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

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

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Rubi [A]  time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 84, normalized size = 0.90 \begin {gather*} \frac {c^3 d \tanh ^{-1}\left (\frac {c x}{a}\right )}{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}-\frac {c^2 e \log \left (a^2-c^2 x^2\right )}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*d/(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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x^4 \left (a^2-c^2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 93, normalized size = 1.00 \begin {gather*} \frac {6 \, a c^{2} e x^{3} \log \relax (x) - 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 {gather*}

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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giac [A]  time = 0.17, size = 104, normalized size = 1.12 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.06, size = 106, normalized size = 1.14 \begin {gather*} \frac {c^{2} e \ln \relax (x )}{a^{4}}-\frac {c^{2} e \ln \left (c x -a \right )}{2 a^{4}}-\frac {c^{2} e \ln \left (c x +a \right )}{2 a^{4}}-\frac {c^{3} d \ln \left (c x -a \right )}{2 a^{5}}+\frac {c^{3} d \ln \left (c x +a \right )}{2 a^{5}}-\frac {c^{2} d}{a^{4} x}-\frac {e}{2 a^{2} x^{2}}-\frac {d}{3 a^{2} x^{3}} \end {gather*}

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]

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

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maxima [A]  time = 0.50, size = 91, normalized size = 0.98 \begin {gather*} \frac {c^{2} e \log \relax (x)}{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 {gather*}

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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mupad [B]  time = 1.11, size = 89, normalized size = 0.96 \begin {gather*} \frac {\ln \left (a+c\,x\right )\,\left (c^3\,d-a\,c^2\,e\right )}{2\,a^5}-\frac {\frac {d}{3\,a^2}+\frac {e\,x}{2\,a^2}+\frac {c^2\,d\,x^2}{a^4}}{x^3}-\frac {\ln \left (a-c\,x\right )\,\left (d\,c^3+a\,e\,c^2\right )}{2\,a^5}+\frac {c^2\,e\,\ln \relax (x)}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 2.00, size = 279, normalized size = 3.00 \begin {gather*} \frac {c^{2} e \log {\relax (x )}}{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 {gather*}

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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