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

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

[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 = 0.193038, 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 \[ \frac{a^4 (d x)^{m+1}}{d (m+1)}+\frac{4 a^3 b (d x)^{m+3}}{d^3 (m+3)}+\frac{6 a^2 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac{4 a b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac{b^4 (d x)^{m+9}}{d^9 (m+9)} \]

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))

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Rubi in Sympy [A]  time = 39.2428, size = 94, normalized size = 0.9 \[ \frac{a^{4} \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{4 a^{3} b \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} + \frac{6 a^{2} b^{2} \left (d x\right )^{m + 5}}{d^{5} \left (m + 5\right )} + \frac{4 a b^{3} \left (d x\right )^{m + 7}}{d^{7} \left (m + 7\right )} + \frac{b^{4} \left (d x\right )^{m + 9}}{d^{9} \left (m + 9\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0469924, size = 73, normalized size = 0.7 \[ (d x)^m \left (\frac{a^4 x}{m+1}+\frac{4 a^3 b x^3}{m+3}+\frac{6 a^2 b^2 x^5}{m+5}+\frac{4 a b^3 x^7}{m+7}+\frac{b^4 x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 292, normalized size = 2.8 \[{\frac{ \left ( dx \right ) ^{m} \left ({b}^{4}{m}^{4}{x}^{8}+16\,{b}^{4}{m}^{3}{x}^{8}+4\,a{b}^{3}{m}^{4}{x}^{6}+86\,{b}^{4}{m}^{2}{x}^{8}+72\,a{b}^{3}{m}^{3}{x}^{6}+176\,{b}^{4}m{x}^{8}+6\,{a}^{2}{b}^{2}{m}^{4}{x}^{4}+416\,a{b}^{3}{m}^{2}{x}^{6}+105\,{b}^{4}{x}^{8}+120\,{a}^{2}{b}^{2}{m}^{3}{x}^{4}+888\,a{b}^{3}m{x}^{6}+4\,{a}^{3}b{m}^{4}{x}^{2}+780\,{a}^{2}{b}^{2}{m}^{2}{x}^{4}+540\,a{b}^{3}{x}^{6}+88\,{a}^{3}b{m}^{3}{x}^{2}+1800\,{a}^{2}{b}^{2}m{x}^{4}+{a}^{4}{m}^{4}+656\,{a}^{3}b{m}^{2}{x}^{2}+1134\,{a}^{2}{b}^{2}{x}^{4}+24\,{a}^{4}{m}^{3}+1832\,{a}^{3}bm{x}^{2}+206\,{a}^{4}{m}^{2}+1260\,{a}^{3}b{x}^{2}+744\,{a}^{4}m+945\,{a}^{4} \right ) x}{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

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+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)*x/(9+m)/(7+m)/(
5+m)/(3+m)/(1+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2*(d*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287238, size = 342, normalized size = 3.29 \[ \frac{{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \,{\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \,{\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \,{\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} +{\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\right )} x\right )} \left (d x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2*(d*x)^m,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 + 2
0*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 = 8.4627, size = 1321, normalized size = 12.7 \[ \text{result too large to display} \]

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 + 945) + 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 + 94
5) + 780*a**2*b**2*d**m*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1
689*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 + 2
5*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 + 1689*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 + 1689*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/XCAS [A]  time = 0.273163, size = 628, normalized size = 6.04 \[ \frac{b^{4} m^{4} x^{9} e^{\left (m{\rm ln}\left (d x\right )\right )} + 16 \, b^{4} m^{3} x^{9} e^{\left (m{\rm ln}\left (d x\right )\right )} + 4 \, a b^{3} m^{4} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 86 \, b^{4} m^{2} x^{9} e^{\left (m{\rm ln}\left (d x\right )\right )} + 72 \, a b^{3} m^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 176 \, b^{4} m x^{9} e^{\left (m{\rm ln}\left (d x\right )\right )} + 6 \, a^{2} b^{2} m^{4} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 416 \, a b^{3} m^{2} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 105 \, b^{4} x^{9} e^{\left (m{\rm ln}\left (d x\right )\right )} + 120 \, a^{2} b^{2} m^{3} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 888 \, a b^{3} m x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 4 \, a^{3} b m^{4} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 780 \, a^{2} b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 540 \, a b^{3} x^{7} e^{\left (m{\rm ln}\left (d x\right )\right )} + 88 \, a^{3} b m^{3} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 1800 \, a^{2} b^{2} m x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + a^{4} m^{4} x e^{\left (m{\rm ln}\left (d x\right )\right )} + 656 \, a^{3} b m^{2} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 1134 \, a^{2} b^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 24 \, a^{4} m^{3} x e^{\left (m{\rm ln}\left (d x\right )\right )} + 1832 \, a^{3} b m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 206 \, a^{4} m^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )} + 1260 \, a^{3} b x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 744 \, a^{4} m x e^{\left (m{\rm ln}\left (d x\right )\right )} + 945 \, a^{4} x e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2*(d*x)^m,x, algorithm="giac")

[Out]

(b^4*m^4*x^9*e^(m*ln(d*x)) + 16*b^4*m^3*x^9*e^(m*ln(d*x)) + 4*a*b^3*m^4*x^7*e^(m
*ln(d*x)) + 86*b^4*m^2*x^9*e^(m*ln(d*x)) + 72*a*b^3*m^3*x^7*e^(m*ln(d*x)) + 176*
b^4*m*x^9*e^(m*ln(d*x)) + 6*a^2*b^2*m^4*x^5*e^(m*ln(d*x)) + 416*a*b^3*m^2*x^7*e^
(m*ln(d*x)) + 105*b^4*x^9*e^(m*ln(d*x)) + 120*a^2*b^2*m^3*x^5*e^(m*ln(d*x)) + 88
8*a*b^3*m*x^7*e^(m*ln(d*x)) + 4*a^3*b*m^4*x^3*e^(m*ln(d*x)) + 780*a^2*b^2*m^2*x^
5*e^(m*ln(d*x)) + 540*a*b^3*x^7*e^(m*ln(d*x)) + 88*a^3*b*m^3*x^3*e^(m*ln(d*x)) +
1800*a^2*b^2*m*x^5*e^(m*ln(d*x)) + a^4*m^4*x*e^(m*ln(d*x)) + 656*a^3*b*m^2*x^3*
e^(m*ln(d*x)) + 1134*a^2*b^2*x^5*e^(m*ln(d*x)) + 24*a^4*m^3*x*e^(m*ln(d*x)) + 18
32*a^3*b*m*x^3*e^(m*ln(d*x)) + 206*a^4*m^2*x*e^(m*ln(d*x)) + 1260*a^3*b*x^3*e^(m
*ln(d*x)) + 744*a^4*m*x*e^(m*ln(d*x)) + 945*a^4*x*e^(m*ln(d*x)))/(m^5 + 25*m^4 +
230*m^3 + 950*m^2 + 1689*m + 945)