3.2870 \(\int \frac{(c+d x)^2}{(a+b (c+d x)^3)^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{3 b d \left (a+b (c+d x)^3\right )} \]

[Out]

-1/(3*b*d*(a + b*(c + d*x)^3))

________________________________________________________________________________________

Rubi [A]  time = 0.0188829, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {372, 261} \[ -\frac{1}{3 b d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*b*d*(a + b*(c + d*x)^3))

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0116112, size = 23, normalized size = 1. \[ -\frac{1}{3 b d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*b*d*(a + b*(c + d*x)^3))

________________________________________________________________________________________

Maple [B]  time = 0., size = 44, normalized size = 1.9 \begin{align*} -{\frac{1}{3\,bd \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.936266, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{3 \,{\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(((d*x + c)^3*b + a)*b*d)

________________________________________________________________________________________

Fricas [B]  time = 1.46935, size = 103, normalized size = 4.48 \begin{align*} -\frac{1}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(b^2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x + (b^2*c^3 + a*b)*d)

________________________________________________________________________________________

Sympy [B]  time = 1.73067, size = 58, normalized size = 2.52 \begin{align*} - \frac{1}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(3*a*b*d + 3*b**2*c**3*d + 9*b**2*c**2*d**2*x + 9*b**2*c*d**3*x**2 + 3*b**2*d**4*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.13199, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{3 \,{\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(((d*x + c)^3*b + a)*b*d)