3.1017 \(\int x^{19} (a+b x^4)^{3/4} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 a^2 \left (a+b x^4\right )^{15/4}}{5 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{a^4 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{\left (a+b x^4\right )^{23/4}}{23 b^5}-\frac{4 a \left (a+b x^4\right )^{19/4}}{19 b^5} \]

[Out]

(a^4*(a + b*x^4)^(7/4))/(7*b^5) - (4*a^3*(a + b*x^4)^(11/4))/(11*b^5) + (2*a^2*(a + b*x^4)^(15/4))/(5*b^5) - (
4*a*(a + b*x^4)^(19/4))/(19*b^5) + (a + b*x^4)^(23/4)/(23*b^5)

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Rubi [A]  time = 0.0569821, 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} \[ \frac{2 a^2 \left (a+b x^4\right )^{15/4}}{5 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{a^4 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{\left (a+b x^4\right )^{23/4}}{23 b^5}-\frac{4 a \left (a+b x^4\right )^{19/4}}{19 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^19*(a + b*x^4)^(3/4),x]

[Out]

(a^4*(a + b*x^4)^(7/4))/(7*b^5) - (4*a^3*(a + b*x^4)^(11/4))/(11*b^5) + (2*a^2*(a + b*x^4)^(15/4))/(5*b^5) - (
4*a*(a + b*x^4)^(19/4))/(19*b^5) + (a + b*x^4)^(23/4)/(23*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} \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^4 (a+b x)^{3/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4 (a+b x)^{3/4}}{b^4}-\frac{4 a^3 (a+b x)^{7/4}}{b^4}+\frac{6 a^2 (a+b x)^{11/4}}{b^4}-\frac{4 a (a+b x)^{15/4}}{b^4}+\frac{(a+b x)^{19/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac{a^4 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{2 a^2 \left (a+b x^4\right )^{15/4}}{5 b^5}-\frac{4 a \left (a+b x^4\right )^{19/4}}{19 b^5}+\frac{\left (a+b x^4\right )^{23/4}}{23 b^5}\\ \end{align*}

Mathematica [A]  time = 0.0308585, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{7/4} \left (4928 a^2 b^2 x^8-3584 a^3 b x^4+2048 a^4-6160 a b^3 x^{12}+7315 b^4 x^{16}\right )}{168245 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^19*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(7/4)*(2048*a^4 - 3584*a^3*b*x^4 + 4928*a^2*b^2*x^8 - 6160*a*b^3*x^12 + 7315*b^4*x^16))/(168245*b
^5)

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Maple [A]  time = 0.007, size = 58, normalized size = 0.6 \begin{align*}{\frac{7315\,{x}^{16}{b}^{4}-6160\,a{x}^{12}{b}^{3}+4928\,{a}^{2}{x}^{8}{b}^{2}-3584\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{168245\,{b}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168245*(b*x^4+a)^(7/4)*(7315*b^4*x^16-6160*a*b^3*x^12+4928*a^2*b^2*x^8-3584*a^3*b*x^4+2048*a^4)/b^5

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Maxima [A]  time = 0.95501, size = 109, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{23}{4}}}{23 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} a}{19 \, b^{5}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a^{2}}{5 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{3}}{11 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{4}}{7 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23*(b*x^4 + a)^(23/4)/b^5 - 4/19*(b*x^4 + a)^(19/4)*a/b^5 + 2/5*(b*x^4 + a)^(15/4)*a^2/b^5 - 4/11*(b*x^4 + a
)^(11/4)*a^3/b^5 + 1/7*(b*x^4 + a)^(7/4)*a^4/b^5

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Fricas [A]  time = 1.70675, size = 180, normalized size = 1.78 \begin{align*} \frac{{\left (7315 \, b^{5} x^{20} + 1155 \, a b^{4} x^{16} - 1232 \, a^{2} b^{3} x^{12} + 1344 \, a^{3} b^{2} x^{8} - 1536 \, a^{4} b x^{4} + 2048 \, a^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{168245 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168245*(7315*b^5*x^20 + 1155*a*b^4*x^16 - 1232*a^2*b^3*x^12 + 1344*a^3*b^2*x^8 - 1536*a^4*b*x^4 + 2048*a^5)*
(b*x^4 + a)^(3/4)/b^5

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Sympy [A]  time = 47.7124, size = 136, normalized size = 1.35 \begin{align*} \begin{cases} \frac{2048 a^{5} \left (a + b x^{4}\right )^{\frac{3}{4}}}{168245 b^{5}} - \frac{1536 a^{4} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{168245 b^{4}} + \frac{192 a^{3} x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{24035 b^{3}} - \frac{16 a^{2} x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{2185 b^{2}} + \frac{3 a x^{16} \left (a + b x^{4}\right )^{\frac{3}{4}}}{437 b} + \frac{x^{20} \left (a + b x^{4}\right )^{\frac{3}{4}}}{23} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} 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)**(3/4),x)

[Out]

Piecewise((2048*a**5*(a + b*x**4)**(3/4)/(168245*b**5) - 1536*a**4*x**4*(a + b*x**4)**(3/4)/(168245*b**4) + 19
2*a**3*x**8*(a + b*x**4)**(3/4)/(24035*b**3) - 16*a**2*x**12*(a + b*x**4)**(3/4)/(2185*b**2) + 3*a*x**16*(a +
b*x**4)**(3/4)/(437*b) + x**20*(a + b*x**4)**(3/4)/23, Ne(b, 0)), (a**(3/4)*x**20/20, True))

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Giac [A]  time = 1.12623, size = 96, normalized size = 0.95 \begin{align*} \frac{7315 \,{\left (b x^{4} + a\right )}^{\frac{23}{4}} - 35420 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} a + 67298 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a^{2} - 61180 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{3} + 24035 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{4}}{168245 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168245*(7315*(b*x^4 + a)^(23/4) - 35420*(b*x^4 + a)^(19/4)*a + 67298*(b*x^4 + a)^(15/4)*a^2 - 61180*(b*x^4 +
a)^(11/4)*a^3 + 24035*(b*x^4 + a)^(7/4)*a^4)/b^5