\(\int x^7 \sqrt {5+3 x^4} \, dx\) [805]

Optimal result

Integrand size = 15, antiderivative size = 31 \[ \int x^7 \sqrt {5+3 x^4} \, dx=-\frac {5}{54} \left (5+3 x^4\right )^{3/2}+\frac {1}{90} \left (5+3 x^4\right )^{5/2} \]

[Out]

-5/54*(3*x^4+5)^(3/2)+1/90*(3*x^4+5)^(5/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, 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} \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{90} \left (3 x^4+5\right )^{5/2}-\frac {5}{54} \left (3 x^4+5\right )^{3/2} \]

[In]

Int[x^7*Sqrt[5 + 3*x^4],x]

[Out]

(-5*(5 + 3*x^4)^(3/2))/54 + (5 + 3*x^4)^(5/2)/90

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 {5+3 x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\frac {5}{3} \sqrt {5+3 x}+\frac {1}{3} (5+3 x)^{3/2}\right ) \, dx,x,x^4\right ) \\ & = -\frac {5}{54} \left (5+3 x^4\right )^{3/2}+\frac {1}{90} \left (5+3 x^4\right )^{5/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{270} \left (5+3 x^4\right )^{3/2} \left (-10+9 x^4\right ) \]

[In]

Integrate[x^7*Sqrt[5 + 3*x^4],x]

[Out]

((5 + 3*x^4)^(3/2)*(-10 + 9*x^4))/270

Maple [A] (verified)

Time = 3.99 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
gosper	\(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\)	\(19\)
default	\(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\)	\(19\)
elliptic	\(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\)	\(19\)
pseudoelliptic	\(\frac {\left (3 x^{4}+5\right )^{\frac {3}{2}} \left (9 x^{4}-10\right )}{270}\)	\(19\)
trager	\(\left (\frac {1}{10} x^{8}+\frac {1}{18} x^{4}-\frac {5}{27}\right ) \sqrt {3 x^{4}+5}\)	\(23\)
risch	\(\frac {\left (27 x^{8}+15 x^{4}-50\right ) \sqrt {3 x^{4}+5}}{270}\)	\(24\)
meijerg	\(-\frac {25 \sqrt {5}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1+\frac {3 x^{4}}{5}\right )^{\frac {3}{2}} \left (-\frac {9 x^{4}}{5}+2\right )}{15}\right )}{72 \sqrt {\pi }}\)	\(36\)

[In]

int(x^7*(3*x^4+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/270*(3*x^4+5)^(3/2)*(9*x^4-10)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{270} \, {\left (27 \, x^{8} + 15 \, x^{4} - 50\right )} \sqrt {3 \, x^{4} + 5} \]

[In]

integrate(x^7*(3*x^4+5)^(1/2),x, algorithm="fricas")

[Out]

1/270*(27*x^8 + 15*x^4 - 50)*sqrt(3*x^4 + 5)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {x^{8} \sqrt {3 x^{4} + 5}}{10} + \frac {x^{4} \sqrt {3 x^{4} + 5}}{18} - \frac {5 \sqrt {3 x^{4} + 5}}{27} \]

[In]

integrate(x**7*(3*x**4+5)**(1/2),x)

[Out]

x**8*sqrt(3*x**4 + 5)/10 + x**4*sqrt(3*x**4 + 5)/18 - 5*sqrt(3*x**4 + 5)/27

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{90} \, {\left (3 \, x^{4} + 5\right )}^{\frac {5}{2}} - \frac {5}{54} \, {\left (3 \, x^{4} + 5\right )}^{\frac {3}{2}} \]

[In]

integrate(x^7*(3*x^4+5)^(1/2),x, algorithm="maxima")

[Out]

1/90*(3*x^4 + 5)^(5/2) - 5/54*(3*x^4 + 5)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {1}{90} \, {\left (3 \, x^{4} + 5\right )}^{\frac {5}{2}} - \frac {5}{54} \, {\left (3 \, x^{4} + 5\right )}^{\frac {3}{2}} \]

[In]

integrate(x^7*(3*x^4+5)^(1/2),x, algorithm="giac")

[Out]

1/90*(3*x^4 + 5)^(5/2) - 5/54*(3*x^4 + 5)^(3/2)

Mupad [B] (verification not implemented)

Time = 5.52 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58 \[ \int x^7 \sqrt {5+3 x^4} \, dx=\frac {{\left (3\,x^4+5\right )}^{3/2}\,\left (9\,x^4-10\right )}{270} \]

[In]

int(x^7*(3*x^4 + 5)^(1/2),x)

[Out]

((3*x^4 + 5)^(3/2)*(9*x^4 - 10))/270