3.1107 \(\int (d x)^m (a+b x^2+c x^4) \, dx\)

Optimal. Leaf size=52 \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+3}}{d^3 (m+3)}+\frac{c (d x)^{m+5}}{d^5 (m+5)} \]

[Out]

(a*(d*x)^(1 + m))/(d*(1 + m)) + (b*(d*x)^(3 + m))/(d^3*(3 + m)) + (c*(d*x)^(5 + m))/(d^5*(5 + m))

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Rubi [A]  time = 0.0195031, 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} \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+3}}{d^3 (m+3)}+\frac{c (d x)^{m+5}}{d^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a + b*x^2 + c*x^4),x]

[Out]

(a*(d*x)^(1 + m))/(d*(1 + m)) + (b*(d*x)^(3 + m))/(d^3*(3 + m)) + (c*(d*x)^(5 + m))/(d^5*(5 + 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^2+c x^4\right ) \, dx &=\int \left (a (d x)^m+\frac{b (d x)^{2+m}}{d^2}+\frac{c (d x)^{4+m}}{d^4}\right ) \, dx\\ &=\frac{a (d x)^{1+m}}{d (1+m)}+\frac{b (d x)^{3+m}}{d^3 (3+m)}+\frac{c (d x)^{5+m}}{d^5 (5+m)}\\ \end{align*}

Mathematica [A]  time = 0.0296029, size = 35, normalized size = 0.67 \[ x (d x)^m \left (\frac{a}{m+1}+\frac{b x^2}{m+3}+\frac{c x^4}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a + b*x^2 + c*x^4),x]

[Out]

x*(d*x)^m*(a/(1 + m) + (b*x^2)/(3 + m) + (c*x^4)/(5 + m))

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Maple [A]  time = 0.043, size = 78, normalized size = 1.5 \begin{align*}{\frac{ \left ( c{m}^{2}{x}^{4}+4\,cm{x}^{4}+b{m}^{2}{x}^{2}+3\,c{x}^{4}+6\,bm{x}^{2}+a{m}^{2}+5\,b{x}^{2}+8\,am+15\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 3+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^4+b*x^2+a),x)

[Out]

x*(c*m^2*x^4+4*c*m*x^4+b*m^2*x^2+3*c*x^4+6*b*m*x^2+a*m^2+5*b*x^2+8*a*m+15*a)*(d*x)^m/(5+m)/(3+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^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29335, size = 159, normalized size = 3.06 \begin{align*} \frac{{\left ({\left (c m^{2} + 4 \, c m + 3 \, c\right )} x^{5} +{\left (b m^{2} + 6 \, b m + 5 \, b\right )} x^{3} +{\left (a m^{2} + 8 \, a m + 15 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

((c*m^2 + 4*c*m + 3*c)*x^5 + (b*m^2 + 6*b*m + 5*b)*x^3 + (a*m^2 + 8*a*m + 15*a)*x)*(d*x)^m/(m^3 + 9*m^2 + 23*m
+ 15)

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Sympy [A]  time = 0.874611, size = 314, normalized size = 6.04 \begin{align*} \begin{cases} \frac{- \frac{a}{4 x^{4}} - \frac{b}{2 x^{2}} + c \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{a}{2 x^{2}} + b \log{\left (x \right )} + \frac{c x^{2}}{2}}{d^{3}} & \text{for}\: m = -3 \\\frac{a \log{\left (x \right )} + \frac{b x^{2}}{2} + \frac{c x^{4}}{4}}{d} & \text{for}\: m = -1 \\\frac{a d^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{8 a d^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{15 a d^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{b d^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{6 b d^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{5 b d^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{c d^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{4 c d^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{3 c d^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(c*x**4+b*x**2+a),x)

[Out]

Piecewise(((-a/(4*x**4) - b/(2*x**2) + c*log(x))/d**5, Eq(m, -5)), ((-a/(2*x**2) + b*log(x) + c*x**2/2)/d**3,
Eq(m, -3)), ((a*log(x) + b*x**2/2 + c*x**4/4)/d, Eq(m, -1)), (a*d**m*m**2*x*x**m/(m**3 + 9*m**2 + 23*m + 15) +
8*a*d**m*m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 15*a*d**m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + b*d**m*m**2*x*
*3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 6*b*d**m*m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 5*b*d**m*x**3*x**m/(m
**3 + 9*m**2 + 23*m + 15) + c*d**m*m**2*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 4*c*d**m*m*x**5*x**m/(m**3 + 9
*m**2 + 23*m + 15) + 3*c*d**m*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15), True))

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Giac [B]  time = 1.10666, size = 161, normalized size = 3.1 \begin{align*} \frac{\left (d x\right )^{m} c m^{2} x^{5} + 4 \, \left (d x\right )^{m} c m x^{5} + \left (d x\right )^{m} b m^{2} x^{3} + 3 \, \left (d x\right )^{m} c x^{5} + 6 \, \left (d x\right )^{m} b m x^{3} + \left (d x\right )^{m} a m^{2} x + 5 \, \left (d x\right )^{m} b x^{3} + 8 \, \left (d x\right )^{m} a m x + 15 \, \left (d x\right )^{m} a x}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

((d*x)^m*c*m^2*x^5 + 4*(d*x)^m*c*m*x^5 + (d*x)^m*b*m^2*x^3 + 3*(d*x)^m*c*x^5 + 6*(d*x)^m*b*m*x^3 + (d*x)^m*a*m
^2*x + 5*(d*x)^m*b*x^3 + 8*(d*x)^m*a*m*x + 15*(d*x)^m*a*x)/(m^3 + 9*m^2 + 23*m + 15)