\(\int x^7 \sqrt [4]{a+b x^4} \, dx\) [991]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=-\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2}+\frac {\left (a+b x^4\right )^{9/4}}{9 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac {a \left (a+b x^4\right )^{5/4}}{5 b^2} \]

[In]

Int[x^7*(a + b*x^4)^(1/4),x]

[Out]

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

Rule 45

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])

Rule 272

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int x \sqrt [4]{a+b x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {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] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {\sqrt [4]{a+b x^4} \left (-4 a^2+a b x^4+5 b^2 x^8\right )}{45 b^2} \]

[In]

Integrate[x^7*(a + b*x^4)^(1/4),x]

[Out]

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

Maple [A] (verified)

Time = 4.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-5 b \,x^{4}+4 a \right )}{45 b^{2}}\)	\(25\)
pseudoelliptic	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (-5 b \,x^{4}+4 a \right )}{45 b^{2}}\)	\(25\)
trager	\(-\frac {\left (-5 b^{2} x^{8}-a b \,x^{4}+4 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{45 b^{2}}\)	\(36\)
risch	\(-\frac {\left (-5 b^{2} x^{8}-a b \,x^{4}+4 a^{2}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{45 b^{2}}\)	\(36\)

[In]

int(x^7*(b*x^4+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {{\left (5 \, b^{2} x^{8} + a b x^{4} - 4 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, b^{2}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.66 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\begin {cases} - \frac {4 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac {a x^{4} \sqrt [4]{a + b x^{4}}}{45 b} + \frac {x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text {for}\: b \neq 0 \\\frac {\sqrt [4]{a} x^{8}}{8} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}}}{9 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}} a}{5 \, b^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx=\frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} - 9 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a}{45 \, b^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 5.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int x^7 \sqrt [4]{a+b x^4} \, dx={\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^8}{9}-\frac {4\,a^2}{45\,b^2}+\frac {a\,x^4}{45\,b}\right ) \]

[In]

int(x^7*(a + b*x^4)^(1/4),x)

[Out]

(a + b*x^4)^(1/4)*(x^8/9 - (4*a^2)/(45*b^2) + (a*x^4)/(45*b))