3.2916 \(\int \frac{(c+d x)^3}{(a+b (c+d x)^4)^3} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{8 b d \left (a+b (c+d x)^4\right )^2} \]

[Out]

-1/(8*b*d*(a + b*(c + d*x)^4)^2)

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Rubi [A]  time = 0.0261986, 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} \[ -\frac{1}{8 b d \left (a+b (c+d x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(8*b*d*(a + b*(c + d*x)^4)^2)

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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (a+b x^4\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{1}{8 b d \left (a+b (c+d x)^4\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0119719, size = 23, normalized size = 1. \[ -\frac{1}{8 b d \left (a+b (c+d x)^4\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(8*b*d*(a + b*(c + d*x)^4)^2)

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Maple [B]  time = 0.001, size = 56, normalized size = 2.4 \begin{align*} -{\frac{1}{8\,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 ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/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)^2

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Maxima [A]  time = 0.935763, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{8 \,{\left ({\left (d x + c\right )}^{4} b + a\right )}^{2} 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)^3,x, algorithm="maxima")

[Out]

-1/8/(((d*x + c)^4*b + a)^2*b*d)

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Fricas [B]  time = 1.65697, size = 342, normalized size = 14.87 \begin{align*} -\frac{1}{8 \,{\left (b^{3} d^{9} x^{8} + 8 \, b^{3} c d^{8} x^{7} + 28 \, b^{3} c^{2} d^{7} x^{6} + 56 \, b^{3} c^{3} d^{6} x^{5} + 2 \,{\left (35 \, b^{3} c^{4} + a b^{2}\right )} d^{5} x^{4} + 8 \,{\left (7 \, b^{3} c^{5} + a b^{2} c\right )} d^{4} x^{3} + 4 \,{\left (7 \, b^{3} c^{6} + 3 \, a b^{2} c^{2}\right )} d^{3} x^{2} + 8 \,{\left (b^{3} c^{7} + a b^{2} c^{3}\right )} d^{2} x +{\left (b^{3} c^{8} + 2 \, a b^{2} c^{4} + a^{2} 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)^3,x, algorithm="fricas")

[Out]

-1/8/(b^3*d^9*x^8 + 8*b^3*c*d^8*x^7 + 28*b^3*c^2*d^7*x^6 + 56*b^3*c^3*d^6*x^5 + 2*(35*b^3*c^4 + a*b^2)*d^5*x^4
+ 8*(7*b^3*c^5 + a*b^2*c)*d^4*x^3 + 4*(7*b^3*c^6 + 3*a*b^2*c^2)*d^3*x^2 + 8*(b^3*c^7 + a*b^2*c^3)*d^2*x + (b^
3*c^8 + 2*a*b^2*c^4 + a^2*b)*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.23129, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{8 \,{\left ({\left (d x + c\right )}^{4} b + a\right )}^{2} 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)^3,x, algorithm="giac")

[Out]

-1/8/(((d*x + c)^4*b + a)^2*b*d)