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

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

[Out]

-(x/(b*c*Sqrt[c*x^2]*(a + b*x)))

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Rubi [A]  time = 0.0038696, 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} \[ -\frac{x}{b c \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0059425, size = 24, normalized size = 0.96 \[ -\frac{x^3}{b \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^3/(b*(c*x^2)^(3/2)*(a + b*x)))

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Maple [A]  time = 0.001, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3}}{ \left ( bx+a \right ) b} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33424, size = 54, normalized size = 2.16 \begin{align*} -\frac{\sqrt{c x^{2}}}{b^{2} c^{2} x^{2} + a b c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.77616, size = 90, normalized size = 3.6 \begin{align*} \begin{cases} \frac{\tilde{\infty } x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{x^{2}}{b^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{4}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} & \text{for}\: b = - \frac{a}{x} \\\frac{x^{4}}{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}} + a b c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x**2/(c**(3/2)*(x**2)**(3/2)), Eq(a, 0) & Eq(b, 0)), (-x**2/(b**2*c**(3/2)*(x**2)**(3/2)), Eq(a
, 0)), (zoo*x**4/(c**(3/2)*(x**2)**(3/2)), Eq(b, -a/x)), (x**4/(a**2*c**(3/2)*(x**2)**(3/2) + a*b*c**(3/2)*x*(
x**2)**(3/2)), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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