3.1545 \(\int x^5 \sqrt{9+x^{12}} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{12} \sqrt{x^{12}+9} x^6+\frac{3}{4} \sinh ^{-1}\left (\frac{x^6}{3}\right ) \]

[Out]

(x^6*Sqrt[9 + x^12])/12 + (3*ArcSinh[x^6/3])/4

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Rubi [A]  time = 0.0090506, antiderivative size = 29, 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} \[ \frac{1}{12} \sqrt{x^{12}+9} x^6+\frac{3}{4} \sinh ^{-1}\left (\frac{x^6}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[9 + x^12],x]

[Out]

(x^6*Sqrt[9 + x^12])/12 + (3*ArcSinh[x^6/3])/4

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^5 \sqrt{9+x^{12}} \, dx &=\frac{1}{6} \operatorname{Subst}\left (\int \sqrt{9+x^2} \, dx,x,x^6\right )\\ &=\frac{1}{12} x^6 \sqrt{9+x^{12}}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{9+x^2}} \, dx,x,x^6\right )\\ &=\frac{1}{12} x^6 \sqrt{9+x^{12}}+\frac{3}{4} \sinh ^{-1}\left (\frac{x^6}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.006709, size = 28, normalized size = 0.97 \[ \frac{1}{12} \left (\sqrt{x^{12}+9} x^6+9 \sinh ^{-1}\left (\frac{x^6}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[9 + x^12],x]

[Out]

(x^6*Sqrt[9 + x^12] + 9*ArcSinh[x^6/3])/12

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Maple [A]  time = 0.032, size = 22, normalized size = 0.8 \begin{align*}{\frac{3}{4}{\it Arcsinh} \left ({\frac{{x}^{6}}{3}} \right ) }+{\frac{{x}^{6}}{12}\sqrt{{x}^{12}+9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^12+9)^(1/2),x)

[Out]

3/4*arcsinh(1/3*x^6)+1/12*x^6*(x^12+9)^(1/2)

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Maxima [B]  time = 0.955351, size = 78, normalized size = 2.69 \begin{align*} \frac{3 \, \sqrt{x^{12} + 9}}{4 \, x^{6}{\left (\frac{x^{12} + 9}{x^{12}} - 1\right )}} + \frac{3}{8} \, \log \left (\frac{\sqrt{x^{12} + 9}}{x^{6}} + 1\right ) - \frac{3}{8} \, \log \left (\frac{\sqrt{x^{12} + 9}}{x^{6}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^12+9)^(1/2),x, algorithm="maxima")

[Out]

3/4*sqrt(x^12 + 9)/(x^6*((x^12 + 9)/x^12 - 1)) + 3/8*log(sqrt(x^12 + 9)/x^6 + 1) - 3/8*log(sqrt(x^12 + 9)/x^6
- 1)

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Fricas [A]  time = 1.4387, size = 78, normalized size = 2.69 \begin{align*} \frac{1}{12} \, \sqrt{x^{12} + 9} x^{6} - \frac{3}{4} \, \log \left (-x^{6} + \sqrt{x^{12} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^12+9)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(x^12 + 9)*x^6 - 3/4*log(-x^6 + sqrt(x^12 + 9))

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Sympy [A]  time = 1.57501, size = 37, normalized size = 1.28 \begin{align*} \frac{x^{18}}{12 \sqrt{x^{12} + 9}} + \frac{3 x^{6}}{4 \sqrt{x^{12} + 9}} + \frac{3 \operatorname{asinh}{\left (\frac{x^{6}}{3} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(x**12+9)**(1/2),x)

[Out]

x**18/(12*sqrt(x**12 + 9)) + 3*x**6/(4*sqrt(x**12 + 9)) + 3*asinh(x**6/3)/4

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Giac [A]  time = 1.14256, size = 39, normalized size = 1.34 \begin{align*} \frac{1}{12} \, \sqrt{x^{12} + 9} x^{6} - \frac{3}{4} \, \log \left (-x^{6} + \sqrt{x^{12} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(x^12+9)^(1/2),x, algorithm="giac")

[Out]

1/12*sqrt(x^12 + 9)*x^6 - 3/4*log(-x^6 + sqrt(x^12 + 9))