3.6.7 \(\int x^3 \sqrt {x+x^4} \, dx\)

Optimal. Leaf size=39 \[ \frac {1}{12} \sqrt {x^4+x} \left (2 x^4+x\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2021, 2024, 2029, 206} \begin {gather*} \frac {1}{6} \sqrt {x^4+x} x^4+\frac {1}{12} \sqrt {x^4+x} x-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[x + x^4],x]

[Out]

(x*Sqrt[x + x^4])/12 + (x^4*Sqrt[x + x^4])/6 - ArcTanh[x^2/Sqrt[x + x^4]]/12

Rule 206

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

Rule 2021

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 2024

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 2029

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^3 \sqrt {x+x^4} \, dx &=\frac {1}{6} x^4 \sqrt {x+x^4}+\frac {1}{4} \int \frac {x^4}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{12} x \sqrt {x+x^4}+\frac {1}{6} x^4 \sqrt {x+x^4}-\frac {1}{8} \int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{12} x \sqrt {x+x^4}+\frac {1}{6} x^4 \sqrt {x+x^4}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=\frac {1}{12} x \sqrt {x+x^4}+\frac {1}{6} x^4 \sqrt {x+x^4}-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 1.23 \begin {gather*} \frac {\sqrt {x^4+x} \left (2 x^{9/2}+x^{3/2}-\frac {\sinh ^{-1}\left (x^{3/2}\right )}{\sqrt {x^3+1}}\right )}{12 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[x + x^4],x]

[Out]

(Sqrt[x + x^4]*(x^(3/2) + 2*x^(9/2) - ArcSinh[x^(3/2)]/Sqrt[1 + x^3]))/(12*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.39, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{12} \sqrt {x+x^4} \left (x+2 x^4\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3*Sqrt[x + x^4],x]

[Out]

(Sqrt[x + x^4]*(x + 2*x^4))/12 - ArcTanh[x^2/Sqrt[x + x^4]]/12

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fricas [A]  time = 0.50, size = 37, normalized size = 0.95 \begin {gather*} \frac {1}{12} \, {\left (2 \, x^{4} + x\right )} \sqrt {x^{4} + x} + \frac {1}{24} \, \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^3*(x^4+x)^(1/2),x, algorithm="fricas")

[Out]

1/12*(2*x^4 + x)*sqrt(x^4 + x) + 1/24*log(2*x^3 - 2*sqrt(x^4 + x)*x + 1)

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giac [A]  time = 0.34, size = 43, normalized size = 1.10 \begin {gather*} \frac {1}{12} \, \sqrt {x^{4} + x} {\left (2 \, x^{3} + 1\right )} x - \frac {1}{24} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{24} \, \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^3*(x^4+x)^(1/2),x, algorithm="giac")

[Out]

1/12*sqrt(x^4 + x)*(2*x^3 + 1)*x - 1/24*log(sqrt(1/x^3 + 1) + 1) + 1/24*log(abs(sqrt(1/x^3 + 1) - 1))

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maple [A]  time = 0.22, size = 38, normalized size = 0.97

method	result	size
meijerg	\(-\frac {-\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \left (6 x^{3}+3\right ) \sqrt {x^{3}+1}}{6}+\frac {\sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )}{2}}{6 \sqrt {\pi }}\)	\(38\)
trager	\(\frac {x \left (2 x^{3}+1\right ) \sqrt {x^{4}+x}}{12}+\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{24}\)	\(39\)
default	\(\frac {x^{4} \sqrt {x^{4}+x}}{6}+\frac {x \sqrt {x^{4}+x}}{12}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(313\)
elliptic	\(\frac {x^{4} \sqrt {x^{4}+x}}{6}+\frac {x \sqrt {x^{4}+x}}{12}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(313\)
risch	\(\frac {x^{2} \left (2 x^{3}+1\right ) \left (x^{3}+1\right )}{12 \sqrt {x \left (x^{3}+1\right )}}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/Pi^(1/2)*(-1/6*Pi^(1/2)*x^(3/2)*(6*x^3+3)*(x^3+1)^(1/2)+1/2*Pi^(1/2)*arcsinh(x^(3/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{4} + x} x^{3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^4 + x)*x^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^3\,\sqrt {x^4+x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x + x^4)^(1/2),x)

[Out]

int(x^3*(x + x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \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**3*(x**4+x)**(1/2),x)

[Out]

Integral(x**3*sqrt(x*(x + 1)*(x**2 - x + 1)), x)

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