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

Optimal. Leaf size=101 \[ \frac{6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac{a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^5}-\frac{4 a \left (a+b x^4\right )^{17/4}}{17 b^5} \]

[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)

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Rubi [A]  time = 0.0587497, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac{a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^5}-\frac{4 a \left (a+b x^4\right )^{17/4}}{17 b^5} \]

Antiderivative was successfully verified.

[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 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int x^{19} \sqrt [4]{a+b x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^4 \sqrt [4]{a+b x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{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]  time = 0.0301539, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{5/4} \left (2880 a^2 b^2 x^8-2560 a^3 b x^4+2048 a^4-3120 a b^3 x^{12}+3315 b^4 x^{16}\right )}{69615 b^5} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.007, size = 58, normalized size = 0.6 \begin{align*}{\frac{3315\,{x}^{16}{b}^{4}-3120\,a{x}^{12}{b}^{3}+2880\,{a}^{2}{x}^{8}{b}^{2}-2560\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{69615\,{b}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 0.967559, size = 109, normalized size = 1.08 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.42265, size = 173, normalized size = 1.71 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 27.3521, size = 134, normalized size = 1.33 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.15191, size = 96, normalized size = 0.95 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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