Integrand size = 16, antiderivative size = 106 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=-\frac {a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac {4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac {6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}+\frac {4 a \left (a-b x^4\right )^{17/4}}{17 b^5}-\frac {\left (a-b x^4\right )^{21/4}}{21 b^5} \] Output:
-1/5*a^4*(-b*x^4+a)^(5/4)/b^5+4/9*a^3*(-b*x^4+a)^(9/4)/b^5-6/13*a^2*(-b*x^ 4+a)^(13/4)/b^5+4/17*a*(-b*x^4+a)^(17/4)/b^5-1/21*(-b*x^4+a)^(21/4)/b^5
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.69 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=\frac {\sqrt [4]{a-b x^4} \left (-2048 a^5-512 a^4 b x^4-320 a^3 b^2 x^8-240 a^2 b^3 x^{12}-195 a b^4 x^{16}+3315 b^5 x^{20}\right )}{69615 b^5} \] Input:
Integrate[x^19*(a - b*x^4)^(1/4),x]
Output:
((a - b*x^4)^(1/4)*(-2048*a^5 - 512*a^4*b*x^4 - 320*a^3*b^2*x^8 - 240*a^2* b^3*x^12 - 195*a*b^4*x^16 + 3315*b^5*x^20))/(69615*b^5)
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {798, 53, 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.
\(\displaystyle \int x^{19} \sqrt [4]{a-b x^4} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int x^{16} \sqrt [4]{a-b x^4}dx^4\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {1}{4} \int \left (\frac {\left (a-b x^4\right )^{17/4}}{b^4}-\frac {4 a \left (a-b x^4\right )^{13/4}}{b^4}+\frac {6 a^2 \left (a-b x^4\right )^{9/4}}{b^4}-\frac {4 a^3 \left (a-b x^4\right )^{5/4}}{b^4}+\frac {a^4 \sqrt [4]{a-b x^4}}{b^4}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {4 a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac {16 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac {24 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}-\frac {4 \left (a-b x^4\right )^{21/4}}{21 b^5}+\frac {16 a \left (a-b x^4\right )^{17/4}}{17 b^5}\right )\) |
Input:
Int[x^19*(a - b*x^4)^(1/4),x]
Output:
((-4*a^4*(a - b*x^4)^(5/4))/(5*b^5) + (16*a^3*(a - b*x^4)^(9/4))/(9*b^5) - (24*a^2*(a - b*x^4)^(13/4))/(13*b^5) + (16*a*(a - b*x^4)^(17/4))/(17*b^5) - (4*(a - b*x^4)^(21/4))/(21*b^5))/4
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.50 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}+3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}+2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\) | \(59\) |
pseudoelliptic | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}+3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}+2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\) | \(59\) |
orering | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}+3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}+2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\) | \(59\) |
trager | \(-\frac {\left (-3315 b^{5} x^{20}+195 a \,b^{4} x^{16}+240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}+512 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{69615 b^{5}}\) | \(70\) |
risch | \(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} {\left (\left (-b \,x^{4}+a \right )^{3}\right )}^{\frac {1}{4}} \left (-3315 b^{5} x^{20}+195 a \,b^{4} x^{16}+240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}+512 a^{4} b \,x^{4}+2048 a^{5}\right )}{69615 b^{5} {\left (-\left (b \,x^{4}-a \right )^{3}\right )}^{\frac {1}{4}}}\) | \(97\) |
Input:
int(x^19*(-b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)
Output:
-1/69615*(-b*x^4+a)^(5/4)*(3315*b^4*x^16+3120*a*b^3*x^12+2880*a^2*b^2*x^8+ 2560*a^3*b*x^4+2048*a^4)/b^5
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=\frac {{\left (3315 \, b^{5} x^{20} - 195 \, a b^{4} x^{16} - 240 \, a^{2} b^{3} x^{12} - 320 \, a^{3} b^{2} x^{8} - 512 \, a^{4} b x^{4} - 2048 \, a^{5}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{69615 \, b^{5}} \] Input:
integrate(x^19*(-b*x^4+a)^(1/4),x, algorithm="fricas")
Output:
1/69615*(3315*b^5*x^20 - 195*a*b^4*x^16 - 240*a^2*b^3*x^12 - 320*a^3*b^2*x ^8 - 512*a^4*b*x^4 - 2048*a^5)*(-b*x^4 + a)^(1/4)/b^5
Time = 0.92 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.26 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=\begin {cases} - \frac {2048 a^{5} \sqrt [4]{a - b x^{4}}}{69615 b^{5}} - \frac {512 a^{4} x^{4} \sqrt [4]{a - b x^{4}}}{69615 b^{4}} - \frac {64 a^{3} x^{8} \sqrt [4]{a - b x^{4}}}{13923 b^{3}} - \frac {16 a^{2} x^{12} \sqrt [4]{a - b x^{4}}}{4641 b^{2}} - \frac {a x^{16} \sqrt [4]{a - b x^{4}}}{357 b} + \frac {x^{20} \sqrt [4]{a - b x^{4}}}{21} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{20}}{20} & \text {otherwise} \end {cases} \] Input:
integrate(x**19*(-b*x**4+a)**(1/4),x)
Output:
Piecewise((-2048*a**5*(a - b*x**4)**(1/4)/(69615*b**5) - 512*a**4*x**4*(a - b*x**4)**(1/4)/(69615*b**4) - 64*a**3*x**8*(a - b*x**4)**(1/4)/(13923*b* *3) - 16*a**2*x**12*(a - b*x**4)**(1/4)/(4641*b**2) - a*x**16*(a - b*x**4) **(1/4)/(357*b) + x**20*(a - b*x**4)**(1/4)/21, Ne(b, 0)), (a**(1/4)*x**20 /20, True))
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=-\frac {{\left (-b x^{4} + a\right )}^{\frac {21}{4}}}{21 \, b^{5}} + \frac {4 \, {\left (-b x^{4} + a\right )}^{\frac {17}{4}} a}{17 \, b^{5}} - \frac {6 \, {\left (-b x^{4} + a\right )}^{\frac {13}{4}} a^{2}}{13 \, b^{5}} + \frac {4 \, {\left (-b x^{4} + a\right )}^{\frac {9}{4}} a^{3}}{9 \, b^{5}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{5 \, b^{5}} \] Input:
integrate(x^19*(-b*x^4+a)^(1/4),x, algorithm="maxima")
Output:
-1/21*(-b*x^4 + a)^(21/4)/b^5 + 4/17*(-b*x^4 + a)^(17/4)*a/b^5 - 6/13*(-b* x^4 + a)^(13/4)*a^2/b^5 + 4/9*(-b*x^4 + a)^(9/4)*a^3/b^5 - 1/5*(-b*x^4 + a )^(5/4)*a^4/b^5
Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=\frac {3315 \, {\left (b x^{4} - a\right )}^{5} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} + 16380 \, {\left (b x^{4} - a\right )}^{4} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a + 32130 \, {\left (b x^{4} - a\right )}^{3} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2} + 30940 \, {\left (b x^{4} - a\right )}^{2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{3} - 13923 \, {\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{69615 \, b^{5}} \] Input:
integrate(x^19*(-b*x^4+a)^(1/4),x, algorithm="giac")
Output:
1/69615*(3315*(b*x^4 - a)^5*(-b*x^4 + a)^(1/4) + 16380*(b*x^4 - a)^4*(-b*x ^4 + a)^(1/4)*a + 32130*(b*x^4 - a)^3*(-b*x^4 + a)^(1/4)*a^2 + 30940*(b*x^ 4 - a)^2*(-b*x^4 + a)^(1/4)*a^3 - 13923*(-b*x^4 + a)^(5/4)*a^4)/b^5
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=-{\left (a-b\,x^4\right )}^{1/4}\,\left (\frac {2048\,a^5}{69615\,b^5}-\frac {x^{20}}{21}+\frac {a\,x^{16}}{357\,b}+\frac {512\,a^4\,x^4}{69615\,b^4}+\frac {64\,a^3\,x^8}{13923\,b^3}+\frac {16\,a^2\,x^{12}}{4641\,b^2}\right ) \] Input:
int(x^19*(a - b*x^4)^(1/4),x)
Output:
-(a - b*x^4)^(1/4)*((2048*a^5)/(69615*b^5) - x^20/21 + (a*x^16)/(357*b) + (512*a^4*x^4)/(69615*b^4) + (64*a^3*x^8)/(13923*b^3) + (16*a^2*x^12)/(4641 *b^2))
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65 \[ \int x^{19} \sqrt [4]{a-b x^4} \, dx=\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (3315 b^{5} x^{20}-195 a \,b^{4} x^{16}-240 a^{2} b^{3} x^{12}-320 a^{3} b^{2} x^{8}-512 a^{4} b \,x^{4}-2048 a^{5}\right )}{69615 b^{5}} \] Input:
int(x^19*(-b*x^4+a)^(1/4),x)
Output:
((a - b*x**4)**(1/4)*( - 2048*a**5 - 512*a**4*b*x**4 - 320*a**3*b**2*x**8 - 240*a**2*b**3*x**12 - 195*a*b**4*x**16 + 3315*b**5*x**20))/(69615*b**5)