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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.05, size = 61, normalized size = 0.60 \[ \frac {\left (a+b x^4\right )^{3/4} \left (2048 a^4-1536 a^3 b x^4+1344 a^2 b^2 x^8-1232 a b^3 x^{12}+1155 b^4 x^{16}\right )}{21945 b^5} \]

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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fricas [A]  time = 0.96, size = 57, normalized size = 0.56 \[ \frac {{\left (1155 \, b^{4} x^{16} - 1232 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1536 \, a^{3} b x^{4} + 2048 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21945 \, b^{5}} \]

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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giac [A]  time = 0.16, size = 71, normalized size = 0.70 \[ \frac {1155 \, {\left (b x^{4} + a\right )}^{\frac {19}{4}} - 5852 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a + 11970 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{2} - 12540 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3} + 7315 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{4}}{21945 \, b^{5}} \]

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

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maple [A]  time = 0.01, size = 58, normalized size = 0.57 \[ \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (1155 x^{16} b^{4}-1232 a \,x^{12} b^{3}+1344 a^{2} x^{8} b^{2}-1536 a^{3} x^{4} b +2048 a^{4}\right )}{21945 b^{5}} \]

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 = 1.38, size = 81, normalized size = 0.80 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {19}{4}}}{19 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {15}{4}} a}{15 \, b^{5}} + \frac {6 \, {\left (b x^{4} + a\right )}^{\frac {11}{4}} a^{2}}{11 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} a^{3}}{7 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} a^{4}}{3 \, b^{5}} \]

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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mupad [B]  time = 1.16, size = 58, normalized size = 0.57 \[ {\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {2048\,a^4}{21945\,b^5}+\frac {x^{16}}{19\,b}-\frac {16\,a\,x^{12}}{285\,b^2}-\frac {512\,a^3\,x^4}{7315\,b^4}+\frac {64\,a^2\,x^8}{1045\,b^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(a + b*x^4)^(1/4),x)

[Out]

(a + b*x^4)^(3/4)*((2048*a^4)/(21945*b^5) + x^16/(19*b) - (16*a*x^12)/(285*b^2) - (512*a^3*x^4)/(7315*b^4) + (
64*a^2*x^8)/(1045*b^3))

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sympy [A]  time = 24.29, size = 116, normalized size = 1.15 \[ \begin {cases} \frac {2048 a^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{21945 b^{5}} - \frac {512 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{7315 b^{4}} + \frac {64 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac {3}{4}}}{1045 b^{3}} - \frac {16 a x^{12} \left (a + b x^{4}\right )^{\frac {3}{4}}}{285 b^{2}} + \frac {x^{16} \left (a + b x^{4}\right )^{\frac {3}{4}}}{19 b} & \text {for}\: b \neq 0 \\\frac {x^{20}}{20 \sqrt [4]{a}} & \text {otherwise} \end {cases} \]

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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