3.2851 \(\int (c+d x) (a+b (c+d x)^2)^p \, dx\)

Optimal. Leaf size=30 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0212933, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {372, 261} \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0089686, size = 30, normalized size = 1. \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 1.3 \begin{align*}{\frac{ \left ( b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a \right ) ^{1+p}}{2\,bd \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d^2*x^2+2*b*c*d*x+b*c^2+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)*(a+b*(d*x+c)^2)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5987, size = 126, normalized size = 4.2 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (b d p + b d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.10946, size = 170, normalized size = 5.67 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b d^{2} x^{2} + 2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b c d x +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b c^{2} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a}{2 \,{\left (b d p + b d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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