3.12.10 \(\int \frac {x^{19}}{(a+b x^4)^{3/4}} \, dx\) [1110]

Optimal. Leaf size=98 \[ \frac {a^4 \sqrt [4]{a+b x^4}}{b^5}-\frac {4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac {2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}-\frac {4 a \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {\left (a+b x^4\right )^{17/4}}{17 b^5} \]

[Out]

a^4*(b*x^4+a)^(1/4)/b^5-4/5*a^3*(b*x^4+a)^(5/4)/b^5+2/3*a^2*(b*x^4+a)^(9/4)/b^5-4/13*a*(b*x^4+a)^(13/4)/b^5+1/
17*(b*x^4+a)^(17/4)/b^5

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Rubi [A]
time = 0.04, antiderivative size = 98, 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 = {272, 45} \begin {gather*} \frac {a^4 \sqrt [4]{a+b x^4}}{b^5}-\frac {4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac {2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}+\frac {\left (a+b x^4\right )^{17/4}}{17 b^5}-\frac {4 a \left (a+b x^4\right )^{13/4}}{13 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^19/(a + b*x^4)^(3/4),x]

[Out]

(a^4*(a + b*x^4)^(1/4))/b^5 - (4*a^3*(a + b*x^4)^(5/4))/(5*b^5) + (2*a^2*(a + b*x^4)^(9/4))/(3*b^5) - (4*a*(a
+ b*x^4)^(13/4))/(13*b^5) + (a + b*x^4)^(17/4)/(17*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*} \int \frac {x^{19}}{\left (a+b x^4\right )^{3/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^4}{(a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^4}{b^4 (a+b x)^{3/4}}-\frac {4 a^3 \sqrt [4]{a+b x}}{b^4}+\frac {6 a^2 (a+b x)^{5/4}}{b^4}-\frac {4 a (a+b x)^{9/4}}{b^4}+\frac {(a+b x)^{13/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac {a^4 \sqrt [4]{a+b x^4}}{b^5}-\frac {4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac {2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}-\frac {4 a \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {\left (a+b x^4\right )^{17/4}}{17 b^5}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.62 \begin {gather*} \frac {\sqrt [4]{a+b x^4} \left (2048 a^4-512 a^3 b x^4+320 a^2 b^2 x^8-240 a b^3 x^{12}+195 b^4 x^{16}\right )}{3315 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^19/(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(1/4)*(2048*a^4 - 512*a^3*b*x^4 + 320*a^2*b^2*x^8 - 240*a*b^3*x^12 + 195*b^4*x^16))/(3315*b^5)

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Maple [A]
time = 0.15, size = 58, normalized size = 0.59

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

1/3315*(b*x^4+a)^(1/4)*(195*b^4*x^16-240*a*b^3*x^12+320*a^2*b^2*x^8-512*a^3*b*x^4+2048*a^4)/b^5

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Maxima [A]
time = 0.30, size = 80, normalized size = 0.82 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {17}{4}}}{17 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a}{13 \, b^{5}} + \frac {2 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{2}}{3 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{3}}{5 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{4}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

1/17*(b*x^4 + a)^(17/4)/b^5 - 4/13*(b*x^4 + a)^(13/4)*a/b^5 + 2/3*(b*x^4 + a)^(9/4)*a^2/b^5 - 4/5*(b*x^4 + a)^
(5/4)*a^3/b^5 + (b*x^4 + a)^(1/4)*a^4/b^5

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Fricas [A]
time = 0.37, size = 57, normalized size = 0.58 \begin {gather*} \frac {{\left (195 \, b^{4} x^{16} - 240 \, a b^{3} x^{12} + 320 \, a^{2} b^{2} x^{8} - 512 \, a^{3} b x^{4} + 2048 \, a^{4}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{3315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

1/3315*(195*b^4*x^16 - 240*a*b^3*x^12 + 320*a^2*b^2*x^8 - 512*a^3*b*x^4 + 2048*a^4)*(b*x^4 + a)^(1/4)/b^5

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Sympy [A]
time = 1.17, size = 116, normalized size = 1.18 \begin {gather*} \begin {cases} \frac {2048 a^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{5}} - \frac {512 a^{3} x^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{4}} + \frac {64 a^{2} x^{8} \sqrt [4]{a + b x^{4}}}{663 b^{3}} - \frac {16 a x^{12} \sqrt [4]{a + b x^{4}}}{221 b^{2}} + \frac {x^{16} \sqrt [4]{a + b x^{4}}}{17 b} & \text {for}\: b \neq 0 \\\frac {x^{20}}{20 a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**19/(b*x**4+a)**(3/4),x)

[Out]

Piecewise((2048*a**4*(a + b*x**4)**(1/4)/(3315*b**5) - 512*a**3*x**4*(a + b*x**4)**(1/4)/(3315*b**4) + 64*a**2
*x**8*(a + b*x**4)**(1/4)/(663*b**3) - 16*a*x**12*(a + b*x**4)**(1/4)/(221*b**2) + x**16*(a + b*x**4)**(1/4)/(
17*b), Ne(b, 0)), (x**20/(20*a**(3/4)), True))

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Giac [A]
time = 1.07, size = 74, normalized size = 0.76 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{4}}{b^{5}} + \frac {195 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} - 1020 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a + 2210 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{2} - 2652 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{3}}{3315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^19/(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

(b*x^4 + a)^(1/4)*a^4/b^5 + 1/3315*(195*(b*x^4 + a)^(17/4) - 1020*(b*x^4 + a)^(13/4)*a + 2210*(b*x^4 + a)^(9/4
)*a^2 - 2652*(b*x^4 + a)^(5/4)*a^3)/b^5

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Mupad [B]
time = 1.17, size = 58, normalized size = 0.59 \begin {gather*} {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {2048\,a^4}{3315\,b^5}+\frac {x^{16}}{17\,b}-\frac {16\,a\,x^{12}}{221\,b^2}-\frac {512\,a^3\,x^4}{3315\,b^4}+\frac {64\,a^2\,x^8}{663\,b^3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^19/(a + b*x^4)^(3/4),x)

[Out]

(a + b*x^4)^(1/4)*((2048*a^4)/(3315*b^5) + x^16/(17*b) - (16*a*x^12)/(221*b^2) - (512*a^3*x^4)/(3315*b^4) + (6
4*a^2*x^8)/(663*b^3))

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