3.2910 \(\int (c+d x)^3 (a+b (c+d x)^4)^p \, dx\)

Optimal. Leaf size=30 \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]

[Out]

(a + b*(c + d*x)^4)^(1 + p)/(4*b*d*(1 + p))

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Rubi [A]  time = 0.0301744, antiderivative size = 30, 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} \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*(c + d*x)^4)^(1 + p)/(4*b*d*(1 + p))

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

Mathematica [A]  time = 0.0166376, size = 30, normalized size = 1. \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*(c + d*x)^4)^(1 + p)/(4*b*d*(1 + p))

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Maple [B]  time = 0.006, size = 63, normalized size = 2.1 \begin{align*}{\frac{ \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 ) ^{1+p}}{4\,bd \left ( 1+p \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)^p,x)

[Out]

1/4*(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)^(1+p)/b/d/(1+p)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.29188, size = 223, normalized size = 7.43 \begin{align*} \frac{{\left (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\right )}{\left (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\right )}^{p}}{4 \,{\left (b d p + b 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)^p,x, algorithm="fricas")

[Out]

1/4*(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)*(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)^p/(b*d*p + 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)**p,x)

[Out]

Timed out

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Giac [A]  time = 1.20282, size = 38, normalized size = 1.27 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{p + 1}}{4 \, b d{\left (p + 1\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)^p,x, algorithm="giac")

[Out]

1/4*((d*x + c)^4*b + a)^(p + 1)/(b*d*(p + 1))