3.3.49 \(\int (d x)^m (a+b x^3+c x^6) \, dx\) [249]

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

[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 = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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*}

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Mathematica [A]
time = 0.05, size = 35, normalized size = 0.67 \begin {gather*} x (d x)^m \left (\frac {a}{1+m}+\frac {b x^3}{4+m}+\frac {c x^6}{7+m}\right ) \end {gather*}

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 = 0.02, size = 51, normalized size = 0.98

method	result	size
norman	\(\frac {a x \,{\mathrm e}^{m \ln \left (d x \right )}}{1+m}+\frac {b \,x^{4} {\mathrm e}^{m \ln \left (d x \right )}}{4+m}+\frac {c \,x^{7} {\mathrm e}^{m \ln \left (d x \right )}}{7+m}\)	\(51\)
gosper	\(\frac {x \left (c \,m^{2} x^{6}+5 c m \,x^{6}+4 c \,x^{6}+b \,m^{2} x^{3}+8 b m \,x^{3}+7 b \,x^{3}+a \,m^{2}+11 a m +28 a \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\)	\(78\)
risch	\(\frac {x \left (c \,m^{2} x^{6}+5 c m \,x^{6}+4 c \,x^{6}+b \,m^{2} x^{3}+8 b m \,x^{3}+7 b \,x^{3}+a \,m^{2}+11 a m +28 a \right ) \left (d x \right )^{m}}{\left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 50, normalized size = 0.96 \begin {gather*} \frac {c d^{m} x^{7} x^{m}}{m + 7} + \frac {b d^{m} x^{4} x^{m}}{m + 4} + \frac {\left (d x\right )^{m + 1} a}{d {\left (m + 1\right )}} \end {gather*}

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]

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

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Fricas [A]
time = 0.36, size = 71, normalized size = 1.37 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (42) = 84\).
time = 0.31, size = 299, normalized size = 5.75 \begin {gather*} \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 m^{2} x \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {11 a m x \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {28 a x \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {b m^{2} x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {8 b m x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {7 b x^{4} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {c m^{2} x^{7} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {5 c m x^{7} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac {4 c x^{7} \left (d x\right )^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (52) = 104\).
time = 3.26, size = 119, normalized size = 2.29 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.36, size = 89, normalized size = 1.71 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b\,x^4\,\left (m^2+8\,m+7\right )}{m^3+12\,m^2+39\,m+28}+\frac {c\,x^7\,\left (m^2+5\,m+4\right )}{m^3+12\,m^2+39\,m+28}+\frac {a\,x\,\left (m^2+11\,m+28\right )}{m^3+12\,m^2+39\,m+28}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*x^3 + c*x^6),x)

[Out]

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

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