3.1207 \(\int \frac{x^7}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a-b x^4\right )^{7/4}}{7 b^2}-\frac{a \left (a-b x^4\right )^{3/4}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0225616, antiderivative size = 40, 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} \[ \frac{\left (a-b x^4\right )^{7/4}}{7 b^2}-\frac{a \left (a-b x^4\right )^{3/4}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a - b*x^4)^(1/4),x]

[Out]

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

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

Mathematica [A]  time = 0.0123611, size = 29, normalized size = 0.72 \[ -\frac{\left (a-b x^4\right )^{3/4} \left (4 a+3 b x^4\right )}{21 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a - b*x^4)^(1/4),x]

[Out]

-((a - b*x^4)^(3/4)*(4*a + 3*b*x^4))/(21*b^2)

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Maple [A]  time = 0.004, size = 26, normalized size = 0.7 \begin{align*} -{\frac{3\,b{x}^{4}+4\,a}{21\,{b}^{2}} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-b*x^4+a)^(1/4),x)

[Out]

-1/21*(-b*x^4+a)^(3/4)*(3*b*x^4+4*a)/b^2

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Maxima [A]  time = 0.984554, size = 43, normalized size = 1.08 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, b^{2}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.41513, size = 62, normalized size = 1.55 \begin{align*} -\frac{{\left (3 \, b x^{4} + 4 \, a\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{21 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/21*(3*b*x^4 + 4*a)*(-b*x^4 + a)^(3/4)/b^2

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Sympy [A]  time = 1.3192, size = 46, normalized size = 1.15 \begin{align*} \begin{cases} - \frac{4 a \left (a - b x^{4}\right )^{\frac{3}{4}}}{21 b^{2}} - \frac{x^{4} \left (a - b x^{4}\right )^{\frac{3}{4}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt [4]{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-4*a*(a - b*x**4)**(3/4)/(21*b**2) - x**4*(a - b*x**4)**(3/4)/(7*b), Ne(b, 0)), (x**8/(8*a**(1/4)),
True))

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Giac [A]  time = 1.16048, size = 42, normalized size = 1.05 \begin{align*} \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{7}{4}} - 7 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} a}{21 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

1/21*(3*(-b*x^4 + a)^(7/4) - 7*(-b*x^4 + a)^(3/4)*a)/b^2