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

Optimal. Leaf size=25 \[ -\frac{c \sqrt{c x^2}}{b x (a+b x)} \]

[Out]

-((c*Sqrt[c*x^2])/(b*x*(a + b*x)))

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Rubi [A]  time = 0.004015, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ -\frac{c \sqrt{c x^2}}{b x (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*Sqrt[c*x^2])/(b*x*(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{\left (c x^2\right )^{3/2}}{x^3 (a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{(a+b x)^2} \, dx}{x}\\ &=-\frac{c \sqrt{c x^2}}{b x (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0055979, size = 24, normalized size = 0.96 \[ -\frac{\left (c x^2\right )^{3/2}}{b x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*x^2)^(3/2)/(b*x^3*(a + b*x)))

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Maple [A]  time = 0.002, size = 23, normalized size = 0.9 \begin{align*} -{\frac{1}{ \left ( bx+a \right ) b{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04378, size = 22, normalized size = 0.88 \begin{align*} -\frac{c^{\frac{3}{2}}}{b^{2} x + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^(3/2)/(b^2*x + a*b)

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Fricas [A]  time = 1.25167, size = 46, normalized size = 1.84 \begin{align*} -\frac{\sqrt{c x^{2}} c}{b^{2} x^{2} + a b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2)*c/(b^2*x^2 + a*b*x)

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Sympy [A]  time = 2.19917, size = 44, normalized size = 1.76 \begin{align*} \begin{cases} - \frac{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{a b x^{3} + b^{2} x^{4}} & \text{for}\: b \neq 0 \\\frac{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{a^{2} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-c**(3/2)*(x**2)**(3/2)/(a*b*x**3 + b**2*x**4), Ne(b, 0)), (c**(3/2)*(x**2)**(3/2)/(a**2*x**2), Tru
e))

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Giac [A]  time = 1.0644, size = 39, normalized size = 1.56 \begin{align*} -c^{\frac{3}{2}}{\left (\frac{\mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b} - \frac{\mathrm{sgn}\left (x\right )}{a b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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