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

Optimal. Leaf size=58 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{b^2 (d x)^{m+5}}{d^5 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.0232997, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {14} \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{b^2 (d x)^{m+5}}{d^5 (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0294711, size = 41, normalized size = 0.71 \[ x (d x)^m \left (\frac{a^2}{m+1}+\frac{2 a b x^2}{m+3}+\frac{b^2 x^4}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 94, normalized size = 1.6 \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ({b}^{2}{m}^{2}{x}^{4}+4\,{b}^{2}m{x}^{4}+2\,ab{m}^{2}{x}^{2}+3\,{b}^{2}{x}^{4}+12\,abm{x}^{2}+{a}^{2}{m}^{2}+10\,ab{x}^{2}+8\,m{a}^{2}+15\,{a}^{2} \right ) x}{ \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*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

(d*x)^m*(b^2*m^2*x^4+4*b^2*m*x^4+2*a*b*m^2*x^2+3*b^2*x^4+12*a*b*m*x^2+a^2*m^2+10*a*b*x^2+8*a^2*m+15*a^2)*x/(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*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57241, size = 186, normalized size = 3.21 \begin{align*} \frac{{\left ({\left (b^{2} m^{2} + 4 \, b^{2} m + 3 \, b^{2}\right )} x^{5} + 2 \,{\left (a b m^{2} + 6 \, a b m + 5 \, a b\right )} x^{3} +{\left (a^{2} m^{2} + 8 \, a^{2} m + 15 \, a^{2}\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*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.894467, size = 345, normalized size = 5.95 \begin{align*} \begin{cases} \frac{- \frac{a^{2}}{4 x^{4}} - \frac{a b}{x^{2}} + b^{2} \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2}}{d^{3}} & \text{for}\: m = -3 \\\frac{a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b^{2} x^{4}}{4}}{d} & \text{for}\: m = -1 \\\frac{a^{2} d^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{8 a^{2} d^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{15 a^{2} d^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{2 a b d^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{12 a b d^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{10 a b d^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{b^{2} d^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{4 b^{2} d^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{3 b^{2} 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*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

Piecewise(((-a**2/(4*x**4) - a*b/x**2 + b**2*log(x))/d**5, Eq(m, -5)), ((-a**2/(2*x**2) + 2*a*b*log(x) + b**2*
x**2/2)/d**3, Eq(m, -3)), ((a**2*log(x) + a*b*x**2 + b**2*x**4/4)/d, Eq(m, -1)), (a**2*d**m*m**2*x*x**m/(m**3
+ 9*m**2 + 23*m + 15) + 8*a**2*d**m*m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 15*a**2*d**m*x*x**m/(m**3 + 9*m**2
+ 23*m + 15) + 2*a*b*d**m*m**2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 12*a*b*d**m*m*x**3*x**m/(m**3 + 9*m**2
+ 23*m + 15) + 10*a*b*d**m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + b**2*d**m*m**2*x**5*x**m/(m**3 + 9*m**2 + 2
3*m + 15) + 4*b**2*d**m*m*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 3*b**2*d**m*x**5*x**m/(m**3 + 9*m**2 + 23*m
+ 15), True))

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Giac [B]  time = 1.28529, size = 182, normalized size = 3.14 \begin{align*} \frac{\left (d x\right )^{m} b^{2} m^{2} x^{5} + 4 \, \left (d x\right )^{m} b^{2} m x^{5} + 2 \, \left (d x\right )^{m} a b m^{2} x^{3} + 3 \, \left (d x\right )^{m} b^{2} x^{5} + 12 \, \left (d x\right )^{m} a b m x^{3} + \left (d x\right )^{m} a^{2} m^{2} x + 10 \, \left (d x\right )^{m} a b x^{3} + 8 \, \left (d x\right )^{m} a^{2} m x + 15 \, \left (d x\right )^{m} a^{2} 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*(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")

[Out]

((d*x)^m*b^2*m^2*x^5 + 4*(d*x)^m*b^2*m*x^5 + 2*(d*x)^m*a*b*m^2*x^3 + 3*(d*x)^m*b^2*x^5 + 12*(d*x)^m*a*b*m*x^3
+ (d*x)^m*a^2*m^2*x + 10*(d*x)^m*a*b*x^3 + 8*(d*x)^m*a^2*m*x + 15*(d*x)^m*a^2*x)/(m^3 + 9*m^2 + 23*m + 15)