### 3.114 $$\int (d x)^m (b x+c x^2) \, dx$$

Optimal. Leaf size=35 $\frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)}$

[Out]

(b*(d*x)^(2 + m))/(d^2*(2 + m)) + (c*(d*x)^(3 + m))/(d^3*(3 + m))

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Rubi [A]  time = 0.0139718, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {14} $\frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d*x)^m*(b*x + c*x^2),x]

[Out]

(b*(d*x)^(2 + m))/(d^2*(2 + m)) + (c*(d*x)^(3 + m))/(d^3*(3 + m))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0147581, size = 25, normalized size = 0.71 $x^2 (d x)^m \left (\frac{b}{m+2}+\frac{c x}{m+3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d*x)^m*(b*x + c*x^2),x]

[Out]

x^2*(d*x)^m*(b/(2 + m) + (c*x)/(3 + m))

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Maple [A]  time = 0.046, size = 35, normalized size = 1. \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ( cmx+bm+2\,cx+3\,b \right ){x}^{2}}{ \left ( 3+m \right ) \left ( 2+m \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x)^m*(c*m*x+b*m+2*c*x+3*b)*x^2/(3+m)/(2+m)

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Maxima [A]  time = 1.15542, size = 45, normalized size = 1.29 \begin{align*} \frac{c d^{m} x^{3} x^{m}}{m + 3} + \frac{b d^{m} x^{2} x^{m}}{m + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^2+b*x),x, algorithm="maxima")

[Out]

c*d^m*x^3*x^m/(m + 3) + b*d^m*x^2*x^m/(m + 2)

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Fricas [A]  time = 2.03065, size = 82, normalized size = 2.34 \begin{align*} \frac{{\left ({\left (c m + 2 \, c\right )} x^{3} +{\left (b m + 3 \, b\right )} x^{2}\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*m + 2*c)*x^3 + (b*m + 3*b)*x^2)*(d*x)^m/(m^2 + 5*m + 6)

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Sympy [A]  time = 0.79318, size = 112, normalized size = 3.2 \begin{align*} \begin{cases} \frac{- \frac{b}{x} + c \log{\left (x \right )}}{d^{3}} & \text{for}\: m = -3 \\\frac{b \log{\left (x \right )} + c x}{d^{2}} & \text{for}\: m = -2 \\\frac{b d^{m} m x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{3 b d^{m} x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{c d^{m} m x^{3} x^{m}}{m^{2} + 5 m + 6} + \frac{2 c d^{m} x^{3} x^{m}}{m^{2} + 5 m + 6} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**2+b*x),x)

[Out]

Piecewise(((-b/x + c*log(x))/d**3, Eq(m, -3)), ((b*log(x) + c*x)/d**2, Eq(m, -2)), (b*d**m*m*x**2*x**m/(m**2 +
5*m + 6) + 3*b*d**m*x**2*x**m/(m**2 + 5*m + 6) + c*d**m*m*x**3*x**m/(m**2 + 5*m + 6) + 2*c*d**m*x**3*x**m/(m*
*2 + 5*m + 6), True))

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Giac [A]  time = 1.33834, size = 76, normalized size = 2.17 \begin{align*} \frac{\left (d x\right )^{m} c m x^{3} + \left (d x\right )^{m} b m x^{2} + 2 \, \left (d x\right )^{m} c x^{3} + 3 \, \left (d x\right )^{m} b x^{2}}{m^{2} + 5 \, m + 6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*x)^m*c*m*x^3 + (d*x)^m*b*m*x^2 + 2*(d*x)^m*c*x^3 + 3*(d*x)^m*b*x^2)/(m^2 + 5*m + 6)