\(\int x^7 \sqrt {a+c x^4} \, dx\) [765]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int x^7 \sqrt {a+c x^4} \, dx=-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2}+\frac {\left (a+c x^4\right )^{5/2}}{10 c^2} \]

[Out]

-1/6*a*(c*x^4+a)^(3/2)/c^2+1/10*(c*x^4+a)^(5/2)/c^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} \[ \int x^7 \sqrt {a+c x^4} \, dx=\frac {\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2} \]

[In]

Int[x^7*Sqrt[a + c*x^4],x]

[Out]

-1/6*(a*(a + c*x^4)^(3/2))/c^2 + (a + c*x^4)^(5/2)/(10*c^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 {a+c x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\frac {a \sqrt {a+c x}}{c}+\frac {(a+c x)^{3/2}}{c}\right ) \, dx,x,x^4\right ) \\ & = -\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2}+\frac {\left (a+c x^4\right )^{5/2}}{10 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int x^7 \sqrt {a+c x^4} \, dx=\frac {\sqrt {a+c x^4} \left (-2 a^2+a c x^4+3 c^2 x^8\right )}{30 c^2} \]

[In]

Integrate[x^7*Sqrt[a + c*x^4],x]

[Out]

(Sqrt[a + c*x^4]*(-2*a^2 + a*c*x^4 + 3*c^2*x^8))/(30*c^2)

Maple [A] (verified)

Time = 4.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\)	\(25\)
default	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\)	\(25\)
elliptic	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\)	\(25\)
pseudoelliptic	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\)	\(25\)
trager	\(-\frac {\left (-3 c^{2} x^{8}-a \,x^{4} c +2 a^{2}\right ) \sqrt {x^{4} c +a}}{30 c^{2}}\)	\(36\)
risch	\(-\frac {\left (-3 c^{2} x^{8}-a \,x^{4} c +2 a^{2}\right ) \sqrt {x^{4} c +a}}{30 c^{2}}\)	\(36\)

[In]

int(x^7*(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/30*(c*x^4+a)^(3/2)*(-3*c*x^4+2*a)/c^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int x^7 \sqrt {a+c x^4} \, dx=\frac {{\left (3 \, c^{2} x^{8} + a c x^{4} - 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{30 \, c^{2}} \]

[In]

integrate(x^7*(c*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

1/30*(3*c^2*x^8 + a*c*x^4 - 2*a^2)*sqrt(c*x^4 + a)/c^2

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int x^7 \sqrt {a+c x^4} \, dx=\begin {cases} - \frac {a^{2} \sqrt {a + c x^{4}}}{15 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{4}}}{30 c} + \frac {x^{8} \sqrt {a + c x^{4}}}{10} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases} \]

[In]

integrate(x**7*(c*x**4+a)**(1/2),x)

[Out]

Piecewise((-a**2*sqrt(a + c*x**4)/(15*c**2) + a*x**4*sqrt(a + c*x**4)/(30*c) + x**8*sqrt(a + c*x**4)/10, Ne(c,
0)), (sqrt(a)*x**8/8, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int x^7 \sqrt {a+c x^4} \, dx=\frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} a}{6 \, c^{2}} \]

[In]

integrate(x^7*(c*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

1/10*(c*x^4 + a)^(5/2)/c^2 - 1/6*(c*x^4 + a)^(3/2)*a/c^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int x^7 \sqrt {a+c x^4} \, dx=\frac {3 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} - 5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} a}{30 \, c^{2}} \]

[In]

integrate(x^7*(c*x^4+a)^(1/2),x, algorithm="giac")

[Out]

1/30*(3*(c*x^4 + a)^(5/2) - 5*(c*x^4 + a)^(3/2)*a)/c^2

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int x^7 \sqrt {a+c x^4} \, dx=\sqrt {c\,x^4+a}\,\left (\frac {x^8}{10}-\frac {a^2}{15\,c^2}+\frac {a\,x^4}{30\,c}\right ) \]

[In]

int(x^7*(a + c*x^4)^(1/2),x)

[Out]

(a + c*x^4)^(1/2)*(x^8/10 - a^2/(15*c^2) + (a*x^4)/(30*c))