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

Optimal. Leaf size=93 \[ -\frac{c^2 (a e+c d) \log (a-c x)}{2 a^5}+\frac{c^2 (c d-a e) \log (a+c x)}{2 a^5}-\frac{c^2 d}{a^4 x}+\frac{c^2 e \log (x)}{a^4}-\frac{d}{3 a^2 x^3}-\frac{e}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0691779, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {801} \[ -\frac{c^2 (a e+c d) \log (a-c x)}{2 a^5}+\frac{c^2 (c d-a e) \log (a+c x)}{2 a^5}-\frac{c^2 d}{a^4 x}+\frac{c^2 e \log (x)}{a^4}-\frac{d}{3 a^2 x^3}-\frac{e}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

\begin{align*} \int \frac{d+e x}{x^4 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac{d}{a^2 x^4}+\frac{e}{a^2 x^3}+\frac{c^2 d}{a^4 x^2}+\frac{c^2 e}{a^4 x}+\frac{c^3 (c d+a e)}{2 a^5 (a-c x)}-\frac{c^3 (-c d+a e)}{2 a^5 (a+c x)}\right ) \, dx\\ &=-\frac{d}{3 a^2 x^3}-\frac{e}{2 a^2 x^2}-\frac{c^2 d}{a^4 x}+\frac{c^2 e \log (x)}{a^4}-\frac{c^2 (c d+a e) \log (a-c x)}{2 a^5}+\frac{c^2 (c d-a e) \log (a+c x)}{2 a^5}\\ \end{align*}

Mathematica [A]  time = 0.0161108, size = 84, normalized size = 0.9 \[ -\frac{c^2 d}{a^4 x}+\frac{c^3 d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a^5}-\frac{c^2 e \log \left (a^2-c^2 x^2\right )}{2 a^4}+\frac{c^2 e \log (x)}{a^4}-\frac{d}{3 a^2 x^3}-\frac{e}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02836, size = 123, normalized size = 1.32 \begin{align*} \frac{c^{2} e \log \left (x\right )}{a^{4}} + \frac{{\left (c^{3} d - a c^{2} e\right )} \log \left (c x + a\right )}{2 \, a^{5}} - \frac{{\left (c^{3} d + a c^{2} e\right )} \log \left (c x - a\right )}{2 \, a^{5}} - \frac{6 \, c^{2} d x^{2} + 3 \, a^{2} e x + 2 \, a^{2} d}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6332, size = 208, normalized size = 2.24 \begin{align*} \frac{6 \, a c^{2} e x^{3} \log \left (x\right ) - 6 \, a c^{2} d x^{2} - 3 \, a^{3} e x + 3 \,{\left (c^{3} d - a c^{2} e\right )} x^{3} \log \left (c x + a\right ) - 3 \,{\left (c^{3} d + a c^{2} e\right )} x^{3} \log \left (c x - a\right ) - 2 \, a^{3} d}{6 \, a^{5} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.2913, size = 279, normalized size = 3. \begin{align*} \frac{c^{2} e \log{\left (x \right )}}{a^{4}} - \frac{2 a^{2} d + 3 a^{2} e x + 6 c^{2} d x^{2}}{6 a^{4} x^{3}} - \frac{c^{2} \left (a e - c d\right ) \log{\left (x + \frac{6 a^{4} c^{4} e^{3} - 3 a^{3} c^{4} e^{2} \left (a e - c d\right ) + 2 a^{2} c^{6} d^{2} e - 3 a^{2} c^{4} e \left (a e - c d\right )^{2} + a c^{6} d^{2} \left (a e - c d\right )}{9 a^{2} c^{6} d e^{2} - c^{8} d^{3}} \right )}}{2 a^{5}} - \frac{c^{2} \left (a e + c d\right ) \log{\left (x + \frac{6 a^{4} c^{4} e^{3} - 3 a^{3} c^{4} e^{2} \left (a e + c d\right ) + 2 a^{2} c^{6} d^{2} e - 3 a^{2} c^{4} e \left (a e + c d\right )^{2} + a c^{6} d^{2} \left (a e + c d\right )}{9 a^{2} c^{6} d e^{2} - c^{8} d^{3}} \right )}}{2 a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*e*log(x)/a**4 - (2*a**2*d + 3*a**2*e*x + 6*c**2*d*x**2)/(6*a**4*x**3) - c**2*(a*e - c*d)*log(x + (6*a**4*
c**4*e**3 - 3*a**3*c**4*e**2*(a*e - c*d) + 2*a**2*c**6*d**2*e - 3*a**2*c**4*e*(a*e - c*d)**2 + a*c**6*d**2*(a*
e - c*d))/(9*a**2*c**6*d*e**2 - c**8*d**3))/(2*a**5) - c**2*(a*e + c*d)*log(x + (6*a**4*c**4*e**3 - 3*a**3*c**
4*e**2*(a*e + c*d) + 2*a**2*c**6*d**2*e - 3*a**2*c**4*e*(a*e + c*d)**2 + a*c**6*d**2*(a*e + c*d))/(9*a**2*c**6
*d*e**2 - c**8*d**3))/(2*a**5)

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Giac [A]  time = 1.12757, size = 140, normalized size = 1.51 \begin{align*} \frac{c^{2} e \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{{\left (c^{4} d - a c^{3} e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{5} c} - \frac{{\left (c^{4} d + a c^{3} e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{5} c} - \frac{6 \, c^{2} d x^{2} + 3 \, a^{2} x e + 2 \, a^{2} d}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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