\(\int x^{19} \sqrt [4]{a+b x^4} \, dx\) [988]

Optimal result

Integrand size = 15, antiderivative size = 101 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\frac {a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}-\frac {4 a \left (a+b x^4\right )^{17/4}}{17 b^5}+\frac {\left (a+b x^4\right )^{21/4}}{21 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\frac {a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {\left (a+b x^4\right )^{21/4}}{21 b^5}-\frac {4 a \left (a+b x^4\right )^{17/4}}{17 b^5} \]

[In]

Int[x^19*(a + b*x^4)^(1/4),x]

[Out]

(a^4*(a + b*x^4)^(5/4))/(5*b^5) - (4*a^3*(a + b*x^4)^(9/4))/(9*b^5) + (6*a^2*(a + b*x^4)^(13/4))/(13*b^5) - (4
*a*(a + b*x^4)^(17/4))/(17*b^5) + (a + b*x^4)^(21/4)/(21*b^5)

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int x^4 \sqrt [4]{a+b x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (\frac {a^4 \sqrt [4]{a+b x}}{b^4}-\frac {4 a^3 (a+b x)^{5/4}}{b^4}+\frac {6 a^2 (a+b x)^{9/4}}{b^4}-\frac {4 a (a+b x)^{13/4}}{b^4}+\frac {(a+b x)^{17/4}}{b^4}\right ) \, dx,x,x^4\right ) \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.60 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\frac {\left (a+b x^4\right )^{5/4} \left (2048 a^4-2560 a^3 b x^4+2880 a^2 b^2 x^8-3120 a b^3 x^{12}+3315 b^4 x^{16}\right )}{69615 b^5} \]

[In]

Integrate[x^19*(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(5/4)*(2048*a^4 - 2560*a^3*b*x^4 + 2880*a^2*b^2*x^8 - 3120*a*b^3*x^12 + 3315*b^4*x^16))/(69615*b^
5)

Maple [A] (verified)

Time = 4.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57

method	result	size
gosper	\(\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}-3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}-2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\)	\(58\)
pseudoelliptic	\(\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}-3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}-2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\)	\(58\)
trager	\(\frac {\left (3315 b^{5} x^{20}+195 a \,b^{4} x^{16}-240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}-512 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{69615 b^{5}}\)	\(69\)
risch	\(\frac {\left (3315 b^{5} x^{20}+195 a \,b^{4} x^{16}-240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}-512 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{69615 b^{5}}\)	\(69\)

[In]

int(x^19*(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\frac {{\left (3315 \, b^{5} x^{20} + 195 \, a b^{4} x^{16} - 240 \, a^{2} b^{3} x^{12} + 320 \, a^{3} b^{2} x^{8} - 512 \, a^{4} b x^{4} + 2048 \, a^{5}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{69615 \, b^{5}} \]

[In]

integrate(x^19*(b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.96 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\begin {cases} \frac {2048 a^{5} \sqrt [4]{a + b x^{4}}}{69615 b^{5}} - \frac {512 a^{4} x^{4} \sqrt [4]{a + b x^{4}}}{69615 b^{4}} + \frac {64 a^{3} x^{8} \sqrt [4]{a + b x^{4}}}{13923 b^{3}} - \frac {16 a^{2} x^{12} \sqrt [4]{a + b x^{4}}}{4641 b^{2}} + \frac {a x^{16} \sqrt [4]{a + b x^{4}}}{357 b} + \frac {x^{20} \sqrt [4]{a + b x^{4}}}{21} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{20}}{20} & \text {otherwise} \end {cases} \]

[In]

integrate(x**19*(b*x**4+a)**(1/4),x)

[Out]

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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\frac {{\left (b x^{4} + a\right )}^{\frac {21}{4}}}{21 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a}{17 \, b^{5}} + \frac {6 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{2}}{13 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3}}{9 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{5 \, b^{5}} \]

[In]

integrate(x^19*(b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.70 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx=\frac {3315 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} - 16380 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a + 32130 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{2} - 30940 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3} + 13923 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{69615 \, b^{5}} \]

[In]

integrate(x^19*(b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

1/69615*(3315*(b*x^4 + a)^(21/4) - 16380*(b*x^4 + a)^(17/4)*a + 32130*(b*x^4 + a)^(13/4)*a^2 - 30940*(b*x^4 +
a)^(9/4)*a^3 + 13923*(b*x^4 + a)^(5/4)*a^4)/b^5

Mupad [B] (verification not implemented)

Time = 5.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65 \[ \int x^{19} \sqrt [4]{a+b x^4} \, dx={\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^{20}}{21}+\frac {2048\,a^5}{69615\,b^5}+\frac {a\,x^{16}}{357\,b}-\frac {512\,a^4\,x^4}{69615\,b^4}+\frac {64\,a^3\,x^8}{13923\,b^3}-\frac {16\,a^2\,x^{12}}{4641\,b^2}\right ) \]

[In]

int(x^19*(a + b*x^4)^(1/4),x)

[Out]

(a + b*x^4)^(1/4)*(x^20/21 + (2048*a^5)/(69615*b^5) + (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))