3.1018 \(\int x^{15} (a+b x^4)^{3/4} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^4\right )^{11/4}}{11 b^4}-\frac{a^3 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^4}-\frac{a \left (a+b x^4\right )^{15/4}}{5 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0465933, antiderivative size = 80, 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} \[ \frac{3 a^2 \left (a+b x^4\right )^{11/4}}{11 b^4}-\frac{a^3 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^4}-\frac{a \left (a+b x^4\right )^{15/4}}{5 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^15*(a + b*x^4)^(3/4),x]

[Out]

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

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

Mathematica [A]  time = 0.0244611, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^4\right )^{7/4} \left (224 a^2 b x^4-128 a^3-308 a b^2 x^8+385 b^3 x^{12}\right )}{7315 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(7/4)*(-128*a^3 + 224*a^2*b*x^4 - 308*a*b^2*x^8 + 385*b^3*x^12))/(7315*b^4)

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Maple [A]  time = 0.003, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-385\,{b}^{3}{x}^{12}+308\,a{b}^{2}{x}^{8}-224\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{7315\,{b}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(b*x^4+a)^(3/4),x)

[Out]

-1/7315*(b*x^4+a)^(7/4)*(-385*b^3*x^12+308*a*b^2*x^8-224*a^2*b*x^4+128*a^3)/b^4

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Maxima [A]  time = 0.982184, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{19}{4}}}{19 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}} a}{5 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2}}{11 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.70094, size = 139, normalized size = 1.74 \begin{align*} \frac{{\left (385 \, b^{4} x^{16} + 77 \, a b^{3} x^{12} - 84 \, a^{2} b^{2} x^{8} + 96 \, a^{3} b x^{4} - 128 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{7315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

1/7315*(385*b^4*x^16 + 77*a*b^3*x^12 - 84*a^2*b^2*x^8 + 96*a^3*b*x^4 - 128*a^4)*(b*x^4 + a)^(3/4)/b^4

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Sympy [A]  time = 26.498, size = 110, normalized size = 1.38 \begin{align*} \begin{cases} - \frac{128 a^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{4}} + \frac{96 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{3}} - \frac{12 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1045 b^{2}} + \frac{a x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{95 b} + \frac{x^{16} \left (a + b x^{4}\right )^{\frac{3}{4}}}{19} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} x^{16}}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15*(b*x**4+a)**(3/4),x)

[Out]

Piecewise((-128*a**4*(a + b*x**4)**(3/4)/(7315*b**4) + 96*a**3*x**4*(a + b*x**4)**(3/4)/(7315*b**3) - 12*a**2*
x**8*(a + b*x**4)**(3/4)/(1045*b**2) + a*x**12*(a + b*x**4)**(3/4)/(95*b) + x**16*(a + b*x**4)**(3/4)/19, Ne(b
, 0)), (a**(3/4)*x**16/16, True))

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Giac [A]  time = 1.13621, size = 77, normalized size = 0.96 \begin{align*} \frac{385 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} - 1463 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a + 1995 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2} - 1045 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7315 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

1/7315*(385*(b*x^4 + a)^(19/4) - 1463*(b*x^4 + a)^(15/4)*a + 1995*(b*x^4 + a)^(11/4)*a^2 - 1045*(b*x^4 + a)^(7
/4)*a^3)/b^4