3.2.14 \(\int (d x)^m (b x+c x^2) \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {b (d x)^{m+2}}{d^2 (m+2)}+\frac {c (d x)^{m+3}}{d^3 (m+3)} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.71 \begin {gather*} x^2 (d x)^m \left (\frac {b}{m+2}+\frac {c x}{m+3}\right ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d*x)^m*(b*x + c*x^2),x]

[Out]

Defer[IntegrateAlgebraic][(d*x)^m*(b*x + c*x^2), x]

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fricas [A]  time = 0.41, size = 39, normalized size = 1.11 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.15, size = 56, normalized size = 1.60 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.04, size = 35, normalized size = 1.00 \begin {gather*} \frac {\left (c m x +b m +2 c x +3 b \right ) x^{2} \left (d x \right )^{m}}{\left (m +3\right ) \left (m +2\right )} \end {gather*}

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/(m+3)/(m+2)

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maxima [A]  time = 1.48, size = 33, normalized size = 0.94 \begin {gather*} \frac {c d^{m} x^{3} x^{m}}{m + 3} + \frac {b d^{m} x^{2} x^{m}}{m + 2} \end {gather*}

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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mupad [B]  time = 0.19, size = 34, normalized size = 0.97 \begin {gather*} \frac {x^2\,{\left (d\,x\right )}^m\,\left (3\,b+b\,m+2\,c\,x+c\,m\,x\right )}{m^2+5\,m+6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)*(d*x)^m,x)

[Out]

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

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sympy [A]  time = 0.45, size = 112, normalized size = 3.20 \begin {gather*} \begin {cases} \frac {- \frac {b}{x} + c \log {\relax (x )}}{d^{3}} & \text {for}\: m = -3 \\\frac {b \log {\relax (x )} + 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 {gather*}

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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