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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]

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)))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{(a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=-\frac {x}{b \sqrt {c x^2} (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]

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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fricas [A]  time = 0.44, size = 25, normalized size = 1.14 \[ -\frac {\sqrt {c x^{2}}}{b^{2} c x^{2} + a b c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(c*x^2)/(b^2*c*x^2 + a*b*c*x)

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giac [A]  time = 1.16, size = 38, normalized size = 1.73 \[ \frac {1}{{\left (b x + a\right )} b \sqrt {c} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 21, normalized size = 0.95 \[ -\frac {x}{\left (b x +a \right ) \sqrt {c \,x^{2}}\, b} \]

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 [A]  time = 1.45, size = 21, normalized size = 0.95 \[ \frac {\sqrt {c x^{2}}}{a b c x + a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

sqrt(c*x^2)/(a*b*c*x + a^2*c)

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mupad [B]  time = 0.16, size = 25, normalized size = 1.14 \[ -\frac {\sqrt {c\,x^2}}{b\,c\,x\,\left (a+b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((c*x^2)^(1/2)*(a + b*x)^2),x)

[Out]

-(c*x^2)^(1/2)/(b*c*x*(a + b*x))

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sympy [A]  time = 1.17, size = 85, normalized size = 3.86 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt {c} \sqrt {x^{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tilde {\infty } x^{2}}{\sqrt {c} \sqrt {x^{2}}} & \text {for}\: a = - b x \\\frac {x^{2}}{a^{2} \sqrt {c} \sqrt {x^{2}}} & \text {for}\: b = 0 \\- \frac {x}{a b \sqrt {c} \sqrt {x^{2}} + b^{2} \sqrt {c} x \sqrt {x^{2}}} & \text {otherwise} \end {cases} \]

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)), (zoo*x**2/(sqrt(c)*sqrt(x**2)), Eq(a, -b*x)), (x**2
/(a**2*sqrt(c)*sqrt(x**2)), Eq(b, 0)), (-x/(a*b*sqrt(c)*sqrt(x**2) + b**2*sqrt(c)*x*sqrt(x**2)), True))

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