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

Optimal. Leaf size=64 \[ -\frac{a^2 x}{b^3 \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 \sqrt{c x^2}}+\frac{x^2}{b^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0518219, antiderivative size = 64, 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 \[ -\frac{a^2 x}{b^3 \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 \sqrt{c x^2}}+\frac{x^2}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \sqrt{c x^{2}}}{b^{3} c x \left (a + b x\right )} - \frac{2 a \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{3} c x} + \frac{\sqrt{c x^{2}} \int \frac{1}{b^{2}}\, dx}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0270059, size = 52, normalized size = 0.81 \[ \frac{x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 60, normalized size = 0.9 \[ -{\frac{x \left ( 2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2} \right ) }{ \left ( bx+a \right ){b}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217433, size = 80, normalized size = 1.25 \[ \frac{{\left (b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} c x^{2} + a b^{3} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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