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

Optimal. Leaf size=104 \[ \frac{6 a^2 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac{4 a^3 b (d x)^{m+3}}{d^3 (m+3)}+\frac{a^4 (d x)^{m+1}}{d (m+1)}+\frac{4 a b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac{b^4 (d x)^{m+9}}{d^9 (m+9)} \]

[Out]

(a^4*(d*x)^(1 + m))/(d*(1 + m)) + (4*a^3*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (6*a^2*b^2*(d*x)^(5 + m))/(d^5*(5 +
m)) + (4*a*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (b^4*(d*x)^(9 + m))/(d^9*(9 + m))

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Rubi [A]  time = 0.074949, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {28, 270} \[ \frac{6 a^2 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac{4 a^3 b (d x)^{m+3}}{d^3 (m+3)}+\frac{a^4 (d x)^{m+1}}{d (m+1)}+\frac{4 a b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac{b^4 (d x)^{m+9}}{d^9 (m+9)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(d*x)^(1 + m))/(d*(1 + m)) + (4*a^3*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (6*a^2*b^2*(d*x)^(5 + m))/(d^5*(5 +
m)) + (4*a*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (b^4*(d*x)^(9 + m))/(d^9*(9 + m))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx &=\frac{\int (d x)^m \left (a b+b^2 x^2\right )^4 \, dx}{b^4}\\ &=\frac{\int \left (a^4 b^4 (d x)^m+\frac{4 a^3 b^5 (d x)^{2+m}}{d^2}+\frac{6 a^2 b^6 (d x)^{4+m}}{d^4}+\frac{4 a b^7 (d x)^{6+m}}{d^6}+\frac{b^8 (d x)^{8+m}}{d^8}\right ) \, dx}{b^4}\\ &=\frac{a^4 (d x)^{1+m}}{d (1+m)}+\frac{4 a^3 b (d x)^{3+m}}{d^3 (3+m)}+\frac{6 a^2 b^2 (d x)^{5+m}}{d^5 (5+m)}+\frac{4 a b^3 (d x)^{7+m}}{d^7 (7+m)}+\frac{b^4 (d x)^{9+m}}{d^9 (9+m)}\\ \end{align*}

Mathematica [A]  time = 0.0349877, size = 73, normalized size = 0.7 \[ x (d x)^m \left (\frac{6 a^2 b^2 x^4}{m+5}+\frac{4 a^3 b x^2}{m+3}+\frac{a^4}{m+1}+\frac{4 a b^3 x^6}{m+7}+\frac{b^4 x^8}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(d*x)^m*(a^4/(1 + m) + (4*a^3*b*x^2)/(3 + m) + (6*a^2*b^2*x^4)/(5 + m) + (4*a*b^3*x^6)/(7 + m) + (b^4*x^8)/(
9 + m))

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Maple [B]  time = 0.051, size = 292, normalized size = 2.8 \begin{align*}{\frac{ \left ( dx \right ) ^{m} \left ({b}^{4}{m}^{4}{x}^{8}+16\,{b}^{4}{m}^{3}{x}^{8}+4\,a{b}^{3}{m}^{4}{x}^{6}+86\,{b}^{4}{m}^{2}{x}^{8}+72\,a{b}^{3}{m}^{3}{x}^{6}+176\,{b}^{4}m{x}^{8}+6\,{a}^{2}{b}^{2}{m}^{4}{x}^{4}+416\,a{b}^{3}{m}^{2}{x}^{6}+105\,{b}^{4}{x}^{8}+120\,{a}^{2}{b}^{2}{m}^{3}{x}^{4}+888\,a{b}^{3}m{x}^{6}+4\,{a}^{3}b{m}^{4}{x}^{2}+780\,{a}^{2}{b}^{2}{m}^{2}{x}^{4}+540\,a{b}^{3}{x}^{6}+88\,{a}^{3}b{m}^{3}{x}^{2}+1800\,{a}^{2}{b}^{2}m{x}^{4}+{a}^{4}{m}^{4}+656\,{a}^{3}b{m}^{2}{x}^{2}+1134\,{a}^{2}{b}^{2}{x}^{4}+24\,{a}^{4}{m}^{3}+1832\,{a}^{3}bm{x}^{2}+206\,{a}^{4}{m}^{2}+1260\,{a}^{3}b{x}^{2}+744\,{a}^{4}m+945\,{a}^{4} \right ) x}{ \left ( 9+m \right ) \left ( 7+m \right ) \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)^2,x)

[Out]

(d*x)^m*(b^4*m^4*x^8+16*b^4*m^3*x^8+4*a*b^3*m^4*x^6+86*b^4*m^2*x^8+72*a*b^3*m^3*x^6+176*b^4*m*x^8+6*a^2*b^2*m^
4*x^4+416*a*b^3*m^2*x^6+105*b^4*x^8+120*a^2*b^2*m^3*x^4+888*a*b^3*m*x^6+4*a^3*b*m^4*x^2+780*a^2*b^2*m^2*x^4+54
0*a*b^3*x^6+88*a^3*b*m^3*x^2+1800*a^2*b^2*m*x^4+a^4*m^4+656*a^3*b*m^2*x^2+1134*a^2*b^2*x^4+24*a^4*m^3+1832*a^3
*b*m*x^2+206*a^4*m^2+1260*a^3*b*x^2+744*a^4*m+945*a^4)*x/(9+m)/(7+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*(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.6283, size = 576, normalized size = 5.54 \begin{align*} \frac{{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \,{\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \,{\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \,{\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} +{\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \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)^2,x, algorithm="fricas")

[Out]

((b^4*m^4 + 16*b^4*m^3 + 86*b^4*m^2 + 176*b^4*m + 105*b^4)*x^9 + 4*(a*b^3*m^4 + 18*a*b^3*m^3 + 104*a*b^3*m^2 +
222*a*b^3*m + 135*a*b^3)*x^7 + 6*(a^2*b^2*m^4 + 20*a^2*b^2*m^3 + 130*a^2*b^2*m^2 + 300*a^2*b^2*m + 189*a^2*b^
2)*x^5 + 4*(a^3*b*m^4 + 22*a^3*b*m^3 + 164*a^3*b*m^2 + 458*a^3*b*m + 315*a^3*b)*x^3 + (a^4*m^4 + 24*a^4*m^3 +
206*a^4*m^2 + 744*a^4*m + 945*a^4)*x)*(d*x)^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)

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Sympy [A]  time = 3.08984, size = 1321, normalized size = 12.7 \begin{align*} \text{result too large to display} \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)**2,x)

[Out]

Piecewise(((-a**4/(8*x**8) - 2*a**3*b/(3*x**6) - 3*a**2*b**2/(2*x**4) - 2*a*b**3/x**2 + b**4*log(x))/d**9, Eq(
m, -9)), ((-a**4/(6*x**6) - a**3*b/x**4 - 3*a**2*b**2/x**2 + 4*a*b**3*log(x) + b**4*x**2/2)/d**7, Eq(m, -7)),
((-a**4/(4*x**4) - 2*a**3*b/x**2 + 6*a**2*b**2*log(x) + 2*a*b**3*x**2 + b**4*x**4/4)/d**5, Eq(m, -5)), ((-a**4
/(2*x**2) + 4*a**3*b*log(x) + 3*a**2*b**2*x**2 + a*b**3*x**4 + b**4*x**6/6)/d**3, Eq(m, -3)), ((a**4*log(x) +
2*a**3*b*x**2 + 3*a**2*b**2*x**4/2 + 2*a*b**3*x**6/3 + b**4*x**8/8)/d, Eq(m, -1)), (a**4*d**m*m**4*x*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 24*a**4*d**m*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*
m**2 + 1689*m + 945) + 206*a**4*d**m*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*a
**4*d**m*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*a**4*d**m*x*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 4*a**3*b*d**m*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 88*a**3*b*d**m*m**3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 656*a**3
*b*d**m*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1832*a**3*b*d**m*m*x**3*x**m/(m
**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1260*a**3*b*d**m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 6*a**2*b**2*d**m*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 120*a**2*b**2*d**m*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 780*a**2*b**2*d
**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1800*a**2*b**2*d**m*m*x**5*x**m/(m*
*5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1134*a**2*b**2*d**m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3
+ 950*m**2 + 1689*m + 945) + 4*a*b**3*d**m*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945
) + 72*a*b**3*d**m*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 416*a*b**3*d**m*m**2
*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 888*a*b**3*d**m*m*x**7*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 540*a*b**3*d**m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + b**4*d**m*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*b**4*d**m*m**
3*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*b**4*d**m*m**2*x**9*x**m/(m**5 + 25*m**
4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 176*b**4*d**m*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 16
89*m + 945) + 105*b**4*d**m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))

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Giac [B]  time = 1.28462, size = 560, normalized size = 5.38 \begin{align*} \frac{\left (d x\right )^{m} b^{4} m^{4} x^{9} + 16 \, \left (d x\right )^{m} b^{4} m^{3} x^{9} + 4 \, \left (d x\right )^{m} a b^{3} m^{4} x^{7} + 86 \, \left (d x\right )^{m} b^{4} m^{2} x^{9} + 72 \, \left (d x\right )^{m} a b^{3} m^{3} x^{7} + 176 \, \left (d x\right )^{m} b^{4} m x^{9} + 6 \, \left (d x\right )^{m} a^{2} b^{2} m^{4} x^{5} + 416 \, \left (d x\right )^{m} a b^{3} m^{2} x^{7} + 105 \, \left (d x\right )^{m} b^{4} x^{9} + 120 \, \left (d x\right )^{m} a^{2} b^{2} m^{3} x^{5} + 888 \, \left (d x\right )^{m} a b^{3} m x^{7} + 4 \, \left (d x\right )^{m} a^{3} b m^{4} x^{3} + 780 \, \left (d x\right )^{m} a^{2} b^{2} m^{2} x^{5} + 540 \, \left (d x\right )^{m} a b^{3} x^{7} + 88 \, \left (d x\right )^{m} a^{3} b m^{3} x^{3} + 1800 \, \left (d x\right )^{m} a^{2} b^{2} m x^{5} + \left (d x\right )^{m} a^{4} m^{4} x + 656 \, \left (d x\right )^{m} a^{3} b m^{2} x^{3} + 1134 \, \left (d x\right )^{m} a^{2} b^{2} x^{5} + 24 \, \left (d x\right )^{m} a^{4} m^{3} x + 1832 \, \left (d x\right )^{m} a^{3} b m x^{3} + 206 \, \left (d x\right )^{m} a^{4} m^{2} x + 1260 \, \left (d x\right )^{m} a^{3} b x^{3} + 744 \, \left (d x\right )^{m} a^{4} m x + 945 \, \left (d x\right )^{m} a^{4} x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \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)^2,x, algorithm="giac")

[Out]

((d*x)^m*b^4*m^4*x^9 + 16*(d*x)^m*b^4*m^3*x^9 + 4*(d*x)^m*a*b^3*m^4*x^7 + 86*(d*x)^m*b^4*m^2*x^9 + 72*(d*x)^m*
a*b^3*m^3*x^7 + 176*(d*x)^m*b^4*m*x^9 + 6*(d*x)^m*a^2*b^2*m^4*x^5 + 416*(d*x)^m*a*b^3*m^2*x^7 + 105*(d*x)^m*b^
4*x^9 + 120*(d*x)^m*a^2*b^2*m^3*x^5 + 888*(d*x)^m*a*b^3*m*x^7 + 4*(d*x)^m*a^3*b*m^4*x^3 + 780*(d*x)^m*a^2*b^2*
m^2*x^5 + 540*(d*x)^m*a*b^3*x^7 + 88*(d*x)^m*a^3*b*m^3*x^3 + 1800*(d*x)^m*a^2*b^2*m*x^5 + (d*x)^m*a^4*m^4*x +
656*(d*x)^m*a^3*b*m^2*x^3 + 1134*(d*x)^m*a^2*b^2*x^5 + 24*(d*x)^m*a^4*m^3*x + 1832*(d*x)^m*a^3*b*m*x^3 + 206*(
d*x)^m*a^4*m^2*x + 1260*(d*x)^m*a^3*b*x^3 + 744*(d*x)^m*a^4*m*x + 945*(d*x)^m*a^4*x)/(m^5 + 25*m^4 + 230*m^3 +
950*m^2 + 1689*m + 945)