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]

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

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.00 \[ \frac {\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]

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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fricas [B]  time = 1.07, size = 104, normalized size = 3.47 \[ \frac {{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )} {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p}}{4 \, {\left (b d p + b d\right )}} \]

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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giac [B]  time = 0.18, size = 62, normalized size = 2.07 \[ \frac {{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p + 1}}{4 \, b d {\left (p + 1\right )}} \]

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*(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 + 1)/(b*d*(p + 1))

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maple [B]  time = 0.01, size = 63, normalized size = 2.10 \[ \frac {\left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )^{p +1}}{4 \left (p +1\right ) b d} \]

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

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maxima [A]  time = 0.44, size = 28, normalized size = 0.93 \[ \frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{p + 1}}{4 \, b d {\left (p + 1\right )}} \]

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]

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

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mupad [B]  time = 1.41, size = 88, normalized size = 2.93 \[ {\left (a+b\,{\left (c+d\,x\right )}^4\right )}^p\,\left (\frac {d^3\,x^4}{4\,\left (p+1\right )}+\frac {c^3\,x}{p+1}+\frac {b\,c^4+a}{4\,b\,d\,\left (p+1\right )}+\frac {3\,c^2\,d\,x^2}{2\,\left (p+1\right )}+\frac {c\,d^2\,x^3}{p+1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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