3.6.89 \(\int x^6 \sqrt {x+x^4} \, dx\) [589]

Optimal. Leaf size=46 \[ \frac {1}{72} \sqrt {x+x^4} \left (-3 x+2 x^4+8 x^7\right )+\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \]

[Out]

1/72*(x^4+x)^(1/2)*(8*x^7+2*x^4-3*x)+1/24*arctanh(x^2/(x^4+x)^(1/2))

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Rubi [A]
time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2046, 2049, 2054, 212} \begin {gather*} \frac {1}{36} \sqrt {x^4+x} x^4-\frac {1}{24} \sqrt {x^4+x} x+\frac {1}{9} \sqrt {x^4+x} x^7+\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*Sqrt[x + x^4],x]

[Out]

-1/24*(x*Sqrt[x + x^4]) + (x^4*Sqrt[x + x^4])/36 + (x^7*Sqrt[x + x^4])/9 + ArcTanh[x^2/Sqrt[x + x^4]]/24

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n +
1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps

\begin {align*} \int x^6 \sqrt {x+x^4} \, dx &=\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{6} \int \frac {x^7}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}-\frac {1}{8} \int \frac {x^4}{\sqrt {x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {x+x^4}+\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{16} \int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {x+x^4}+\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{24} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=-\frac {1}{24} x \sqrt {x+x^4}+\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 64, normalized size = 1.39 \begin {gather*} \frac {\sqrt {x+x^4} \left (x^{3/2} \left (-3+2 x^3+8 x^6\right )+\frac {3 \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{\sqrt {1+x^3}}\right )}{72 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*Sqrt[x + x^4],x]

[Out]

(Sqrt[x + x^4]*(x^(3/2)*(-3 + 2*x^3 + 8*x^6) + (3*ArcTanh[x^(3/2)/Sqrt[1 + x^3]])/Sqrt[1 + x^3]))/(72*Sqrt[x])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.43, size = 325, normalized size = 7.07

method	result	size
meijerg	\(-\frac {\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \left (-40 x^{6}-10 x^{3}+15\right ) \sqrt {x^{3}+1}}{60}-\frac {\sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )}{4}}{6 \sqrt {\pi }}\)	\(43\)
trager	\(\frac {x \left (8 x^{6}+2 x^{3}-3\right ) \sqrt {x^{4}+x}}{72}+\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{48}\)	\(44\)
risch	\(\frac {x^{2} \left (8 x^{6}+2 x^{3}-3\right ) \left (x^{3}+1\right )}{72 \sqrt {x \left (x^{3}+1\right )}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(322\)
default	\(\frac {x^{7} \sqrt {x^{4}+x}}{9}+\frac {x^{4} \sqrt {x^{4}+x}}{36}-\frac {x \sqrt {x^{4}+x}}{24}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(325\)
elliptic	\(\frac {x^{7} \sqrt {x^{4}+x}}{9}+\frac {x^{4} \sqrt {x^{4}+x}}{36}-\frac {x \sqrt {x^{4}+x}}{24}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(325\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^4+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*x^7*(x^4+x)^(1/2)+1/36*x^4*(x^4+x)^(1/2)-1/24*x*(x^4+x)^(1/2)-1/8*(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2)
)*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)*(1+x)^2*(-(x-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x))^(1/2)*(-(x-1/2
-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)/(3/2+1/2*I*3^(1/2))/(x*(1+x)*(x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2
*I*3^(1/2)))^(1/2)*(-EllipticF(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(
-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+EllipticPi(((3/2+1/2*I*3^(1/2))*x/(1/2+1
/2*I*3^(1/2))/(1+x))^(1/2),(1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/
(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^4+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + x)*x^6, x)

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Fricas [A]
time = 0.37, size = 44, normalized size = 0.96 \begin {gather*} \frac {1}{72} \, {\left (8 \, x^{7} + 2 \, x^{4} - 3 \, x\right )} \sqrt {x^{4} + x} + \frac {1}{48} \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^4+x)^(1/2),x, algorithm="fricas")

[Out]

1/72*(8*x^7 + 2*x^4 - 3*x)*sqrt(x^4 + x) + 1/48*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(x**4+x)**(1/2),x)

[Out]

Integral(x**6*sqrt(x*(x + 1)*(x**2 - x + 1)), x)

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Giac [A]
time = 0.40, size = 50, normalized size = 1.09 \begin {gather*} \frac {1}{72} \, {\left (2 \, {\left (4 \, x^{3} + 1\right )} x^{3} - 3\right )} \sqrt {x^{4} + x} x + \frac {1}{48} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{48} \, \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^4+x)^(1/2),x, algorithm="giac")

[Out]

1/72*(2*(4*x^3 + 1)*x^3 - 3)*sqrt(x^4 + x)*x + 1/48*log(sqrt(1/x^3 + 1) + 1) - 1/48*log(abs(sqrt(1/x^3 + 1) -
1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^6\,\sqrt {x^4+x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x + x^4)^(1/2),x)

[Out]

int(x^6*(x + x^4)^(1/2), x)

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