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

Optimal. Leaf size=79 \[ \frac{a^4 x^{m+1}}{m+1}+\frac{4 a^3 b x^{m+3}}{m+3}+\frac{6 a^2 b^2 x^{m+5}}{m+5}+\frac{4 a b^3 x^{m+7}}{m+7}+\frac{b^4 x^{m+9}}{m+9} \]

[Out]

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

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Rubi [A]  time = 0.0862479, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^4 x^{m+1}}{m+1}+\frac{4 a^3 b x^{m+3}}{m+3}+\frac{6 a^2 b^2 x^{m+5}}{m+5}+\frac{4 a b^3 x^{m+7}}{m+7}+\frac{b^4 x^{m+9}}{m+9} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^2)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.8013, size = 70, normalized size = 0.89 \[ \frac{a^{4} x^{m + 1}}{m + 1} + \frac{4 a^{3} b x^{m + 3}}{m + 3} + \frac{6 a^{2} b^{2} x^{m + 5}}{m + 5} + \frac{4 a b^{3} x^{m + 7}}{m + 7} + \frac{b^{4} x^{m + 9}}{m + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0421853, size = 71, normalized size = 0.9 \[ x^m \left (\frac{a^4 x}{m+1}+\frac{4 a^3 b x^3}{m+3}+\frac{6 a^2 b^2 x^5}{m+5}+\frac{4 a b^3 x^7}{m+7}+\frac{b^4 x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^2)^4,x]

[Out]

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 = 0.009, size = 291, normalized size = 3.7 \[{\frac{{x}^{1+m} \left ({b}^{4}{m}^{4}{x}^{8}+16\,{b}^{4}{m}^{3}{x}^{8}+4\,a{b}^{3}{m}^{4}{x}^{6}+86\,{b}^{4}{m}^{2}{x}^{8}+72\,a{b}^{3}{m}^{3}{x}^{6}+176\,{b}^{4}m{x}^{8}+6\,{a}^{2}{b}^{2}{m}^{4}{x}^{4}+416\,a{b}^{3}{m}^{2}{x}^{6}+105\,{b}^{4}{x}^{8}+120\,{a}^{2}{b}^{2}{m}^{3}{x}^{4}+888\,a{b}^{3}m{x}^{6}+4\,{a}^{3}b{m}^{4}{x}^{2}+780\,{a}^{2}{b}^{2}{m}^{2}{x}^{4}+540\,a{b}^{3}{x}^{6}+88\,{a}^{3}b{m}^{3}{x}^{2}+1800\,{a}^{2}{b}^{2}m{x}^{4}+{a}^{4}{m}^{4}+656\,{a}^{3}b{m}^{2}{x}^{2}+1134\,{a}^{2}{b}^{2}{x}^{4}+24\,{a}^{4}{m}^{3}+1832\,{a}^{3}bm{x}^{2}+206\,{a}^{4}{m}^{2}+1260\,{a}^{3}b{x}^{2}+744\,{a}^{4}m+945\,{a}^{4} \right ) }{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^2+a)^4,x)

[Out]

x^(1+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)/(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*x^2 + a)^4*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249321, size = 339, normalized size = 4.29 \[ \frac{{\left ({\left (b^{4} m^{4} + 16 \, b^{4} m^{3} + 86 \, b^{4} m^{2} + 176 \, b^{4} m + 105 \, b^{4}\right )} x^{9} + 4 \,{\left (a b^{3} m^{4} + 18 \, a b^{3} m^{3} + 104 \, a b^{3} m^{2} + 222 \, a b^{3} m + 135 \, a b^{3}\right )} x^{7} + 6 \,{\left (a^{2} b^{2} m^{4} + 20 \, a^{2} b^{2} m^{3} + 130 \, a^{2} b^{2} m^{2} + 300 \, a^{2} b^{2} m + 189 \, a^{2} b^{2}\right )} x^{5} + 4 \,{\left (a^{3} b m^{4} + 22 \, a^{3} b m^{3} + 164 \, a^{3} b m^{2} + 458 \, a^{3} b m + 315 \, a^{3} b\right )} x^{3} +{\left (a^{4} m^{4} + 24 \, a^{4} m^{3} + 206 \, a^{4} m^{2} + 744 \, a^{4} m + 945 \, a^{4}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^4*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)*x^m/(m^5 + 25*m^4 + 230*m^3 + 95
0*m^2 + 1689*m + 945)

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Sympy [A]  time = 8.15085, size = 1221, normalized size = 15.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**2+a)**4,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), 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, 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, 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, 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, E
q(m, -1)), (a**4*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 24*a**4*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
206*a**4*m**2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 74
4*a**4*m*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*a**4
*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 4*a**3*b*m**4*x*
*3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 88*a**3*b*m**3*x
**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 656*a**3*b*m**2
*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1832*a**3*b*m
*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1260*a**3*b*x
**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 6*a**2*b**2*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*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*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 18
00*a**2*b**2*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
1134*a**2*b**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 4*a*b**3*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 72*a*b**3*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945)
+ 416*a*b**3*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 888*a*b**3*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945
) + 540*a*b**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
b**4*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*
b**4*m**3*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*b
**4*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 176*b
**4*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 105*b**4
*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 = 0.214517, size = 560, normalized size = 7.09 \[ \frac{b^{4} m^{4} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 16 \, b^{4} m^{3} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, a b^{3} m^{4} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 86 \, b^{4} m^{2} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 72 \, a b^{3} m^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 176 \, b^{4} m x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, a^{2} b^{2} m^{4} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 416 \, a b^{3} m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 105 \, b^{4} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 120 \, a^{2} b^{2} m^{3} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 888 \, a b^{3} m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, a^{3} b m^{4} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 780 \, a^{2} b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 540 \, a b^{3} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 88 \, a^{3} b m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 1800 \, a^{2} b^{2} m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{4} m^{4} x e^{\left (m{\rm ln}\left (x\right )\right )} + 656 \, a^{3} b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 1134 \, a^{2} b^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a^{4} m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 1832 \, a^{3} b m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 206 \, a^{4} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 1260 \, a^{3} b x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 744 \, a^{4} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 945 \, a^{4} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^4*m^4*x^9*e^(m*ln(x)) + 16*b^4*m^3*x^9*e^(m*ln(x)) + 4*a*b^3*m^4*x^7*e^(m*ln(
x)) + 86*b^4*m^2*x^9*e^(m*ln(x)) + 72*a*b^3*m^3*x^7*e^(m*ln(x)) + 176*b^4*m*x^9*
e^(m*ln(x)) + 6*a^2*b^2*m^4*x^5*e^(m*ln(x)) + 416*a*b^3*m^2*x^7*e^(m*ln(x)) + 10
5*b^4*x^9*e^(m*ln(x)) + 120*a^2*b^2*m^3*x^5*e^(m*ln(x)) + 888*a*b^3*m*x^7*e^(m*l
n(x)) + 4*a^3*b*m^4*x^3*e^(m*ln(x)) + 780*a^2*b^2*m^2*x^5*e^(m*ln(x)) + 540*a*b^
3*x^7*e^(m*ln(x)) + 88*a^3*b*m^3*x^3*e^(m*ln(x)) + 1800*a^2*b^2*m*x^5*e^(m*ln(x)
) + a^4*m^4*x*e^(m*ln(x)) + 656*a^3*b*m^2*x^3*e^(m*ln(x)) + 1134*a^2*b^2*x^5*e^(
m*ln(x)) + 24*a^4*m^3*x*e^(m*ln(x)) + 1832*a^3*b*m*x^3*e^(m*ln(x)) + 206*a^4*m^2
*x*e^(m*ln(x)) + 1260*a^3*b*x^3*e^(m*ln(x)) + 744*a^4*m*x*e^(m*ln(x)) + 945*a^4*
x*e^(m*ln(x)))/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)