac{x ((a+b x) \log (a+b x)+a)}{b^2 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 39, normalized size = 0.9 \[{\frac{x \left ( b\ln \left ( bx+a \right ) x+a\ln \left ( bx+a \right ) +a \right ) }{ \left ( bx+a \right ){b}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210533, size = 54, normalized size = 1.26 \[ \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c x^{2} + a b^{2} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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