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)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = 0.03, 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))

________________________________________________________________________________________

fricas [A]  time = 1.02, size = 87, normalized size = 1.50 \[ \frac {{\left ({\left (b^{2} m^{2} + 4 \, b^{2} m + 3 \, b^{2}\right )} x^{5} + 2 \, {\left (a b m^{2} + 6 \, a b m + 5 \, a b\right )} x^{3} + {\left (a^{2} m^{2} + 8 \, a^{2} m + 15 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.16, size = 135, normalized size = 2.33 \[ \frac {\left (d x\right )^{m} b^{2} m^{2} x^{5} + 4 \, \left (d x\right )^{m} b^{2} m x^{5} + 2 \, \left (d x\right )^{m} a b m^{2} x^{3} + 3 \, \left (d x\right )^{m} b^{2} x^{5} + 12 \, \left (d x\right )^{m} a b m x^{3} + \left (d x\right )^{m} a^{2} m^{2} x + 10 \, \left (d x\right )^{m} a b x^{3} + 8 \, \left (d x\right )^{m} a^{2} m x + 15 \, \left (d x\right )^{m} a^{2} x}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 94, normalized size = 1.62 \[ \frac {\left (b^{2} m^{2} x^{4}+4 b^{2} m \,x^{4}+2 a b \,m^{2} x^{2}+3 b^{2} x^{4}+12 a b m \,x^{2}+a^{2} m^{2}+10 a b \,x^{2}+8 a^{2} m +15 a^{2}\right ) x \left (d x \right )^{m}}{\left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 56, normalized size = 0.97 \[ \frac {b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 4.27, size = 95, normalized size = 1.64 \[ {\left (d\,x\right )}^m\,\left (\frac {a^2\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}+\frac {b^2\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {2\,a\,b\,x^3\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.01, size = 345, normalized size = 5.95 \[ \begin {cases} \frac {- \frac {a^{2}}{4 x^{4}} - \frac {a b}{x^{2}} + b^{2} \log {\relax (x )}}{d^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a^{2}}{2 x^{2}} + 2 a b \log {\relax (x )} + \frac {b^{2} x^{2}}{2}}{d^{3}} & \text {for}\: m = -3 \\\frac {a^{2} \log {\relax (x )} + a b x^{2} + \frac {b^{2} x^{4}}{4}}{d} & \text {for}\: m = -1 \\\frac {a^{2} d^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 a^{2} d^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 a^{2} d^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {2 a b d^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {12 a b d^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {10 a b d^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {b^{2} d^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 b^{2} d^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 b^{2} d^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________