3.297 \(\int \frac{x^4 (d+e x)}{a^2-c^2 x^2} \, dx\)

Optimal. Leaf size=95 \[ -\frac{a^3 (a e+c d) \log (a-c x)}{2 c^6}+\frac{a^3 (c d-a e) \log (a+c x)}{2 c^6}-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2} \]

[Out]

-((a^2*d*x)/c^4) - (a^2*e*x^2)/(2*c^4) - (d*x^3)/(3*c^2) - (e*x^4)/(4*c^2) - (a^3*(c*d + a*e)*Log[a - c*x])/(2
*c^6) + (a^3*(c*d - a*e)*Log[a + c*x])/(2*c^6)

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Rubi [A]  time = 0.0700957, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {801, 633, 31} \[ -\frac{a^3 (a e+c d) \log (a-c x)}{2 c^6}+\frac{a^3 (c d-a e) \log (a+c x)}{2 c^6}-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x))/(a^2 - c^2*x^2),x]

[Out]

-((a^2*d*x)/c^4) - (a^2*e*x^2)/(2*c^4) - (d*x^3)/(3*c^2) - (e*x^4)/(4*c^2) - (a^3*(c*d + a*e)*Log[a - c*x])/(2
*c^6) + (a^3*(c*d - a*e)*Log[a + c*x])/(2*c^6)

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^4 (d+e x)}{a^2-c^2 x^2} \, dx &=\int \left (-\frac{a^2 d}{c^4}-\frac{a^2 e x}{c^4}-\frac{d x^2}{c^2}-\frac{e x^3}{c^2}+\frac{a^4 d+a^4 e x}{c^4 \left (a^2-c^2 x^2\right )}\right ) \, dx\\ &=-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2}+\frac{\int \frac{a^4 d+a^4 e x}{a^2-c^2 x^2} \, dx}{c^4}\\ &=-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2}-\frac{\left (a^3 (c d-a e)\right ) \int \frac{1}{-a c-c^2 x} \, dx}{2 c^4}+\frac{\left (a^3 (c d+a e)\right ) \int \frac{1}{a c-c^2 x} \, dx}{2 c^4}\\ &=-\frac{a^2 d x}{c^4}-\frac{a^2 e x^2}{2 c^4}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2}-\frac{a^3 (c d+a e) \log (a-c x)}{2 c^6}+\frac{a^3 (c d-a e) \log (a+c x)}{2 c^6}\\ \end{align*}

Mathematica [A]  time = 0.0158961, size = 86, normalized size = 0.91 \[ -\frac{a^2 d x}{c^4}+\frac{a^3 d \tanh ^{-1}\left (\frac{c x}{a}\right )}{c^5}-\frac{a^2 e x^2}{2 c^4}-\frac{a^4 e \log \left (a^2-c^2 x^2\right )}{2 c^6}-\frac{d x^3}{3 c^2}-\frac{e x^4}{4 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x))/(a^2 - c^2*x^2),x]

[Out]

-((a^2*d*x)/c^4) - (a^2*e*x^2)/(2*c^4) - (d*x^3)/(3*c^2) - (e*x^4)/(4*c^2) + (a^3*d*ArcTanh[(c*x)/a])/c^5 - (a
^4*e*Log[a^2 - c^2*x^2])/(2*c^6)

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Maple [A]  time = 0.009, size = 106, normalized size = 1.1 \begin{align*} -{\frac{e{x}^{4}}{4\,{c}^{2}}}-{\frac{d{x}^{3}}{3\,{c}^{2}}}-{\frac{{a}^{2}e{x}^{2}}{2\,{c}^{4}}}-{\frac{{a}^{2}dx}{{c}^{4}}}-{\frac{{a}^{4}\ln \left ( cx+a \right ) e}{2\,{c}^{6}}}+{\frac{{a}^{3}\ln \left ( cx+a \right ) d}{2\,{c}^{5}}}-{\frac{{a}^{4}\ln \left ( cx-a \right ) e}{2\,{c}^{6}}}-{\frac{{a}^{3}\ln \left ( cx-a \right ) d}{2\,{c}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)/(-c^2*x^2+a^2),x)

[Out]

-1/4*e*x^4/c^2-1/3*d*x^3/c^2-1/2*a^2*e*x^2/c^4-a^2*d*x/c^4-1/2/c^6*a^4*ln(c*x+a)*e+1/2/c^5*a^3*ln(c*x+a)*d-1/2
/c^6*a^4*ln(c*x-a)*e-1/2/c^5*a^3*ln(c*x-a)*d

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Maxima [A]  time = 1.0776, size = 122, normalized size = 1.28 \begin{align*} -\frac{3 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} + 6 \, a^{2} e x^{2} + 12 \, a^{2} d x}{12 \, c^{4}} + \frac{{\left (a^{3} c d - a^{4} e\right )} \log \left (c x + a\right )}{2 \, c^{6}} - \frac{{\left (a^{3} c d + a^{4} e\right )} \log \left (c x - a\right )}{2 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(-c^2*x^2+a^2),x, algorithm="maxima")

[Out]

-1/12*(3*c^2*e*x^4 + 4*c^2*d*x^3 + 6*a^2*e*x^2 + 12*a^2*d*x)/c^4 + 1/2*(a^3*c*d - a^4*e)*log(c*x + a)/c^6 - 1/
2*(a^3*c*d + a^4*e)*log(c*x - a)/c^6

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Fricas [A]  time = 1.49947, size = 194, normalized size = 2.04 \begin{align*} -\frac{3 \, c^{4} e x^{4} + 4 \, c^{4} d x^{3} + 6 \, a^{2} c^{2} e x^{2} + 12 \, a^{2} c^{2} d x - 6 \,{\left (a^{3} c d - a^{4} e\right )} \log \left (c x + a\right ) + 6 \,{\left (a^{3} c d + a^{4} e\right )} \log \left (c x - a\right )}{12 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(-c^2*x^2+a^2),x, algorithm="fricas")

[Out]

-1/12*(3*c^4*e*x^4 + 4*c^4*d*x^3 + 6*a^2*c^2*e*x^2 + 12*a^2*c^2*d*x - 6*(a^3*c*d - a^4*e)*log(c*x + a) + 6*(a^
3*c*d + a^4*e)*log(c*x - a))/c^6

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Sympy [A]  time = 0.580899, size = 129, normalized size = 1.36 \begin{align*} - \frac{a^{3} \left (a e - c d\right ) \log{\left (x + \frac{a^{4} e - a^{3} \left (a e - c d\right )}{a^{2} c^{2} d} \right )}}{2 c^{6}} - \frac{a^{3} \left (a e + c d\right ) \log{\left (x + \frac{a^{4} e - a^{3} \left (a e + c d\right )}{a^{2} c^{2} d} \right )}}{2 c^{6}} - \frac{a^{2} d x}{c^{4}} - \frac{a^{2} e x^{2}}{2 c^{4}} - \frac{d x^{3}}{3 c^{2}} - \frac{e x^{4}}{4 c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(-c**2*x**2+a**2),x)

[Out]

-a**3*(a*e - c*d)*log(x + (a**4*e - a**3*(a*e - c*d))/(a**2*c**2*d))/(2*c**6) - a**3*(a*e + c*d)*log(x + (a**4
*e - a**3*(a*e + c*d))/(a**2*c**2*d))/(2*c**6) - a**2*d*x/c**4 - a**2*e*x**2/(2*c**4) - d*x**3/(3*c**2) - e*x*
*4/(4*c**2)

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Giac [A]  time = 1.11857, size = 138, normalized size = 1.45 \begin{align*} \frac{{\left (a^{3} c d - a^{4} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, c^{6}} - \frac{{\left (a^{3} c d + a^{4} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, c^{6}} - \frac{3 \, c^{6} x^{4} e + 4 \, c^{6} d x^{3} + 6 \, a^{2} c^{4} x^{2} e + 12 \, a^{2} c^{4} d x}{12 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(-c^2*x^2+a^2),x, algorithm="giac")

[Out]

1/2*(a^3*c*d - a^4*e)*log(abs(c*x + a))/c^6 - 1/2*(a^3*c*d + a^4*e)*log(abs(c*x - a))/c^6 - 1/12*(3*c^6*x^4*e
+ 4*c^6*d*x^3 + 6*a^2*c^4*x^2*e + 12*a^2*c^4*d*x)/c^8