3.991 \(\int x^7 \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2} \]

[Out]

-(a*(a + b*x^4)^(5/4))/(5*b^2) + (a + b*x^4)^(9/4)/(9*b^2)

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Rubi [A]  time = 0.0225679, antiderivative size = 38, 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} \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^4)^(1/4),x]

[Out]

-(a*(a + b*x^4)^(5/4))/(5*b^2) + (a + b*x^4)^(9/4)/(9*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 x^7 \sqrt [4]{a+b x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x \sqrt [4]{a+b x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt [4]{a+b x}}{b}+\frac{(a+b x)^{5/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2}+\frac{\left (a+b x^4\right )^{9/4}}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0136701, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^4\right )^{5/4} \left (5 b x^4-4 a\right )}{45 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(5/4)*(-4*a + 5*b*x^4))/(45*b^2)

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Maple [A]  time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-5\,b{x}^{4}+4\,a}{45\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{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]

-1/45*(b*x^4+a)^(5/4)*(-5*b*x^4+4*a)/b^2

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Maxima [A]  time = 0.949801, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{5}{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)^(1/4),x, algorithm="maxima")

[Out]

1/9*(b*x^4 + a)^(9/4)/b^2 - 1/5*(b*x^4 + a)^(5/4)*a/b^2

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Fricas [A]  time = 1.42787, size = 78, normalized size = 2.05 \begin{align*} \frac{{\left (5 \, b^{2} x^{8} + a b x^{4} - 4 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, 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]

1/45*(5*b^2*x^8 + a*b*x^4 - 4*a^2)*(b*x^4 + a)^(1/4)/b^2

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Sympy [A]  time = 2.2081, size = 63, normalized size = 1.66 \begin{align*} \begin{cases} - \frac{4 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac{a x^{4} \sqrt [4]{a + b x^{4}}}{45 b} + \frac{x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{8}}{8} & \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**2*(a + b*x**4)**(1/4)/(45*b**2) + a*x**4*(a + b*x**4)**(1/4)/(45*b) + x**8*(a + b*x**4)**(1/4
)/9, Ne(b, 0)), (a**(1/4)*x**8/8, True))

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Giac [A]  time = 1.13719, size = 39, normalized size = 1.03 \begin{align*} \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} - 9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a}{45 \, 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]

1/45*(5*(b*x^4 + a)^(9/4) - 9*(b*x^4 + a)^(5/4)*a)/b^2