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

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

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Rubi [A]  time = 0.09, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {823, 801} \begin {gather*} \frac {d+e x}{2 a^2 x \left (a^2-c^2 x^2\right )}-\frac {(2 a e+3 c d) \log (a-c x)}{4 a^5}+\frac {(3 c d-2 a e) \log (a+c x)}{4 a^5}-\frac {3 d}{2 a^4 x}+\frac {e \log (x)}{a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 77, normalized size = 0.83 \begin {gather*} \frac {-a e \log \left (a^2-c^2 x^2\right )+\frac {a^3 e+a c^2 d x}{a^2-c^2 x^2}+3 c d \tanh ^{-1}\left (\frac {c x}{a}\right )-\frac {2 a d}{x}+2 a e \log (x)}{2 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 155, normalized size = 1.67 \begin {gather*} -\frac {6 \, a c^{2} d x^{2} + 2 \, a^{3} e x - 4 \, a^{3} d - {\left ({\left (3 \, c^{3} d - 2 \, a c^{2} e\right )} x^{3} - {\left (3 \, a^{2} c d - 2 \, a^{3} e\right )} x\right )} \log \left (c x + a\right ) + {\left ({\left (3 \, c^{3} d + 2 \, a c^{2} e\right )} x^{3} - {\left (3 \, a^{2} c d + 2 \, a^{3} e\right )} x\right )} \log \left (c x - a\right ) - 4 \, {\left (a c^{2} e x^{3} - a^{3} e x\right )} \log \relax (x)}{4 \, {\left (a^{5} c^{2} x^{3} - a^{7} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(6*a*c^2*d*x^2 + 2*a^3*e*x - 4*a^3*d - ((3*c^3*d - 2*a*c^2*e)*x^3 - (3*a^2*c*d - 2*a^3*e)*x)*log(c*x + a)
+ ((3*c^3*d + 2*a*c^2*e)*x^3 - (3*a^2*c*d + 2*a^3*e)*x)*log(c*x - a) - 4*(a*c^2*e*x^3 - a^3*e*x)*log(x))/(a^5
*c^2*x^3 - a^7*x)

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giac [A]  time = 0.16, size = 112, normalized size = 1.20 \begin {gather*} \frac {e \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {3 \, c^{2} d x^{2} + a^{2} x e - 2 \, a^{2} d}{2 \, {\left (c^{2} x^{3} - a^{2} x\right )} a^{4}} + \frac {{\left (3 \, c^{2} d - 2 \, a c e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{5} c} - \frac {{\left (3 \, c^{2} d + 2 \, a c e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{5} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 130, normalized size = 1.40 \begin {gather*} \frac {e}{4 \left (c x +a \right ) a^{3}}-\frac {e}{4 \left (c x -a \right ) a^{3}}-\frac {c d}{4 \left (c x +a \right ) a^{4}}-\frac {c d}{4 \left (c x -a \right ) a^{4}}+\frac {e \ln \relax (x )}{a^{4}}-\frac {e \ln \left (c x -a \right )}{2 a^{4}}-\frac {e \ln \left (c x +a \right )}{2 a^{4}}-\frac {3 c d \ln \left (c x -a \right )}{4 a^{5}}+\frac {3 c d \ln \left (c x +a \right )}{4 a^{5}}-\frac {d}{a^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.47, size = 93, normalized size = 1.00 \begin {gather*} -\frac {3 \, c^{2} d x^{2} + a^{2} e x - 2 \, a^{2} d}{2 \, {\left (a^{4} c^{2} x^{3} - a^{6} x\right )}} + \frac {e \log \relax (x)}{a^{4}} + \frac {{\left (3 \, c d - 2 \, a e\right )} \log \left (c x + a\right )}{4 \, a^{5}} - \frac {{\left (3 \, c d + 2 \, a e\right )} \log \left (c x - a\right )}{4 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.11, size = 92, normalized size = 0.99 \begin {gather*} \frac {\frac {e\,x}{2\,a^2}-\frac {d}{a^2}+\frac {3\,c^2\,d\,x^2}{2\,a^4}}{a^2\,x-c^2\,x^3}-\frac {\ln \left (a+c\,x\right )\,\left (2\,a\,e-3\,c\,d\right )}{4\,a^5}-\frac {\ln \left (a-c\,x\right )\,\left (2\,a\,e+3\,c\,d\right )}{4\,a^5}+\frac {e\,\ln \relax (x)}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.95, size = 291, normalized size = 3.13 \begin {gather*} \frac {2 a^{2} d - a^{2} e x - 3 c^{2} d x^{2}}{- 2 a^{6} x + 2 a^{4} c^{2} x^{3}} + \frac {e \log {\relax (x )}}{a^{4}} - \frac {\left (2 a e - 3 c d\right ) \log {\left (x + \frac {16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e - 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e - 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e - 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} - \frac {\left (2 a e + 3 c d\right ) \log {\left (x + \frac {16 a^{4} e^{3} - 4 a^{3} e^{2} \left (2 a e + 3 c d\right ) + 12 a^{2} c^{2} d^{2} e - 2 a^{2} e \left (2 a e + 3 c d\right )^{2} + 3 a c^{2} d^{2} \left (2 a e + 3 c d\right )}{36 a^{2} c^{2} d e^{2} - 9 c^{4} d^{3}} \right )}}{4 a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a**2*d - a**2*e*x - 3*c**2*d*x**2)/(-2*a**6*x + 2*a**4*c**2*x**3) + e*log(x)/a**4 - (2*a*e - 3*c*d)*log(x +
(16*a**4*e**3 - 4*a**3*e**2*(2*a*e - 3*c*d) + 12*a**2*c**2*d**2*e - 2*a**2*e*(2*a*e - 3*c*d)**2 + 3*a*c**2*d*
*2*(2*a*e - 3*c*d))/(36*a**2*c**2*d*e**2 - 9*c**4*d**3))/(4*a**5) - (2*a*e + 3*c*d)*log(x + (16*a**4*e**3 - 4*
a**3*e**2*(2*a*e + 3*c*d) + 12*a**2*c**2*d**2*e - 2*a**2*e*(2*a*e + 3*c*d)**2 + 3*a*c**2*d**2*(2*a*e + 3*c*d))
/(36*a**2*c**2*d*e**2 - 9*c**4*d**3))/(4*a**5)

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