3.4.13 \(\int \frac {d+e x}{x^3 (a^2-c^2 x^2)^2} \, dx\)

Optimal. Leaf size=108 \[ \frac {2 c^2 d \log (x)}{a^6}-\frac {c (3 a e+4 c d) \log (a-c x)}{4 a^6}-\frac {c (4 c d-3 a e) \log (a+c x)}{4 a^6}-\frac {d}{a^4 x^2}-\frac {3 e}{2 a^4 x}+\frac {d+e x}{2 a^2 x^2 \left (a^2-c^2 x^2\right )} \]

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Rubi [A]  time = 0.11, antiderivative size = 108, 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^2 \left (a^2-c^2 x^2\right )}+\frac {2 c^2 d \log (x)}{a^6}-\frac {c (3 a e+4 c d) \log (a-c x)}{4 a^6}-\frac {c (4 c d-3 a e) \log (a+c x)}{4 a^6}-\frac {d}{a^4 x^2}-\frac {3 e}{2 a^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(x^3*(a^2 - c^2*x^2)^2),x]

[Out]

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

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Mathematica [A]  time = 0.07, size = 91, normalized size = 0.84 \begin {gather*} \frac {\frac {a^2 c^2 (d+e x)}{a^2-c^2 x^2}-2 c^2 d \log \left (a^2-c^2 x^2\right )-\frac {a^2 d}{x^2}-\frac {2 a^2 e}{x}+3 a c e \tanh ^{-1}\left (\frac {c x}{a}\right )+4 c^2 d \log (x)}{2 a^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(x^3*(a^2 - c^2*x^2)^2),x]

[Out]

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

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

[Out]

IntegrateAlgebraic[(d + e*x)/(x^3*(a^2 - c^2*x^2)^2), x]

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fricas [A]  time = 0.43, size = 184, normalized size = 1.70 \begin {gather*} -\frac {6 \, a^{2} c^{2} e x^{3} + 4 \, a^{2} c^{2} d x^{2} - 4 \, a^{4} e x - 2 \, a^{4} d + {\left ({\left (4 \, c^{4} d - 3 \, a c^{3} e\right )} x^{4} - {\left (4 \, a^{2} c^{2} d - 3 \, a^{3} c e\right )} x^{2}\right )} \log \left (c x + a\right ) + {\left ({\left (4 \, c^{4} d + 3 \, a c^{3} e\right )} x^{4} - {\left (4 \, a^{2} c^{2} d + 3 \, a^{3} c e\right )} x^{2}\right )} \log \left (c x - a\right ) - 8 \, {\left (c^{4} d x^{4} - a^{2} c^{2} d x^{2}\right )} \log \relax (x)}{4 \, {\left (a^{6} c^{2} x^{4} - a^{8} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-c^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 139, normalized size = 1.29 \begin {gather*} \frac {2 \, c^{2} d \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {{\left (4 \, c^{3} d - 3 \, a c^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, a^{6} c} - \frac {{\left (4 \, c^{3} d + 3 \, a c^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, a^{6} c} - \frac {3 \, a^{2} c^{2} x^{3} e + 2 \, a^{2} c^{2} d x^{2} - 2 \, a^{4} x e - a^{4} d}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} a^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-c^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 155, normalized size = 1.44 \begin {gather*} -\frac {c e}{4 \left (c x +a \right ) a^{4}}-\frac {c e}{4 \left (c x -a \right ) a^{4}}+\frac {c^{2} d}{4 \left (c x +a \right ) a^{5}}-\frac {c^{2} d}{4 \left (c x -a \right ) a^{5}}-\frac {3 c e \ln \left (c x -a \right )}{4 a^{5}}+\frac {3 c e \ln \left (c x +a \right )}{4 a^{5}}+\frac {2 c^{2} d \ln \relax (x )}{a^{6}}-\frac {c^{2} d \ln \left (c x -a \right )}{a^{6}}-\frac {c^{2} d \ln \left (c x +a \right )}{a^{6}}-\frac {e}{a^{4} x}-\frac {d}{2 a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x^3/(-c^2*x^2+a^2)^2,x)

[Out]

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

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maxima [A]  time = 0.53, size = 115, normalized size = 1.06 \begin {gather*} -\frac {3 \, c^{2} e x^{3} + 2 \, c^{2} d x^{2} - 2 \, a^{2} e x - a^{2} d}{2 \, {\left (a^{4} c^{2} x^{4} - a^{6} x^{2}\right )}} + \frac {2 \, c^{2} d \log \relax (x)}{a^{6}} - \frac {{\left (4 \, c^{2} d - 3 \, a c e\right )} \log \left (c x + a\right )}{4 \, a^{6}} - \frac {{\left (4 \, c^{2} d + 3 \, a c e\right )} \log \left (c x - a\right )}{4 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^3/(-c^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.11, size = 116, normalized size = 1.07 \begin {gather*} \frac {2\,c^2\,d\,\ln \relax (x)}{a^6}-\frac {\ln \left (a+c\,x\right )\,\left (4\,c^2\,d-3\,a\,c\,e\right )}{4\,a^6}-\frac {\ln \left (a-c\,x\right )\,\left (4\,d\,c^2+3\,a\,e\,c\right )}{4\,a^6}-\frac {\frac {d}{2\,a^2}+\frac {e\,x}{a^2}-\frac {c^2\,d\,x^2}{a^4}-\frac {3\,c^2\,e\,x^3}{2\,a^4}}{a^2\,x^2-c^2\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(x^3*(a^2 - c^2*x^2)^2),x)

[Out]

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

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sympy [B]  time = 2.30, size = 311, normalized size = 2.88 \begin {gather*} \frac {a^{2} d + 2 a^{2} e x - 2 c^{2} d x^{2} - 3 c^{2} e x^{3}}{- 2 a^{6} x^{2} + 2 a^{4} c^{2} x^{4}} + \frac {2 c^{2} d \log {\relax (x )}}{a^{6}} + \frac {c \left (3 a e - 4 c d\right ) \log {\left (x + \frac {- 24 a^{2} c^{2} d e^{2} + 3 a^{2} c e^{2} \left (3 a e - 4 c d\right ) - 128 c^{4} d^{3} - 16 c^{3} d^{2} \left (3 a e - 4 c d\right ) + 4 c^{2} d \left (3 a e - 4 c d\right )^{2}}{9 a^{2} c^{2} e^{3} - 144 c^{4} d^{2} e} \right )}}{4 a^{6}} - \frac {c \left (3 a e + 4 c d\right ) \log {\left (x + \frac {- 24 a^{2} c^{2} d e^{2} - 3 a^{2} c e^{2} \left (3 a e + 4 c d\right ) - 128 c^{4} d^{3} + 16 c^{3} d^{2} \left (3 a e + 4 c d\right ) + 4 c^{2} d \left (3 a e + 4 c d\right )^{2}}{9 a^{2} c^{2} e^{3} - 144 c^{4} d^{2} e} \right )}}{4 a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x**3/(-c**2*x**2+a**2)**2,x)

[Out]

(a**2*d + 2*a**2*e*x - 2*c**2*d*x**2 - 3*c**2*e*x**3)/(-2*a**6*x**2 + 2*a**4*c**2*x**4) + 2*c**2*d*log(x)/a**6
+ c*(3*a*e - 4*c*d)*log(x + (-24*a**2*c**2*d*e**2 + 3*a**2*c*e**2*(3*a*e - 4*c*d) - 128*c**4*d**3 - 16*c**3*d
**2*(3*a*e - 4*c*d) + 4*c**2*d*(3*a*e - 4*c*d)**2)/(9*a**2*c**2*e**3 - 144*c**4*d**2*e))/(4*a**6) - c*(3*a*e +
4*c*d)*log(x + (-24*a**2*c**2*d*e**2 - 3*a**2*c*e**2*(3*a*e + 4*c*d) - 128*c**4*d**3 + 16*c**3*d**2*(3*a*e +
4*c*d) + 4*c**2*d*(3*a*e + 4*c*d)**2)/(9*a**2*c**2*e**3 - 144*c**4*d**2*e))/(4*a**6)

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