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

Optimal. Leaf size=35 \[ \frac {2}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )-\frac {2 \sqrt {x^4+x}}{3 x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {2020, 2029, 206} \begin {gather*} \frac {2}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )-\frac {2 \sqrt {x^4+x}}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x + x^4]/x^3,x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2) + (2*ArcTanh[x^2/Sqrt[x + x^4]])/3

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 2020

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 + j*p + 1)), x] - Dist[(b*p*(n - j))/(c^n*(m + j*p + 1)), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*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 \frac {\sqrt {x+x^4}}{x^3} \, dx &=-\frac {2 \sqrt {x+x^4}}{3 x^2}+\int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=-\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=-\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.43 \begin {gather*} \frac {2 \sqrt {x^4+x} \sinh ^{-1}\left (x^{3/2}\right )}{3 \sqrt {x} \sqrt {x^3+1}}-\frac {2 \sqrt {x^4+x}}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x + x^4]/x^3,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.34, size = 35, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2}{3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[x + x^4])/(3*x^2) + (2*ArcTanh[x^2/Sqrt[x + x^4]])/3

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fricas [A]  time = 0.52, size = 37, normalized size = 1.06 \begin {gather*} \frac {x^{2} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) - 2 \, \sqrt {x^{4} + x}}{3 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 35, normalized size = 1.00 \begin {gather*} -\frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} + \frac {1}{3} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{3} \, \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^4+x)^(1/2)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.25, size = 31, normalized size = 0.89

method	result	size
meijerg	\(-\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{\frac {3}{2}}}-4 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )}{6 \sqrt {\pi }}\)	\(31\)
trager	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3}\)	\(34\)
default	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}+\frac {2 \sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, -\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(300\)
elliptic	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}+\frac {2 \sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, -\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(300\)
risch	\(-\frac {2 \left (x^{3}+1\right )}{3 x \sqrt {x \left (x^{3}+1\right )}}-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/Pi^(1/2)*(4*Pi^(1/2)/x^(3/2)*(x^3+1)^(1/2)-4*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 \frac {\sqrt {x^{4} + x}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/2)/x^3,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 \frac {\sqrt {x^4+x}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^4)^(1/2)/x^3,x)

[Out]

int((x + x^4)^(1/2)/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+x)**(1/2)/x**3,x)

[Out]

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

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