3.1205 \(\int \frac{x^{15}}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=84 \[ \frac{3 a^2 \left (a-b x^4\right )^{7/4}}{7 b^4}-\frac{a^3 \left (a-b x^4\right )^{3/4}}{3 b^4}+\frac{\left (a-b x^4\right )^{15/4}}{15 b^4}-\frac{3 a \left (a-b x^4\right )^{11/4}}{11 b^4} \]

[Out]

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

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Rubi [A]  time = 0.0460536, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {266, 43} \[ \frac{3 a^2 \left (a-b x^4\right )^{7/4}}{7 b^4}-\frac{a^3 \left (a-b x^4\right )^{3/4}}{3 b^4}+\frac{\left (a-b x^4\right )^{15/4}}{15 b^4}-\frac{3 a \left (a-b x^4\right )^{11/4}}{11 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^15/(a - b*x^4)^(1/4),x]

[Out]

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

Mathematica [A]  time = 0.0252194, size = 51, normalized size = 0.61 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (96 a^2 b x^4+128 a^3+84 a b^2 x^8+77 b^3 x^{12}\right )}{1155 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15/(a - b*x^4)^(1/4),x]

[Out]

-((a - b*x^4)^(3/4)*(128*a^3 + 96*a^2*b*x^4 + 84*a*b^2*x^8 + 77*b^3*x^12))/(1155*b^4)

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Maple [A]  time = 0.007, size = 48, normalized size = 0.6 \begin{align*} -{\frac{77\,{b}^{3}{x}^{12}+84\,a{b}^{2}{x}^{8}+96\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{1155\,{b}^{4}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(-b*x^4+a)^(1/4),x)

[Out]

-1/1155*(-b*x^4+a)^(3/4)*(77*b^3*x^12+84*a*b^2*x^8+96*a^2*b*x^4+128*a^3)/b^4

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Maxima [A]  time = 0.951553, size = 92, normalized size = 1.1 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{15}{4}}}{15 \, b^{4}} - \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{11}{4}} a}{11 \, b^{4}} + \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{7 \, b^{4}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a^{3}}{3 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46485, size = 116, normalized size = 1.38 \begin{align*} -\frac{{\left (77 \, b^{3} x^{12} + 84 \, a b^{2} x^{8} + 96 \, a^{2} b x^{4} + 128 \, a^{3}\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/1155*(77*b^3*x^12 + 84*a*b^2*x^8 + 96*a^2*b*x^4 + 128*a^3)*(-b*x^4 + a)^(3/4)/b^4

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Sympy [A]  time = 8.46692, size = 94, normalized size = 1.12 \begin{align*} \begin{cases} - \frac{128 a^{3} \left (a - b x^{4}\right )^{\frac{3}{4}}}{1155 b^{4}} - \frac{32 a^{2} x^{4} \left (a - b x^{4}\right )^{\frac{3}{4}}}{385 b^{3}} - \frac{4 a x^{8} \left (a - b x^{4}\right )^{\frac{3}{4}}}{55 b^{2}} - \frac{x^{12} \left (a - b x^{4}\right )^{\frac{3}{4}}}{15 b} & \text{for}\: b \neq 0 \\\frac{x^{16}}{16 \sqrt [4]{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-128*a**3*(a - b*x**4)**(3/4)/(1155*b**4) - 32*a**2*x**4*(a - b*x**4)**(3/4)/(385*b**3) - 4*a*x**8*
(a - b*x**4)**(3/4)/(55*b**2) - x**12*(a - b*x**4)**(3/4)/(15*b), Ne(b, 0)), (x**16/(16*a**(1/4)), True))

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Giac [A]  time = 1.13507, size = 112, normalized size = 1.33 \begin{align*} -\frac{77 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{3}{4}} + 315 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a - 495 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} a^{2} + 385 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a^{3}}{1155 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

-1/1155*(77*(b*x^4 - a)^3*(-b*x^4 + a)^(3/4) + 315*(b*x^4 - a)^2*(-b*x^4 + a)^(3/4)*a - 495*(-b*x^4 + a)^(7/4)
*a^2 + 385*(-b*x^4 + a)^(3/4)*a^3)/b^4