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

Optimal. Leaf size=22 \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]

[Out]

-(x/(b*Sqrt[c*x^2]*(a + b*x)))

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Rubi [A]  time = 0.0122026, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(x/(b*Sqrt[c*x^2]*(a + b*x)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00797302, size = 22, normalized size = 1. \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(x/(b*Sqrt[c*x^2]*(a + b*x)))

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Maple [A]  time = 0.004, size = 21, normalized size = 1. \[ -{\frac{x}{b \left ( bx+a \right ) }{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.209756, size = 34, normalized size = 1.55 \[ -\frac{\sqrt{c x^{2}}}{b^{2} c x^{2} + a b c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*x^2)/(b^2*c*x^2 + a*b*c*x)

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Sympy [A]  time = 3.80443, size = 85, normalized size = 3.86 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt{c} \sqrt{x^{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{b^{2} \sqrt{c} \sqrt{x^{2}}} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{2}}{\sqrt{c} \sqrt{x^{2}}} & \text{for}\: b = - \frac{a}{x} \\\frac{x^{2}}{a^{2} \sqrt{c} \sqrt{x^{2}} + a b \sqrt{c} x \sqrt{x^{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/(sqrt(c)*sqrt(x**2)), Eq(a, 0) & Eq(b, 0)), (-1/(b**2*sqrt(c)*sqr
t(x**2)), Eq(a, 0)), (zoo*x**2/(sqrt(c)*sqrt(x**2)), Eq(b, -a/x)), (x**2/(a**2*s
qrt(c)*sqrt(x**2) + a*b*sqrt(c)*x*sqrt(x**2)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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