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

Optimal. Leaf size=94 \[ \frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {819, 801, 633, 31} \begin {gather*} \frac {x^3 (d+e x)}{2 c^2 \left (a^2-c^2 x^2\right )}+\frac {a (4 a e+3 c d) \log (a-c x)}{4 c^6}-\frac {a (3 c d-4 a e) \log (a+c x)}{4 c^6}+\frac {3 d x}{2 c^4}+\frac {e x^2}{c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

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 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 819

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 84, normalized size = 0.89 \begin {gather*} \frac {2 a^2 e \log \left (a^2-c^2 x^2\right )+\frac {a^4 e+a^2 c^2 d x}{a^2-c^2 x^2}-3 a c d \tanh ^{-1}\left (\frac {c x}{a}\right )+2 c^2 d x+c^2 e x^2}{2 c^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 156, normalized size = 1.66 \begin {gather*} \frac {2 \, c^{4} e x^{4} + 4 \, c^{4} d x^{3} - 2 \, a^{2} c^{2} e x^{2} - 6 \, a^{2} c^{2} d x - 2 \, a^{4} e + {\left (3 \, a^{3} c d - 4 \, a^{4} e - {\left (3 \, a c^{3} d - 4 \, a^{2} c^{2} e\right )} x^{2}\right )} \log \left (c x + a\right ) - {\left (3 \, a^{3} c d + 4 \, a^{4} e - {\left (3 \, a c^{3} d + 4 \, a^{2} c^{2} e\right )} x^{2}\right )} \log \left (c x - a\right )}{4 \, {\left (c^{8} x^{2} - a^{2} c^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 112, normalized size = 1.19 \begin {gather*} -\frac {{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left ({\left | c x + a \right |}\right )}{4 \, c^{6}} + \frac {{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left ({\left | c x - a \right |}\right )}{4 \, c^{6}} + \frac {c^{4} x^{2} e + 2 \, c^{4} d x}{2 \, c^{8}} - \frac {a^{2} c^{2} d x + a^{4} e}{2 \, {\left (c x + a\right )} {\left (c x - a\right )} c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 143, normalized size = 1.52 \begin {gather*} \frac {e \,x^{2}}{2 c^{4}}+\frac {a^{3} e}{4 \left (c x +a \right ) c^{6}}-\frac {a^{3} e}{4 \left (c x -a \right ) c^{6}}-\frac {a^{2} d}{4 \left (c x +a \right ) c^{5}}-\frac {a^{2} d}{4 \left (c x -a \right ) c^{5}}+\frac {a^{2} e \ln \left (c x -a \right )}{c^{6}}+\frac {a^{2} e \ln \left (c x +a \right )}{c^{6}}+\frac {3 a d \ln \left (c x -a \right )}{4 c^{5}}-\frac {3 a d \ln \left (c x +a \right )}{4 c^{5}}+\frac {d x}{c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 99, normalized size = 1.05 \begin {gather*} -\frac {a^{2} c^{2} d x + a^{4} e}{2 \, {\left (c^{8} x^{2} - a^{2} c^{6}\right )}} + \frac {e x^{2} + 2 \, d x}{2 \, c^{4}} - \frac {{\left (3 \, a c d - 4 \, a^{2} e\right )} \log \left (c x + a\right )}{4 \, c^{6}} + \frac {{\left (3 \, a c d + 4 \, a^{2} e\right )} \log \left (c x - a\right )}{4 \, c^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.09, size = 99, normalized size = 1.05 \begin {gather*} \frac {\frac {a^4\,e}{2\,c^2}+\frac {a^2\,d\,x}{2}}{a^2\,c^4-c^6\,x^2}+\frac {e\,x^2}{2\,c^4}+\frac {\ln \left (a+c\,x\right )\,\left (4\,a^2\,e-3\,a\,c\,d\right )}{4\,c^6}+\frac {\ln \left (a-c\,x\right )\,\left (4\,e\,a^2+3\,c\,d\,a\right )}{4\,c^6}+\frac {d\,x}{c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.80, size = 141, normalized size = 1.50 \begin {gather*} \frac {a \left (4 a e - 3 c d\right ) \log {\left (x + \frac {4 a^{2} e - a \left (4 a e - 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac {a \left (4 a e + 3 c d\right ) \log {\left (x + \frac {4 a^{2} e - a \left (4 a e + 3 c d\right )}{3 c^{2} d} \right )}}{4 c^{6}} + \frac {- a^{4} e - a^{2} c^{2} d x}{- 2 a^{2} c^{6} + 2 c^{8} x^{2}} + \frac {d x}{c^{4}} + \frac {e x^{2}}{2 c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________