3.1080 \(\int \frac{x^{19}}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=101 \[ \frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]

[Out]

(a^4*(a + b*x^4)^(3/4))/(3*b^5) - (4*a^3*(a + b*x^4)^(7/4))/(7*b^5) + (6*a^2*(a + b*x^4)^(11/4))/(11*b^5) - (4
*a*(a + b*x^4)^(15/4))/(15*b^5) + (a + b*x^4)^(19/4)/(19*b^5)

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Rubi [A]  time = 0.0534013, 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} \[ \frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^19/(a + b*x^4)^(1/4),x]

[Out]

(a^4*(a + b*x^4)^(3/4))/(3*b^5) - (4*a^3*(a + b*x^4)^(7/4))/(7*b^5) + (6*a^2*(a + b*x^4)^(11/4))/(11*b^5) - (4
*a*(a + b*x^4)^(15/4))/(15*b^5) + (a + b*x^4)^(19/4)/(19*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 \frac{x^{19}}{\sqrt [4]{a+b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^4}{\sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 \sqrt [4]{a+b x}}-\frac{4 a^3 (a+b x)^{3/4}}{b^4}+\frac{6 a^2 (a+b x)^{7/4}}{b^4}-\frac{4 a (a+b x)^{11/4}}{b^4}+\frac{(a+b x)^{15/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0291817, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{3/4} \left (1344 a^2 b^2 x^8-1536 a^3 b x^4+2048 a^4-1232 a b^3 x^{12}+1155 b^4 x^{16}\right )}{21945 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^19/(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(3/4)*(2048*a^4 - 1536*a^3*b*x^4 + 1344*a^2*b^2*x^8 - 1232*a*b^3*x^12 + 1155*b^4*x^16))/(21945*b^
5)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.6 \begin{align*}{\frac{1155\,{x}^{16}{b}^{4}-1232\,a{x}^{12}{b}^{3}+1344\,{a}^{2}{x}^{8}{b}^{2}-1536\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{21945\,{b}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{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]

1/21945*(b*x^4+a)^(3/4)*(1155*b^4*x^16-1232*a*b^3*x^12+1344*a^2*b^2*x^8-1536*a^3*b*x^4+2048*a^4)/b^5

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Maxima [A]  time = 1.02766, size = 109, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{19}{4}}}{19 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a}{15 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2}}{11 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{4}}{3 \, 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]

1/19*(b*x^4 + a)^(19/4)/b^5 - 4/15*(b*x^4 + a)^(15/4)*a/b^5 + 6/11*(b*x^4 + a)^(11/4)*a^2/b^5 - 4/7*(b*x^4 + a
)^(7/4)*a^3/b^5 + 1/3*(b*x^4 + a)^(3/4)*a^4/b^5

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Fricas [A]  time = 1.42504, size = 151, normalized size = 1.5 \begin{align*} \frac{{\left (1155 \, b^{4} x^{16} - 1232 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1536 \, a^{3} b x^{4} + 2048 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21945 \, 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]

1/21945*(1155*b^4*x^16 - 1232*a*b^3*x^12 + 1344*a^2*b^2*x^8 - 1536*a^3*b*x^4 + 2048*a^4)*(b*x^4 + a)^(3/4)/b^5

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Sympy [A]  time = 21.0584, size = 116, normalized size = 1.15 \begin{align*} \begin{cases} \frac{2048 a^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{21945 b^{5}} - \frac{512 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{4}} + \frac{64 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1045 b^{3}} - \frac{16 a x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{285 b^{2}} + \frac{x^{16} \left (a + b x^{4}\right )^{\frac{3}{4}}}{19 b} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 \sqrt [4]{a}} & \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**4*(a + b*x**4)**(3/4)/(21945*b**5) - 512*a**3*x**4*(a + b*x**4)**(3/4)/(7315*b**4) + 64*a**
2*x**8*(a + b*x**4)**(3/4)/(1045*b**3) - 16*a*x**12*(a + b*x**4)**(3/4)/(285*b**2) + x**16*(a + b*x**4)**(3/4)
/(19*b), Ne(b, 0)), (x**20/(20*a**(1/4)), True))

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Giac [A]  time = 1.59247, size = 96, normalized size = 0.95 \begin{align*} \frac{1155 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} - 5852 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a + 11970 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2} - 12540 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3} + 7315 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{4}}{21945 \, 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]

1/21945*(1155*(b*x^4 + a)^(19/4) - 5852*(b*x^4 + a)^(15/4)*a + 11970*(b*x^4 + a)^(11/4)*a^2 - 12540*(b*x^4 + a
)^(7/4)*a^3 + 7315*(b*x^4 + a)^(3/4)*a^4)/b^5