3.114 \(\int (d x)^m \left (b x+c x^2\right ) \, dx\)

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 = 0.0398129, 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 \[ \frac{b (d x)^{m+2}}{d^2 (m+2)}+\frac{c (d x)^{m+3}}{d^3 (m+3)} \]

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))

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Rubi in Sympy [A]  time = 9.34904, size = 29, normalized size = 0.83 \[ \frac{b \left (d x\right )^{m + 2}}{d^{2} \left (m + 2\right )} + \frac{c \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.0237165, size = 27, normalized size = 0.77 \[ (d x)^m \left (\frac{b x^2}{m+2}+\frac{c x^3}{m+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 1. \[{\frac{ \left ( dx \right ) ^{m} \left ( cmx+bm+2\,cx+3\,b \right ){x}^{2}}{ \left ( 3+m \right ) \left ( 2+m \right ) }} \]

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 = 0.711024, size = 45, normalized size = 1.29 \[ \frac{c d^{m} x^{3} x^{m}}{m + 3} + \frac{b d^{m} x^{2} x^{m}}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(d*x)^m,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 = 0.232397, size = 53, normalized size = 1.51 \[ \frac{{\left ({\left (c m + 2 \, c\right )} x^{3} +{\left (b m + 3 \, b\right )} x^{2}\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(d*x)^m,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 = 1.13931, size = 112, normalized size = 3.2 \[ \begin{cases} \frac{- \frac{b}{x} + c \log{\left (x \right )}}{d^{3}} & \text{for}\: m = -3 \\\frac{b \log{\left (x \right )} + c x}{d^{2}} & \text{for}\: m = -2 \\\frac{b d^{m} m x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{3 b d^{m} x^{2} x^{m}}{m^{2} + 5 m + 6} + \frac{c d^{m} m x^{3} x^{m}}{m^{2} + 5 m + 6} + \frac{2 c d^{m} x^{3} x^{m}}{m^{2} + 5 m + 6} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.210961, size = 86, normalized size = 2.46 \[ \frac{c m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + b m x^{2} e^{\left (m{\rm ln}\left (d x\right )\right )} + 2 \, c x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 3 \, b x^{2} e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{2} + 5 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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