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

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 = 0.0469034, 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 \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+4}}{d^4 (m+4)}+\frac{c (d x)^{m+7}}{d^7 (m+7)} \]

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))

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Rubi in Sympy [A]  time = 11.1272, size = 42, normalized size = 0.81 \[ \frac{a \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{b \left (d x\right )^{m + 4}}{d^{4} \left (m + 4\right )} + \frac{c \left (d x\right )^{m + 7}}{d^{7} \left (m + 7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**6+b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.0319807, size = 35, normalized size = 0.67 \[ (d x)^m \left (\frac{a x}{m+1}+\frac{b x^4}{m+4}+\frac{c x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 78, normalized size = 1.5 \[{\frac{ \left ( c{m}^{2}{x}^{6}+5\,cm{x}^{6}+4\,c{x}^{6}+b{m}^{2}{x}^{3}+8\,bm{x}^{3}+7\,b{x}^{3}+a{m}^{2}+11\,am+28\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)*(d*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269514, size = 96, normalized size = 1.85 \[ \frac{{\left ({\left (c m^{2} + 5 \, c m + 4 \, c\right )} x^{7} +{\left (b m^{2} + 8 \, b m + 7 \, b\right )} x^{4} +{\left (a m^{2} + 11 \, a m + 28 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)*(d*x)^m,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 = 4.40074, size = 314, normalized size = 6.04 \[ \begin{cases} \frac{- \frac{a}{6 x^{6}} - \frac{b}{3 x^{3}} + c \log{\left (x \right )}}{d^{7}} & \text{for}\: m = -7 \\\frac{- \frac{a}{3 x^{3}} + b \log{\left (x \right )} + \frac{c x^{3}}{3}}{d^{4}} & \text{for}\: m = -4 \\\frac{a \log{\left (x \right )} + \frac{b x^{3}}{3} + \frac{c x^{6}}{6}}{d} & \text{for}\: m = -1 \\\frac{a d^{m} m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 a d^{m} m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 a d^{m} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{b d^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{8 b d^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{7 b d^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{c d^{m} m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 c d^{m} m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 c d^{m} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \]

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, E
q(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/XCAS [A]  time = 0.266169, size = 185, normalized size = 3.56 \[ \frac{c m^{2} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 5 \, c m x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 4 \, c x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + b m^{2} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 8 \, b m x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 7 \, b x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + a m^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )} + 11 \, a m x e^{\left (m{\rm ln}\left (d x\right )\right )} + 28 \, a x e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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