Integrand size = 15, antiderivative size = 81 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx=\frac {a^3 (c x)^{1+m}}{c (1+m)}+\frac {3 a^2 b (c x)^{5+m}}{c^5 (5+m)}+\frac {3 a b^2 (c x)^{9+m}}{c^9 (9+m)}+\frac {b^3 (c x)^{13+m}}{c^{13} (13+m)} \] Output:
a^3*(c*x)^(1+m)/c/(1+m)+3*a^2*b*(c*x)^(5+m)/c^5/(5+m)+3*a*b^2*(c*x)^(9+m)/ c^9/(9+m)+b^3*(c*x)^(13+m)/c^13/(13+m)
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.70 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx=x (c x)^m \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^4}{5+m}+\frac {3 a b^2 x^8}{9+m}+\frac {b^3 x^{12}}{13+m}\right ) \] Input:
Integrate[(c*x)^m*(a + b*x^4)^3,x]
Output:
x*(c*x)^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))
Time = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {802, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+b x^4\right )^3 (c x)^m \, dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \int \left (a^3 (c x)^m+\frac {3 a^2 b (c x)^{m+4}}{c^4}+\frac {3 a b^2 (c x)^{m+8}}{c^8}+\frac {b^3 (c x)^{m+12}}{c^{12}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 (c x)^{m+1}}{c (m+1)}+\frac {3 a^2 b (c x)^{m+5}}{c^5 (m+5)}+\frac {3 a b^2 (c x)^{m+9}}{c^9 (m+9)}+\frac {b^3 (c x)^{m+13}}{c^{13} (m+13)}\) |
Input:
Int[(c*x)^m*(a + b*x^4)^3,x]
Output:
(a^3*(c*x)^(1 + m))/(c*(1 + m)) + (3*a^2*b*(c*x)^(5 + m))/(c^5*(5 + m)) + (3*a*b^2*(c*x)^(9 + m))/(c^9*(9 + m)) + (b^3*(c*x)^(13 + m))/(c^13*(13 + m ))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(81)=162\).
Time = 0.58 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.21
method | result | size |
gosper | \(\frac {x \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right ) \left (c x \right )^{m}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(179\) |
risch | \(\frac {x \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right ) \left (c x \right )^{m}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(179\) |
orering | \(\frac {x \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 m \,x^{12} b^{3}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 m \,x^{8} a \,b^{2}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 m \,x^{4} a^{2} b +351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 m \,a^{3}+585 a^{3}\right ) \left (c x \right )^{m}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(179\) |
parallelrisch | \(\frac {x^{13} \left (c x \right )^{m} b^{3} m^{3}+15 x^{13} \left (c x \right )^{m} b^{3} m^{2}+59 x^{13} \left (c x \right )^{m} b^{3} m +45 x^{13} \left (c x \right )^{m} b^{3}+3 x^{9} \left (c x \right )^{m} a \,b^{2} m^{3}+57 x^{9} \left (c x \right )^{m} a \,b^{2} m^{2}+249 x^{9} \left (c x \right )^{m} a \,b^{2} m +195 x^{9} \left (c x \right )^{m} a \,b^{2}+3 x^{5} \left (c x \right )^{m} a^{2} b \,m^{3}+69 x^{5} \left (c x \right )^{m} a^{2} b \,m^{2}+417 x^{5} \left (c x \right )^{m} a^{2} b m +351 x^{5} \left (c x \right )^{m} a^{2} b +x \left (c x \right )^{m} a^{3} m^{3}+27 x \left (c x \right )^{m} a^{3} m^{2}+227 x \left (c x \right )^{m} a^{3} m +585 x \left (c x \right )^{m} a^{3}}{\left (13+m \right ) \left (9+m \right ) \left (5+m \right ) \left (1+m \right )}\) | \(257\) |
Input:
int((c*x)^m*(b*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
x*(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)* (c*x)^m/(13+m)/(9+m)/(5+m)/(1+m)
Time = 0.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.96 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx=\frac {{\left ({\left (b^{3} m^{3} + 15 \, b^{3} m^{2} + 59 \, b^{3} m + 45 \, b^{3}\right )} x^{13} + 3 \, {\left (a b^{2} m^{3} + 19 \, a b^{2} m^{2} + 83 \, a b^{2} m + 65 \, a b^{2}\right )} x^{9} + 3 \, {\left (a^{2} b m^{3} + 23 \, a^{2} b m^{2} + 139 \, a^{2} b m + 117 \, a^{2} b\right )} x^{5} + {\left (a^{3} m^{3} + 27 \, a^{3} m^{2} + 227 \, a^{3} m + 585 \, a^{3}\right )} x\right )} \left (c x\right )^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \] Input:
integrate((c*x)^m*(b*x^4+a)^3,x, algorithm="fricas")
Output:
((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)*(c* x)^m/(m^4 + 28*m^3 + 254*m^2 + 812*m + 585)
Leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (71) = 142\).
Time = 0.75 (sec) , antiderivative size = 722, normalized size of antiderivative = 8.91 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((c*x)**m*(b*x**4+a)**3,x)
Output:
Piecewise(((-a**3/(12*x**12) - 3*a**2*b/(8*x**8) - 3*a*b**2/(4*x**4) + b** 3*log(x))/c**13, 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)/c**9, Eq(m, -9)), ((-a**3/(4*x**4) + 3*a**2*b*lo g(x) + 3*a*b**2*x**4/4 + b**3*x**8/8)/c**5, 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)/c, Eq(m, -1)), (a**3*m**3 *x*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 27*a**3*m**2*x*(c* x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 227*a**3*m*x*(c*x)**m/(m **4 + 28*m**3 + 254*m**2 + 812*m + 585) + 585*a**3*x*(c*x)**m/(m**4 + 28*m **3 + 254*m**2 + 812*m + 585) + 3*a**2*b*m**3*x**5*(c*x)**m/(m**4 + 28*m** 3 + 254*m**2 + 812*m + 585) + 69*a**2*b*m**2*x**5*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 417*a**2*b*m*x**5*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 351*a**2*b*x**5*(c*x)**m/(m**4 + 28*m**3 + 254*m **2 + 812*m + 585) + 3*a*b**2*m**3*x**9*(c*x)**m/(m**4 + 28*m**3 + 254*m** 2 + 812*m + 585) + 57*a*b**2*m**2*x**9*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 249*a*b**2*m*x**9*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 195*a*b**2*x**9*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + b**3*m**3*x**13*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 58 5) + 15*b**3*m**2*x**13*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 59*b**3*m*x**13*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585) + 45 *b**3*x**13*(c*x)**m/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585), True))
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx=\frac {b^{3} c^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, a b^{2} c^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, a^{2} b c^{m} x^{5} x^{m}}{m + 5} + \frac {\left (c x\right )^{m + 1} a^{3}}{c {\left (m + 1\right )}} \] Input:
integrate((c*x)^m*(b*x^4+a)^3,x, algorithm="maxima")
Output:
b^3*c^m*x^13*x^m/(m + 13) + 3*a*b^2*c^m*x^9*x^m/(m + 9) + 3*a^2*b*c^m*x^5* x^m/(m + 5) + (c*x)^(m + 1)*a^3/(c*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (81) = 162\).
Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 3.16 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx=\frac {\left (c x\right )^{m} b^{3} m^{3} x^{13} + 15 \, \left (c x\right )^{m} b^{3} m^{2} x^{13} + 59 \, \left (c x\right )^{m} b^{3} m x^{13} + 45 \, \left (c x\right )^{m} b^{3} x^{13} + 3 \, \left (c x\right )^{m} a b^{2} m^{3} x^{9} + 57 \, \left (c x\right )^{m} a b^{2} m^{2} x^{9} + 249 \, \left (c x\right )^{m} a b^{2} m x^{9} + 195 \, \left (c x\right )^{m} a b^{2} x^{9} + 3 \, \left (c x\right )^{m} a^{2} b m^{3} x^{5} + 69 \, \left (c x\right )^{m} a^{2} b m^{2} x^{5} + 417 \, \left (c x\right )^{m} a^{2} b m x^{5} + 351 \, \left (c x\right )^{m} a^{2} b x^{5} + \left (c x\right )^{m} a^{3} m^{3} x + 27 \, \left (c x\right )^{m} a^{3} m^{2} x + 227 \, \left (c x\right )^{m} a^{3} m x + 585 \, \left (c x\right )^{m} a^{3} x}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \] Input:
integrate((c*x)^m*(b*x^4+a)^3,x, algorithm="giac")
Output:
((c*x)^m*b^3*m^3*x^13 + 15*(c*x)^m*b^3*m^2*x^13 + 59*(c*x)^m*b^3*m*x^13 + 45*(c*x)^m*b^3*x^13 + 3*(c*x)^m*a*b^2*m^3*x^9 + 57*(c*x)^m*a*b^2*m^2*x^9 + 249*(c*x)^m*a*b^2*m*x^9 + 195*(c*x)^m*a*b^2*x^9 + 3*(c*x)^m*a^2*b*m^3*x^5 + 69*(c*x)^m*a^2*b*m^2*x^5 + 417*(c*x)^m*a^2*b*m*x^5 + 351*(c*x)^m*a^2*b* x^5 + (c*x)^m*a^3*m^3*x + 27*(c*x)^m*a^3*m^2*x + 227*(c*x)^m*a^3*m*x + 585 *(c*x)^m*a^3*x)/(m^4 + 28*m^3 + 254*m^2 + 812*m + 585)
Time = 0.36 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.09 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx={\left (c\,x\right )}^m\,\left (\frac {a^3\,x\,\left (m^3+27\,m^2+227\,m+585\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {b^3\,x^{13}\,\left (m^3+15\,m^2+59\,m+45\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a\,b^2\,x^9\,\left (m^3+19\,m^2+83\,m+65\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}+\frac {3\,a^2\,b\,x^5\,\left (m^3+23\,m^2+139\,m+117\right )}{m^4+28\,m^3+254\,m^2+812\,m+585}\right ) \] Input:
int((c*x)^m*(a + b*x^4)^3,x)
Output:
(c*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))
Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.21 \[ \int (c x)^m \left (a+b x^4\right )^3 \, dx=\frac {x^{m} c^{m} x \left (b^{3} m^{3} x^{12}+15 b^{3} m^{2} x^{12}+59 b^{3} m \,x^{12}+45 b^{3} x^{12}+3 a \,b^{2} m^{3} x^{8}+57 a \,b^{2} m^{2} x^{8}+249 a \,b^{2} m \,x^{8}+195 a \,b^{2} x^{8}+3 a^{2} b \,m^{3} x^{4}+69 a^{2} b \,m^{2} x^{4}+417 a^{2} b m \,x^{4}+351 a^{2} b \,x^{4}+a^{3} m^{3}+27 a^{3} m^{2}+227 a^{3} m +585 a^{3}\right )}{m^{4}+28 m^{3}+254 m^{2}+812 m +585} \] Input:
int((c*x)^m*(b*x^4+a)^3,x)
Output:
(x**m*c**m*x*(a**3*m**3 + 27*a**3*m**2 + 227*a**3*m + 585*a**3 + 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 + 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 + b**3*m**3*x**12 + 15*b**3*m**2*x**12 + 59*b**3*m*x**12 + 45*b**3*x**1 2))/(m**4 + 28*m**3 + 254*m**2 + 812*m + 585)