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

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

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Rubi [A]  time = 0.02, 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+3}}{d^3 (m+3)}+\frac {c (d x)^{m+5}}{d^5 (m+5)} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 35, normalized size = 0.67 \begin {gather*} x (d x)^m \left (\frac {a}{m+1}+\frac {b x^2}{m+3}+\frac {c x^4}{m+5}\right ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b x^2+c x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.79, size = 71, normalized size = 1.37 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.16, size = 119, normalized size = 2.29 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.00, size = 78, normalized size = 1.50 \begin {gather*} \frac {\left (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 \right ) x \left (d x \right )^{m}}{\left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \end {gather*}

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/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 1.09, size = 50, normalized size = 0.96 \begin {gather*} \frac {c d^{m} x^{5} x^{m}}{m + 5} + \frac {b d^{m} x^{3} x^{m}}{m + 3} + \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^4+b*x^2+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.40, size = 89, normalized size = 1.71 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b\,x^3\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}+\frac {c\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {a\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*x^2 + c*x^4),x)

[Out]

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

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sympy [A]  time = 0.99, size = 314, normalized size = 6.04 \begin {gather*} \begin {cases} \frac {- \frac {a}{4 x^{4}} - \frac {b}{2 x^{2}} + c \log {\relax (x )}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a}{2 x^{2}} + b \log {\relax (x )} + \frac {c x^{2}}{2}}{d^{3}} & \text {for}\: m = -3 \\\frac {a \log {\relax (x )} + \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 {gather*}

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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