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

Optimal. Leaf size=31 \[ \frac{1}{90} \left (3 x^4+5\right )^{5/2}-\frac{5}{54} \left (3 x^4+5\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0148999, antiderivative size = 31, 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} \[ \frac{1}{90} \left (3 x^4+5\right )^{5/2}-\frac{5}{54} \left (3 x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

[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 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{5+3 x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x \sqrt{5+3 x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{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]  time = 0.0080551, size = 22, normalized size = 0.71 \[ \frac{1}{270} \left (3 x^4+5\right )^{3/2} \left (9 x^4-10\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*Sqrt[5 + 3*x^4],x]

[Out]

((5 + 3*x^4)^(3/2)*(-10 + 9*x^4))/270

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Maple [A]  time = 0.004, size = 19, normalized size = 0.6 \begin{align*}{\frac{9\,{x}^{4}-10}{270} \left ( 3\,{x}^{4}+5 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.955752, size = 31, normalized size = 1. \begin{align*} \frac{1}{90} \,{\left (3 \, x^{4} + 5\right )}^{\frac{5}{2}} - \frac{5}{54} \,{\left (3 \, x^{4} + 5\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.38996, size = 62, normalized size = 2. \begin{align*} \frac{1}{270} \,{\left (27 \, x^{8} + 15 \, x^{4} - 50\right )} \sqrt{3 \, x^{4} + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.796639, size = 42, normalized size = 1.35 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.09694, size = 31, normalized size = 1. \begin{align*} \frac{1}{90} \,{\left (3 \, x^{4} + 5\right )}^{\frac{5}{2}} - \frac{5}{54} \,{\left (3 \, x^{4} + 5\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)