3.6.33 \(\int \frac {x^2 (3 e+2 f x^2)}{e^2+4 e f x^2+4 f^2 x^4-4 d f x^6} \, dx\) [533]

Optimal. Leaf size=40 \[ \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \]

[Out]

1/2*arctanh(2*x^3*d^(1/2)*f^(1/2)/(2*f*x^2+e))/d^(1/2)/f^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2119, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6),x]

[Out]

ArcTanh[(2*Sqrt[d]*Sqrt[f]*x^3)/(e + 2*f*x^2)]/(2*Sqrt[d]*Sqrt[f])

Rule 214

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

Rule 2119

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(n_.)))/((a_) + (b_.)*(x_)^(k_.) + (c_.)*(x_)^(n_.) + (d_.)*(x_)^(n2_)), x_
Symbol] :> Dist[A^2*((m - n + 1)/(m + 1)), Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m -
n + 1) + B*(m + 1)*x^n)], x] /; FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && Eq
Q[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4-4 d f x^6} \, dx &=\left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{e^2-36 d e^2 f x^2} \, dx,x,\frac {x^3}{3 e+6 f x^2}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.04, size = 85, normalized size = 2.12 \begin {gather*} \frac {\text {RootSum}\left [e^2+4 e f \text {$\#$1}^2+4 f^2 \text {$\#$1}^4-4 d f \text {$\#$1}^6\&,\frac {3 e \log (x-\text {$\#$1}) \text {$\#$1}+2 f \log (x-\text {$\#$1}) \text {$\#$1}^3}{e+2 f \text {$\#$1}^2-3 d \text {$\#$1}^4}\&\right ]}{8 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6),x]

[Out]

RootSum[e^2 + 4*e*f*#1^2 + 4*f^2*#1^4 - 4*d*f*#1^6 & , (3*e*Log[x - #1]*#1 + 2*f*Log[x - #1]*#1^3)/(e + 2*f*#1
^2 - 3*d*#1^4) & ]/(8*f)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.10, size = 77, normalized size = 1.92

method	result	size
default	\(-\frac {\munderset {\textit {\_R} =\RootOf \left (4 d f \,\textit {\_Z}^{6}-4 f^{2} \textit {\_Z}^{4}-4 e f \,\textit {\_Z}^{2}-e^{2}\right )}{\sum }\frac {\left (2 \textit {\_R}^{4} f +3 e \,\textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{3 d \,\textit {\_R}^{5}-2 \textit {\_R}^{3} f -e \textit {\_R}}}{8 f}\)	\(77\)
risch	\(\frac {\ln \left (-2 d^{2} f^{2} x^{3}-2 \left (d f \right )^{\frac {3}{2}} f \,x^{2}-\left (d f \right )^{\frac {3}{2}} e \right )}{4 \sqrt {d f}}-\frac {\ln \left (2 d^{2} f^{2} x^{3}-2 \left (d f \right )^{\frac {3}{2}} f \,x^{2}-\left (d f \right )^{\frac {3}{2}} e \right )}{4 \sqrt {d f}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(2*f*x^2+3*e)/(-4*d*f*x^6+4*f^2*x^4+4*e*f*x^2+e^2),x,method=_RETURNVERBOSE)

[Out]

-1/8/f*sum((2*_R^4*f+3*_R^2*e)/(3*_R^5*d-2*_R^3*f-_R*e)*ln(x-_R),_R=RootOf(4*_Z^6*d*f-4*_Z^4*f^2-4*_Z^2*e*f-e^
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^2*(2*f*x^2+3*e)/(-4*d*f*x^6+4*f^2*x^4+4*e*f*x^2+e^2),x, algorithm="maxima")

[Out]

-integrate((2*f*x^2 + 3*e)*x^2/(4*d*f*x^6 - 4*f^2*x^4 - 4*f*x^2*e - e^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (31) = 62\).
time = 0.36, size = 215, normalized size = 5.38 \begin {gather*} \left [\frac {\sqrt {d f} \log \left (\frac {4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, f x^{2} e + 4 \, {\left (2 \, f x^{5} + x^{3} e\right )} \sqrt {d f} + e^{2}}{4 \, d f x^{6} - 4 \, f^{2} x^{4} - 4 \, f x^{2} e - e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {-d f} \arctan \left (-\frac {2 \, {\left (2 \, d f x^{5} - 2 \, f^{2} x^{3} - {\left (d x^{3} + f x\right )} e\right )} \sqrt {-d f} e^{\left (-2\right )}}{d}\right ) + \sqrt {-d f} \arctan \left (\frac {\sqrt {-d f} x}{f}\right ) - \sqrt {-d f} \arctan \left (-\frac {{\left (2 \, d f x^{3} - 2 \, f^{2} x - d x e\right )} \sqrt {-d f} e^{\left (-1\right )}}{d f}\right )}{2 \, d f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*f*x^2+3*e)/(-4*d*f*x^6+4*f^2*x^4+4*e*f*x^2+e^2),x, algorithm="fricas")

[Out]

[1/4*sqrt(d*f)*log((4*d*f*x^6 + 4*f^2*x^4 + 4*f*x^2*e + 4*(2*f*x^5 + x^3*e)*sqrt(d*f) + e^2)/(4*d*f*x^6 - 4*f^
2*x^4 - 4*f*x^2*e - e^2))/(d*f), -1/2*(sqrt(-d*f)*arctan(-2*(2*d*f*x^5 - 2*f^2*x^3 - (d*x^3 + f*x)*e)*sqrt(-d*
f)*e^(-2)/d) + sqrt(-d*f)*arctan(sqrt(-d*f)*x/f) - sqrt(-d*f)*arctan(-(2*d*f*x^3 - 2*f^2*x - d*x*e)*sqrt(-d*f)
*e^(-1)/(d*f)))/(d*f)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (36) = 72\).
time = 0.70, size = 80, normalized size = 2.00 \begin {gather*} - \frac {\sqrt {\frac {1}{d f}} \log {\left (- \frac {e \sqrt {\frac {1}{d f}}}{2} - f x^{2} \sqrt {\frac {1}{d f}} + x^{3} \right )}}{4} + \frac {\sqrt {\frac {1}{d f}} \log {\left (\frac {e \sqrt {\frac {1}{d f}}}{2} + f x^{2} \sqrt {\frac {1}{d f}} + x^{3} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(2*f*x**2+3*e)/(-4*d*f*x**6+4*f**2*x**4+4*e*f*x**2+e**2),x)

[Out]

-sqrt(1/(d*f))*log(-e*sqrt(1/(d*f))/2 - f*x**2*sqrt(1/(d*f)) + x**3)/4 + sqrt(1/(d*f))*log(e*sqrt(1/(d*f))/2 +
f*x**2*sqrt(1/(d*f)) + x**3)/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (31) = 62\).
time = 6.52, size = 69, normalized size = 1.72 \begin {gather*} \frac {\sqrt {d f} \log \left ({\left | 2 \, \sqrt {d f} x^{3} + 2 \, f x^{2} + e \right |}\right )}{4 \, d f} - \frac {\sqrt {d f} \log \left ({\left | -2 \, \sqrt {d f} x^{3} + 2 \, f x^{2} + e \right |}\right )}{4 \, d f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(2*f*x^2+3*e)/(-4*d*f*x^6+4*f^2*x^4+4*e*f*x^2+e^2),x, algorithm="giac")

[Out]

1/4*sqrt(d*f)*log(abs(2*sqrt(d*f)*x^3 + 2*f*x^2 + e))/(d*f) - 1/4*sqrt(d*f)*log(abs(-2*sqrt(d*f)*x^3 + 2*f*x^2
+ e))/(d*f)

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Mupad [B]
time = 3.14, size = 30, normalized size = 0.75 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {2\,\sqrt {d}\,\sqrt {f}\,x^3}{2\,f\,x^2+e}\right )}{2\,\sqrt {d}\,\sqrt {f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(3*e + 2*f*x^2))/(e^2 + 4*f^2*x^4 - 4*d*f*x^6 + 4*e*f*x^2),x)

[Out]

atanh((2*d^(1/2)*f^(1/2)*x^3)/(e + 2*f*x^2))/(2*d^(1/2)*f^(1/2))

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