3.1237 \(\int \frac{x^7}{(a-b x^4)^{3/4}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a-b x^4\right )^{5/4}}{5 b^2}-\frac{a \sqrt [4]{a-b x^4}}{b^2} \]

[Out]

-((a*(a - b*x^4)^(1/4))/b^2) + (a - b*x^4)^(5/4)/(5*b^2)

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Rubi [A]  time = 0.0228438, antiderivative size = 38, 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} \[ \frac{\left (a-b x^4\right )^{5/4}}{5 b^2}-\frac{a \sqrt [4]{a-b x^4}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a - b*x^4)^(3/4),x]

[Out]

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

Mathematica [A]  time = 0.0128191, size = 28, normalized size = 0.74 \[ -\frac{\sqrt [4]{a-b x^4} \left (4 a+b x^4\right )}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a - b*x^4)^(3/4),x]

[Out]

-((a - b*x^4)^(1/4)*(4*a + b*x^4))/(5*b^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{b{x}^{4}+4\,a}{5\,{b}^{2}}\sqrt [4]{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(-b*x^4+a)^(1/4)*(b*x^4+4*a)/b^2

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Maxima [A]  time = 1.15496, size = 43, normalized size = 1.13 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b^{2}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

1/5*(-b*x^4 + a)^(5/4)/b^2 - (-b*x^4 + a)^(1/4)*a/b^2

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Fricas [A]  time = 1.65492, size = 58, normalized size = 1.53 \begin{align*} -\frac{{\left (b x^{4} + 4 \, a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{5 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

-1/5*(b*x^4 + 4*a)*(-b*x^4 + a)^(1/4)/b^2

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Sympy [A]  time = 1.52441, size = 46, normalized size = 1.21 \begin{align*} \begin{cases} - \frac{4 a \sqrt [4]{a - b x^{4}}}{5 b^{2}} - \frac{x^{4} \sqrt [4]{a - b x^{4}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((-4*a*(a - b*x**4)**(1/4)/(5*b**2) - x**4*(a - b*x**4)**(1/4)/(5*b), Ne(b, 0)), (x**8/(8*a**(3/4)),
True))

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Giac [A]  time = 1.18654, size = 39, normalized size = 1.03 \begin{align*} \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} - 5 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}{5 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

1/5*((-b*x^4 + a)^(5/4) - 5*(-b*x^4 + a)^(1/4)*a)/b^2