3.2915 \(\int \frac{(c+d x)^3}{(a+b (c+d x)^4)^2} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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Rubi [A]  time = 0.0255701, 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} \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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)^3}{\left (a+b (c+d x)^4\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x^4\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac{1}{4 b d \left (a+b (c+d x)^4\right )}\\ \end{align*}

Mathematica [A]  time = 0.0112555, size = 23, normalized size = 1. \[ -\frac{1}{4 b d \left (a+b (c+d x)^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3/(a + b*(c + d*x)^4)^2,x]

[Out]

-1/(4*b*d*(a + b*(c + d*x)^4))

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Maple [B]  time = 0.001, size = 56, normalized size = 2.4 \begin{align*} -{\frac{1}{4\,bd \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3/(a+b*(d*x+c)^4)^2,x)

[Out]

-1/4/b/d/(b*d^4*x^4+4*b*c*d^3*x^3+6*b*c^2*d^2*x^2+4*b*c^3*d*x+b*c^4+a)

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Maxima [A]  time = 0.976203, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{4 \,{\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

-1/4/(((d*x + c)^4*b + a)*b*d)

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Fricas [B]  time = 1.23135, size = 130, normalized size = 5.65 \begin{align*} -\frac{1}{4 \,{\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x +{\left (b^{2} c^{4} + a b\right )} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

-1/4/(b^2*d^5*x^4 + 4*b^2*c*d^4*x^3 + 6*b^2*c^2*d^3*x^2 + 4*b^2*c^3*d^2*x + (b^2*c^4 + a*b)*d)

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Sympy [B]  time = 5.23872, size = 73, normalized size = 3.17 \begin{align*} - \frac{1}{4 a b d + 4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3/(a+b*(d*x+c)**4)**2,x)

[Out]

-1/(4*a*b*d + 4*b**2*c**4*d + 16*b**2*c**3*d**2*x + 24*b**2*c**2*d**3*x**2 + 16*b**2*c*d**4*x**3 + 4*b**2*d**5
*x**4)

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Giac [A]  time = 1.16967, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{4 \,{\left ({\left (d x + c\right )}^{4} b + a\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/4/(((d*x + c)^4*b + a)*b*d)