3.249 \(\int (d x)^m (a+b x^3+c x^6) \, dx\)

Optimal. Leaf size=52 \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+4}}{d^4 (m+4)}+\frac{c (d x)^{m+7}}{d^7 (m+7)} \]

[Out]

(a*(d*x)^(1 + m))/(d*(1 + m)) + (b*(d*x)^(4 + m))/(d^4*(4 + m)) + (c*(d*x)^(7 + m))/(d^7*(7 + m))

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Rubi [A]  time = 0.0206515, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {14} \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+4}}{d^4 (m+4)}+\frac{c (d x)^{m+7}}{d^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*x^3 + c*x^6),x]

[Out]

(a*(d*x)^(1 + m))/(d*(1 + m)) + (b*(d*x)^(4 + m))/(d^4*(4 + m)) + (c*(d*x)^(7 + m))/(d^7*(7 + 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 (a+b x^3+c x^6\right ) \, dx &=\int \left (a (d x)^m+\frac{b (d x)^{3+m}}{d^3}+\frac{c (d x)^{6+m}}{d^6}\right ) \, dx\\ &=\frac{a (d x)^{1+m}}{d (1+m)}+\frac{b (d x)^{4+m}}{d^4 (4+m)}+\frac{c (d x)^{7+m}}{d^7 (7+m)}\\ \end{align*}

Mathematica [A]  time = 0.0295413, size = 35, normalized size = 0.67 \[ x (d x)^m \left (\frac{a}{m+1}+\frac{b x^3}{m+4}+\frac{c x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*x^3 + c*x^6),x]

[Out]

x*(d*x)^m*(a/(1 + m) + (b*x^3)/(4 + m) + (c*x^6)/(7 + m))

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Maple [A]  time = 0.003, size = 78, normalized size = 1.5 \begin{align*}{\frac{ \left ( c{m}^{2}{x}^{6}+5\,cm{x}^{6}+4\,c{x}^{6}+b{m}^{2}{x}^{3}+8\,bm{x}^{3}+7\,b{x}^{3}+a{m}^{2}+11\,am+28\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(c*x^6+b*x^3+a),x)

[Out]

x*(c*m^2*x^6+5*c*m*x^6+4*c*x^6+b*m^2*x^3+8*b*m*x^3+7*b*x^3+a*m^2+11*a*m+28*a)*(d*x)^m/(7+m)/(4+m)/(1+m)

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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)^m*(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57497, size = 162, normalized size = 3.12 \begin{align*} \frac{{\left ({\left (c m^{2} + 5 \, c m + 4 \, c\right )} x^{7} +{\left (b m^{2} + 8 \, b m + 7 \, b\right )} x^{4} +{\left (a m^{2} + 11 \, a m + 28 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

((c*m^2 + 5*c*m + 4*c)*x^7 + (b*m^2 + 8*b*m + 7*b)*x^4 + (a*m^2 + 11*a*m + 28*a)*x)*(d*x)^m/(m^3 + 12*m^2 + 39
*m + 28)

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Sympy [A]  time = 1.50703, size = 314, normalized size = 6.04 \begin{align*} \begin{cases} \frac{- \frac{a}{6 x^{6}} - \frac{b}{3 x^{3}} + c \log{\left (x \right )}}{d^{7}} & \text{for}\: m = -7 \\\frac{- \frac{a}{3 x^{3}} + b \log{\left (x \right )} + \frac{c x^{3}}{3}}{d^{4}} & \text{for}\: m = -4 \\\frac{a \log{\left (x \right )} + \frac{b x^{3}}{3} + \frac{c x^{6}}{6}}{d} & \text{for}\: m = -1 \\\frac{a d^{m} m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 a d^{m} m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 a d^{m} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{b d^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{8 b d^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{7 b d^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{c d^{m} m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 c d^{m} m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 c d^{m} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**6+b*x**3+a),x)

[Out]

Piecewise(((-a/(6*x**6) - b/(3*x**3) + c*log(x))/d**7, Eq(m, -7)), ((-a/(3*x**3) + b*log(x) + c*x**3/3)/d**4,
Eq(m, -4)), ((a*log(x) + b*x**3/3 + c*x**6/6)/d, Eq(m, -1)), (a*d**m*m**2*x*x**m/(m**3 + 12*m**2 + 39*m + 28)
+ 11*a*d**m*m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 28*a*d**m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + b*d**m*m**
2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 8*b*d**m*m*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 7*b*d**m*x**4*x
**m/(m**3 + 12*m**2 + 39*m + 28) + c*d**m*m**2*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28) + 5*c*d**m*m*x**7*x**m/(
m**3 + 12*m**2 + 39*m + 28) + 4*c*d**m*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28), True))

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Giac [B]  time = 1.09214, size = 161, normalized size = 3.1 \begin{align*} \frac{\left (d x\right )^{m} c m^{2} x^{7} + 5 \, \left (d x\right )^{m} c m x^{7} + 4 \, \left (d x\right )^{m} c x^{7} + \left (d x\right )^{m} b m^{2} x^{4} + 8 \, \left (d x\right )^{m} b m x^{4} + 7 \, \left (d x\right )^{m} b x^{4} + \left (d x\right )^{m} a m^{2} x + 11 \, \left (d x\right )^{m} a m x + 28 \, \left (d x\right )^{m} a x}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

((d*x)^m*c*m^2*x^7 + 5*(d*x)^m*c*m*x^7 + 4*(d*x)^m*c*x^7 + (d*x)^m*b*m^2*x^4 + 8*(d*x)^m*b*m*x^4 + 7*(d*x)^m*b
*x^4 + (d*x)^m*a*m^2*x + 11*(d*x)^m*a*m*x + 28*(d*x)^m*a*x)/(m^3 + 12*m^2 + 39*m + 28)