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

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

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Rubi [A]  time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 73, normalized size = 0.70 \begin {gather*} x (d x)^m \left (\frac {a^4}{m+1}+\frac {4 a^3 b x^2}{m+3}+\frac {6 a^2 b^2 x^4}{m+5}+\frac {4 a b^3 x^6}{m+7}+\frac {b^4 x^8}{m+9}\right ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.98, size = 253, normalized size = 2.43 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.18, size = 415, normalized size = 3.99 \begin {gather*} \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 {gather*}

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)

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maple [B]  time = 0.01, size = 292, normalized size = 2.81 \begin {gather*} \frac {\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} b m \,x^{2}+206 a^{4} m^{2}+1260 a^{3} b \,x^{2}+744 a^{4} m +945 a^{4}\right ) x \left (d x \right )^{m}}{\left (m +9\right ) \left (m +7\right ) \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*(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/(m+9)/(m+7)/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 1.49, size = 100, normalized size = 0.96 \begin {gather*} \frac {b^{4} d^{m} x^{9} x^{m}}{m + 9} + \frac {4 \, a b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, a^{2} b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {4 \, a^{3} b d^{m} x^{3} x^{m}}{m + 3} + \frac {\left (d x\right )^{m + 1} a^{4}}{d {\left (m + 1\right )}} \end {gather*}

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]

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

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mupad [B]  time = 4.51, size = 263, normalized size = 2.53 \begin {gather*} {\left (d\,x\right )}^m\,\left (\frac {b^4\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^4\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {4\,a\,b^3\,x^7\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {4\,a^3\,b\,x^3\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {6\,a^2\,b^2\,x^5\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \end {gather*}

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)^2,x)

[Out]

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

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sympy [A]  time = 3.20, size = 1321, normalized size = 12.70

result too large to display

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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