3.1509 \(\int x^3 \sqrt{1+x^8} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{8} \sqrt{x^8+1} x^4+\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]

[Out]

(x^4*Sqrt[1 + x^8])/8 + ArcSinh[x^4]/8

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Rubi [A]  time = 0.0076866, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 195, 215} \[ \frac{1}{8} \sqrt{x^8+1} x^4+\frac{1}{8} \sinh ^{-1}\left (x^4\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[1 + x^8],x]

[Out]

(x^4*Sqrt[1 + x^8])/8 + ArcSinh[x^4]/8

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int x^3 \sqrt{1+x^8} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \sqrt{1+x^2} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{1+x^8}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^4\right )\\ &=\frac{1}{8} x^4 \sqrt{1+x^8}+\frac{1}{8} \sinh ^{-1}\left (x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.006428, size = 22, normalized size = 0.88 \[ \frac{1}{8} \left (\sqrt{x^8+1} x^4+\sinh ^{-1}\left (x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[1 + x^8],x]

[Out]

(x^4*Sqrt[1 + x^8] + ArcSinh[x^4])/8

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Maple [A]  time = 0.028, size = 20, normalized size = 0.8 \begin{align*}{\frac{{\it Arcsinh} \left ({x}^{4} \right ) }{8}}+{\frac{{x}^{4}}{8}\sqrt{{x}^{8}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^8+1)^(1/2),x)

[Out]

1/8*arcsinh(x^4)+1/8*x^4*(x^8+1)^(1/2)

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Maxima [B]  time = 0.969173, size = 78, normalized size = 3.12 \begin{align*} \frac{\sqrt{x^{8} + 1}}{8 \, x^{4}{\left (\frac{x^{8} + 1}{x^{8}} - 1\right )}} + \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} + 1\right ) - \frac{1}{16} \, \log \left (\frac{\sqrt{x^{8} + 1}}{x^{4}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

1/8*sqrt(x^8 + 1)/(x^4*((x^8 + 1)/x^8 - 1)) + 1/16*log(sqrt(x^8 + 1)/x^4 + 1) - 1/16*log(sqrt(x^8 + 1)/x^4 - 1
)

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Fricas [A]  time = 1.23788, size = 74, normalized size = 2.96 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{8} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(x^8 + 1)*x^4 - 1/8*log(-x^4 + sqrt(x^8 + 1))

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Sympy [A]  time = 1.45465, size = 19, normalized size = 0.76 \begin{align*} \frac{x^{4} \sqrt{x^{8} + 1}}{8} + \frac{\operatorname{asinh}{\left (x^{4} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(x**8+1)**(1/2),x)

[Out]

x**4*sqrt(x**8 + 1)/8 + asinh(x**4)/8

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Giac [A]  time = 1.19585, size = 39, normalized size = 1.56 \begin{align*} \frac{1}{8} \, \sqrt{x^{8} + 1} x^{4} - \frac{1}{8} \, \log \left (-x^{4} + \sqrt{x^{8} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/8*sqrt(x^8 + 1)*x^4 - 1/8*log(-x^4 + sqrt(x^8 + 1))