3.6.25 \(\int \frac {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx\)

Optimal. Leaf size=40 \[ \log \left (x^3+2 x^2+\sqrt {x^6+4 x^5+4 x^4+2 x^3+4 x^2-5}+1\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.42, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x + 3*x^2)/Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6],x]

[Out]

4*Defer[Int][x/Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6], x] + 3*Defer[Int][x^2/Sqrt[-5 + 4*x^2 + 2*x^3 +
4*x^4 + 4*x^5 + x^6], x]

Rubi steps

\begin {align*} \int \frac {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx &=\int \frac {x (4+3 x)}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx\\ &=\int \left (\frac {4 x}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}}+\frac {3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}}\right ) \, dx\\ &=3 \int \frac {x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx+4 \int \frac {x}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x+3 x^2}{\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4*x + 3*x^2)/Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6],x]

[Out]

Integrate[(4*x + 3*x^2)/Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6], x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.17, size = 40, normalized size = 1.00 \begin {gather*} \log \left (1+2 x^2+x^3+\sqrt {-5+4 x^2+2 x^3+4 x^4+4 x^5+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(4*x + 3*x^2)/Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6],x]

[Out]

Log[1 + 2*x^2 + x^3 + Sqrt[-5 + 4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6]]

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 38, normalized size = 0.95 \begin {gather*} \log \left (x^{3} + 2 \, x^{2} + \sqrt {x^{6} + 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} + 4 \, x^{2} - 5} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+4*x)/(x^6+4*x^5+4*x^4+2*x^3+4*x^2-5)^(1/2),x, algorithm="fricas")

[Out]

log(x^3 + 2*x^2 + sqrt(x^6 + 4*x^5 + 4*x^4 + 2*x^3 + 4*x^2 - 5) + 1)

________________________________________________________________________________________

giac [A]  time = 0.35, size = 41, normalized size = 1.02 \begin {gather*} -\log \left ({\left | -x^{3} - 2 \, x^{2} + \sqrt {2 \, x^{3} + {\left (x^{3} + 2 \, x^{2}\right )}^{2} + 4 \, x^{2} - 5} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+4*x)/(x^6+4*x^5+4*x^4+2*x^3+4*x^2-5)^(1/2),x, algorithm="giac")

[Out]

-log(abs(-x^3 - 2*x^2 + sqrt(2*x^3 + (x^3 + 2*x^2)^2 + 4*x^2 - 5) - 1))

________________________________________________________________________________________

maple [A]  time = 0.19, size = 39, normalized size = 0.98

method	result	size
trager	\(\ln \left (1+2 x^{2}+x^{3}+\sqrt {x^{6}+4 x^{5}+4 x^{4}+2 x^{3}+4 x^{2}-5}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+4*x)/(x^6+4*x^5+4*x^4+2*x^3+4*x^2-5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

ln(1+2*x^2+x^3+(x^6+4*x^5+4*x^4+2*x^3+4*x^2-5)^(1/2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{2} + 4 \, x}{\sqrt {x^{6} + 4 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} + 4 \, x^{2} - 5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+4*x)/(x^6+4*x^5+4*x^4+2*x^3+4*x^2-5)^(1/2),x, algorithm="maxima")

[Out]

integrate((3*x^2 + 4*x)/sqrt(x^6 + 4*x^5 + 4*x^4 + 2*x^3 + 4*x^2 - 5), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {3\,x^2+4\,x}{\sqrt {x^6+4\,x^5+4\,x^4+2\,x^3+4\,x^2-5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + 3*x^2)/(4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6 - 5)^(1/2),x)

[Out]

int((4*x + 3*x^2)/(4*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + x^6 - 5)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (3 x + 4\right )}{\sqrt {x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} - 5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+4*x)/(x**6+4*x**5+4*x**4+2*x**3+4*x**2-5)**(1/2),x)

[Out]

Integral(x*(3*x + 4)/sqrt(x**6 + 4*x**5 + 4*x**4 + 2*x**3 + 4*x**2 - 5), x)

________________________________________________________________________________________