3.183 \(\int (b+3 d x^2) (a+b x+d x^3)^n \, dx\)

Optimal. Leaf size=20 \[ \frac{\left (a+b x+d x^3\right )^{n+1}}{n+1} \]

[Out]

(a + b*x + d*x^3)^(1 + n)/(1 + n)

________________________________________________________________________________________

Rubi [A]  time = 0.0099577, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1588} \[ \frac{\left (a+b x+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[(b + 3*d*x^2)*(a + b*x + d*x^3)^n,x]

[Out]

(a + b*x + d*x^3)^(1 + n)/(1 + n)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (b+3 d x^2\right ) \left (a+b x+d x^3\right )^n \, dx &=\frac{\left (a+b x+d x^3\right )^{1+n}}{1+n}\\ \end{align*}

Mathematica [A]  time = 0.0085864, size = 20, normalized size = 1. \[ \frac{\left (a+b x+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 3*d*x^2)*(a + b*x + d*x^3)^n,x]

[Out]

(a + b*x + d*x^3)^(1 + n)/(1 + n)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 21, normalized size = 1.1 \begin{align*}{\frac{ \left ( d{x}^{3}+bx+a \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*d*x^2+b)*(d*x^3+b*x+a)^n,x)

[Out]

(d*x^3+b*x+a)^(1+n)/(1+n)

________________________________________________________________________________________

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((3*d*x^2+b)*(d*x^3+b*x+a)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.46924, size = 63, normalized size = 3.15 \begin{align*} \frac{{\left (d x^{3} + b x + a\right )}{\left (d x^{3} + b x + a\right )}^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x^3 + b*x + a)*(d*x^3 + b*x + a)^n/(n + 1)

________________________________________________________________________________________

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((3*d*x**2+b)*(d*x**3+b*x+a)**n,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.18805, size = 72, normalized size = 3.6 \begin{align*} \frac{{\left (d x^{3} + b x + a\right )}^{n} d x^{3} +{\left (d x^{3} + b x + a\right )}^{n} b x +{\left (d x^{3} + b x + a\right )}^{n} a}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*x^3 + b*x + a)^n*d*x^3 + (d*x^3 + b*x + a)^n*b*x + (d*x^3 + b*x + a)^n*a)/(n + 1)