3.632 \(\int x^m (a+b x^4)^3 \, dx\)

Optimal. Leaf size=61 \[ \frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+5}}{m+5}+\frac {3 a b^2 x^{m+9}}{m+9}+\frac {b^3 x^{m+13}}{m+13} \]

[Out]

a^3*x^(1+m)/(1+m)+3*a^2*b*x^(5+m)/(5+m)+3*a*b^2*x^(9+m)/(9+m)+b^3*x^(13+m)/(13+m)

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Rubi [A]  time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {3 a^2 b x^{m+5}}{m+5}+\frac {a^3 x^{m+1}}{m+1}+\frac {3 a b^2 x^{m+9}}{m+9}+\frac {b^3 x^{m+13}}{m+13} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^4)^3,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(5 + m))/(5 + m) + (3*a*b^2*x^(9 + m))/(9 + m) + (b^3*x^(13 + m))/(13 + m
)

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 x^m \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{4+m}+3 a b^2 x^{8+m}+b^3 x^{12+m}\right ) \, dx\\ &=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{5+m}}{5+m}+\frac {3 a b^2 x^{9+m}}{9+m}+\frac {b^3 x^{13+m}}{13+m}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 56, normalized size = 0.92 \[ x^{m+1} \left (\frac {a^3}{m+1}+\frac {3 a^2 b x^4}{m+5}+\frac {3 a b^2 x^8}{m+9}+\frac {b^3 x^{12}}{m+13}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^4)^3,x]

[Out]

x^(1 + m)*(a^3/(1 + m) + (3*a^2*b*x^4)/(5 + m) + (3*a*b^2*x^8)/(9 + m) + (b^3*x^12)/(13 + m))

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fricas [B]  time = 0.58, size = 157, normalized size = 2.57 \[ \frac {{\left ({\left (b^{3} m^{3} + 15 \, b^{3} m^{2} + 59 \, b^{3} m + 45 \, b^{3}\right )} x^{13} + 3 \, {\left (a b^{2} m^{3} + 19 \, a b^{2} m^{2} + 83 \, a b^{2} m + 65 \, a b^{2}\right )} x^{9} + 3 \, {\left (a^{2} b m^{3} + 23 \, a^{2} b m^{2} + 139 \, a^{2} b m + 117 \, a^{2} b\right )} x^{5} + {\left (a^{3} m^{3} + 27 \, a^{3} m^{2} + 227 \, a^{3} m + 585 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a)^3,x, algorithm="fricas")

[Out]

((b^3*m^3 + 15*b^3*m^2 + 59*b^3*m + 45*b^3)*x^13 + 3*(a*b^2*m^3 + 19*a*b^2*m^2 + 83*a*b^2*m + 65*a*b^2)*x^9 +
3*(a^2*b*m^3 + 23*a^2*b*m^2 + 139*a^2*b*m + 117*a^2*b)*x^5 + (a^3*m^3 + 27*a^3*m^2 + 227*a^3*m + 585*a^3)*x)*x
^m/(m^4 + 28*m^3 + 254*m^2 + 812*m + 585)

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giac [B]  time = 0.18, size = 224, normalized size = 3.67 \[ \frac {b^{3} m^{3} x^{13} x^{m} + 15 \, b^{3} m^{2} x^{13} x^{m} + 59 \, b^{3} m x^{13} x^{m} + 45 \, b^{3} x^{13} x^{m} + 3 \, a b^{2} m^{3} x^{9} x^{m} + 57 \, a b^{2} m^{2} x^{9} x^{m} + 249 \, a b^{2} m x^{9} x^{m} + 195 \, a b^{2} x^{9} x^{m} + 3 \, a^{2} b m^{3} x^{5} x^{m} + 69 \, a^{2} b m^{2} x^{5} x^{m} + 417 \, a^{2} b m x^{5} x^{m} + 351 \, a^{2} b x^{5} x^{m} + a^{3} m^{3} x x^{m} + 27 \, a^{3} m^{2} x x^{m} + 227 \, a^{3} m x x^{m} + 585 \, a^{3} x x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a)^3,x, algorithm="giac")

[Out]

(b^3*m^3*x^13*x^m + 15*b^3*m^2*x^13*x^m + 59*b^3*m*x^13*x^m + 45*b^3*x^13*x^m + 3*a*b^2*m^3*x^9*x^m + 57*a*b^2
*m^2*x^9*x^m + 249*a*b^2*m*x^9*x^m + 195*a*b^2*x^9*x^m + 3*a^2*b*m^3*x^5*x^m + 69*a^2*b*m^2*x^5*x^m + 417*a^2*
b*m*x^5*x^m + 351*a^2*b*x^5*x^m + a^3*m^3*x*x^m + 27*a^3*m^2*x*x^m + 227*a^3*m*x*x^m + 585*a^3*x*x^m)/(m^4 + 2
8*m^3 + 254*m^2 + 812*m + 585)

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maple [B]  time = 0.01, size = 178, normalized size = 2.92 \[ \frac {\left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 b^{3} m \,x^{12}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 a \,b^{2} m \,x^{8}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 a^{2} b m \,x^{4}+351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 a^{3} m +585 a^{3}\right ) x^{m +1}}{\left (m +13\right ) \left (m +9\right ) \left (m +5\right ) \left (m +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^4+a)^3,x)

[Out]

x^(m+1)*(b^3*m^3*x^12+15*b^3*m^2*x^12+59*b^3*m*x^12+45*b^3*x^12+3*a*b^2*m^3*x^8+57*a*b^2*m^2*x^8+249*a*b^2*m*x
^8+195*a*b^2*x^8+3*a^2*b*m^3*x^4+69*a^2*b*m^2*x^4+417*a^2*b*m*x^4+351*a^2*b*x^4+a^3*m^3+27*a^3*m^2+227*a^3*m+5
85*a^3)/(m+13)/(m+9)/(m+5)/(m+1)

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maxima [A]  time = 1.34, size = 61, normalized size = 1.00 \[ \frac {b^{3} x^{m + 13}}{m + 13} + \frac {3 \, a b^{2} x^{m + 9}}{m + 9} + \frac {3 \, a^{2} b x^{m + 5}}{m + 5} + \frac {a^{3} x^{m + 1}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a)^3,x, algorithm="maxima")

[Out]

b^3*x^(m + 13)/(m + 13) + 3*a*b^2*x^(m + 9)/(m + 9) + 3*a^2*b*x^(m + 5)/(m + 5) + a^3*x^(m + 1)/(m + 1)

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mupad [B]  time = 1.33, size = 167, normalized size = 2.74 \[ x^m\,\left (\frac {a^3\,x\,\left (m^3+27\,m^2+227\,m+585\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {b^3\,x^{13}\,\left (m^3+15\,m^2+59\,m+45\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a\,b^2\,x^9\,\left (m^3+19\,m^2+83\,m+65\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a^2\,b\,x^5\,\left (m^3+23\,m^2+139\,m+117\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^4)^3,x)

[Out]

x^m*((a^3*x*(227*m + 27*m^2 + m^3 + 585))/(812*m + 254*m^2 + 28*m^3 + m^4 + 585) + (b^3*x^13*(59*m + 15*m^2 +
m^3 + 45))/(812*m + 254*m^2 + 28*m^3 + m^4 + 585) + (3*a*b^2*x^9*(83*m + 19*m^2 + m^3 + 65))/(812*m + 254*m^2
+ 28*m^3 + m^4 + 585) + (3*a^2*b*x^5*(139*m + 23*m^2 + m^3 + 117))/(812*m + 254*m^2 + 28*m^3 + m^4 + 585))

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sympy [A]  time = 7.36, size = 683, normalized size = 11.20 \[ \begin {cases} - \frac {a^{3}}{12 x^{12}} - \frac {3 a^{2} b}{8 x^{8}} - \frac {3 a b^{2}}{4 x^{4}} + b^{3} \log {\relax (x )} & \text {for}\: m = -13 \\- \frac {a^{3}}{8 x^{8}} - \frac {3 a^{2} b}{4 x^{4}} + 3 a b^{2} \log {\relax (x )} + \frac {b^{3} x^{4}}{4} & \text {for}\: m = -9 \\- \frac {a^{3}}{4 x^{4}} + 3 a^{2} b \log {\relax (x )} + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{8}}{8} & \text {for}\: m = -5 \\a^{3} \log {\relax (x )} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{8}}{8} + \frac {b^{3} x^{12}}{12} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {27 a^{3} m^{2} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {227 a^{3} m x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {585 a^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {3 a^{2} b m^{3} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {69 a^{2} b m^{2} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {417 a^{2} b m x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {351 a^{2} b x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {3 a b^{2} m^{3} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {57 a b^{2} m^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {249 a b^{2} m x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {195 a b^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {b^{3} m^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {15 b^{3} m^{2} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {59 b^{3} m x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac {45 b^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**4+a)**3,x)

[Out]

Piecewise((-a**3/(12*x**12) - 3*a**2*b/(8*x**8) - 3*a*b**2/(4*x**4) + b**3*log(x), Eq(m, -13)), (-a**3/(8*x**8
) - 3*a**2*b/(4*x**4) + 3*a*b**2*log(x) + b**3*x**4/4, Eq(m, -9)), (-a**3/(4*x**4) + 3*a**2*b*log(x) + 3*a*b**
2*x**4/4 + b**3*x**8/8, Eq(m, -5)), (a**3*log(x) + 3*a**2*b*x**4/4 + 3*a*b**2*x**8/8 + b**3*x**12/12, Eq(m, -1
)), (a**3*m**3*x*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 27*a**3*m**2*x*x**m/(m**4 + 28*m**3 + 254*m*
*2 + 812*m + 585) + 227*a**3*m*x*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 585*a**3*x*x**m/(m**4 + 28*m
**3 + 254*m**2 + 812*m + 585) + 3*a**2*b*m**3*x**5*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 69*a**2*b*
m**2*x**5*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 417*a**2*b*m*x**5*x**m/(m**4 + 28*m**3 + 254*m**2 +
812*m + 585) + 351*a**2*b*x**5*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 3*a*b**2*m**3*x**9*x**m/(m**4
+ 28*m**3 + 254*m**2 + 812*m + 585) + 57*a*b**2*m**2*x**9*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 24
9*a*b**2*m*x**9*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 195*a*b**2*x**9*x**m/(m**4 + 28*m**3 + 254*m*
*2 + 812*m + 585) + b**3*m**3*x**13*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 15*b**3*m**2*x**13*x**m/(
m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 59*b**3*m*x**13*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 45
*b**3*x**13*x**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585), True))

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