3.775 \(\int \frac{1}{(a+b x) (a^2-b^2 x^2)^3} \, dx\)

Optimal. Leaf size=105 \[ \frac{1}{8 a^5 b (a-b x)}-\frac{3}{16 a^5 b (a+b x)}+\frac{1}{32 a^4 b (a-b x)^2}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{1}{24 a^3 b (a+b x)^3}+\frac{5 \tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^6 b} \]

[Out]

1/(32*a^4*b*(a - b*x)^2) + 1/(8*a^5*b*(a - b*x)) - 1/(24*a^3*b*(a + b*x)^3) - 3/(32*a^4*b*(a + b*x)^2) - 3/(16
*a^5*b*(a + b*x)) + (5*ArcTanh[(b*x)/a])/(16*a^6*b)

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Rubi [A]  time = 0.0655311, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {627, 44, 208} \[ \frac{1}{8 a^5 b (a-b x)}-\frac{3}{16 a^5 b (a+b x)}+\frac{1}{32 a^4 b (a-b x)^2}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{1}{24 a^3 b (a+b x)^3}+\frac{5 \tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^6 b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(a^2 - b^2*x^2)^3),x]

[Out]

1/(32*a^4*b*(a - b*x)^2) + 1/(8*a^5*b*(a - b*x)) - 1/(24*a^3*b*(a + b*x)^3) - 3/(32*a^4*b*(a + b*x)^2) - 3/(16
*a^5*b*(a + b*x)) + (5*ArcTanh[(b*x)/a])/(16*a^6*b)

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x) \left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{1}{(a-b x)^3 (a+b x)^4} \, dx\\ &=\int \left (\frac{1}{16 a^4 (a-b x)^3}+\frac{1}{8 a^5 (a-b x)^2}+\frac{1}{8 a^3 (a+b x)^4}+\frac{3}{16 a^4 (a+b x)^3}+\frac{3}{16 a^5 (a+b x)^2}+\frac{5}{16 a^5 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{32 a^4 b (a-b x)^2}+\frac{1}{8 a^5 b (a-b x)}-\frac{1}{24 a^3 b (a+b x)^3}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{3}{16 a^5 b (a+b x)}+\frac{5 \int \frac{1}{a^2-b^2 x^2} \, dx}{16 a^5}\\ &=\frac{1}{32 a^4 b (a-b x)^2}+\frac{1}{8 a^5 b (a-b x)}-\frac{1}{24 a^3 b (a+b x)^3}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{3}{16 a^5 b (a+b x)}+\frac{5 \tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^6 b}\\ \end{align*}

Mathematica [A]  time = 0.0429631, size = 87, normalized size = 0.83 \[ \frac{-\frac{2 a \left (-25 a^2 b^2 x^2-25 a^3 b x+8 a^4+15 a b^3 x^3+15 b^4 x^4\right )}{(a-b x)^2 (a+b x)^3}-15 \log (a-b x)+15 \log (a+b x)}{96 a^6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(a^2 - b^2*x^2)^3),x]

[Out]

((-2*a*(8*a^4 - 25*a^3*b*x - 25*a^2*b^2*x^2 + 15*a*b^3*x^3 + 15*b^4*x^4))/((a - b*x)^2*(a + b*x)^3) - 15*Log[a
- b*x] + 15*Log[a + b*x])/(96*a^6*b)

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Maple [A]  time = 0.051, size = 111, normalized size = 1.1 \begin{align*}{\frac{5\,\ln \left ( bx+a \right ) }{32\,{a}^{6}b}}-{\frac{3}{16\,{a}^{5}b \left ( bx+a \right ) }}-{\frac{3}{32\,{a}^{4}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{24\,{a}^{3}b \left ( bx+a \right ) ^{3}}}-{\frac{5\,\ln \left ( bx-a \right ) }{32\,{a}^{6}b}}-{\frac{1}{8\,{a}^{5}b \left ( bx-a \right ) }}+{\frac{1}{32\,{a}^{4}b \left ( bx-a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(-b^2*x^2+a^2)^3,x)

[Out]

5/32/a^6/b*ln(b*x+a)-3/16/a^5/b/(b*x+a)-3/32/a^4/b/(b*x+a)^2-1/24/a^3/b/(b*x+a)^3-5/32/a^6/b*ln(b*x-a)-1/8/a^5
/b/(b*x-a)+1/32/a^4/b/(b*x-a)^2

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Maxima [A]  time = 1.29289, size = 178, normalized size = 1.7 \begin{align*} -\frac{15 \, b^{4} x^{4} + 15 \, a b^{3} x^{3} - 25 \, a^{2} b^{2} x^{2} - 25 \, a^{3} b x + 8 \, a^{4}}{48 \,{\left (a^{5} b^{6} x^{5} + a^{6} b^{5} x^{4} - 2 \, a^{7} b^{4} x^{3} - 2 \, a^{8} b^{3} x^{2} + a^{9} b^{2} x + a^{10} b\right )}} + \frac{5 \, \log \left (b x + a\right )}{32 \, a^{6} b} - \frac{5 \, \log \left (b x - a\right )}{32 \, a^{6} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/48*(15*b^4*x^4 + 15*a*b^3*x^3 - 25*a^2*b^2*x^2 - 25*a^3*b*x + 8*a^4)/(a^5*b^6*x^5 + a^6*b^5*x^4 - 2*a^7*b^4
*x^3 - 2*a^8*b^3*x^2 + a^9*b^2*x + a^10*b) + 5/32*log(b*x + a)/(a^6*b) - 5/32*log(b*x - a)/(a^6*b)

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Fricas [B]  time = 1.81665, size = 454, normalized size = 4.32 \begin{align*} -\frac{30 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 50 \, a^{3} b^{2} x^{2} - 50 \, a^{4} b x + 16 \, a^{5} - 15 \,{\left (b^{5} x^{5} + a b^{4} x^{4} - 2 \, a^{2} b^{3} x^{3} - 2 \, a^{3} b^{2} x^{2} + a^{4} b x + a^{5}\right )} \log \left (b x + a\right ) + 15 \,{\left (b^{5} x^{5} + a b^{4} x^{4} - 2 \, a^{2} b^{3} x^{3} - 2 \, a^{3} b^{2} x^{2} + a^{4} b x + a^{5}\right )} \log \left (b x - a\right )}{96 \,{\left (a^{6} b^{6} x^{5} + a^{7} b^{5} x^{4} - 2 \, a^{8} b^{4} x^{3} - 2 \, a^{9} b^{3} x^{2} + a^{10} b^{2} x + a^{11} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/96*(30*a*b^4*x^4 + 30*a^2*b^3*x^3 - 50*a^3*b^2*x^2 - 50*a^4*b*x + 16*a^5 - 15*(b^5*x^5 + a*b^4*x^4 - 2*a^2*
b^3*x^3 - 2*a^3*b^2*x^2 + a^4*b*x + a^5)*log(b*x + a) + 15*(b^5*x^5 + a*b^4*x^4 - 2*a^2*b^3*x^3 - 2*a^3*b^2*x^
2 + a^4*b*x + a^5)*log(b*x - a))/(a^6*b^6*x^5 + a^7*b^5*x^4 - 2*a^8*b^4*x^3 - 2*a^9*b^3*x^2 + a^10*b^2*x + a^1
1*b)

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Sympy [A]  time = 0.858839, size = 134, normalized size = 1.28 \begin{align*} - \frac{8 a^{4} - 25 a^{3} b x - 25 a^{2} b^{2} x^{2} + 15 a b^{3} x^{3} + 15 b^{4} x^{4}}{48 a^{10} b + 48 a^{9} b^{2} x - 96 a^{8} b^{3} x^{2} - 96 a^{7} b^{4} x^{3} + 48 a^{6} b^{5} x^{4} + 48 a^{5} b^{6} x^{5}} - \frac{\frac{5 \log{\left (- \frac{a}{b} + x \right )}}{32} - \frac{5 \log{\left (\frac{a}{b} + x \right )}}{32}}{a^{6} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b**2*x**2+a**2)**3,x)

[Out]

-(8*a**4 - 25*a**3*b*x - 25*a**2*b**2*x**2 + 15*a*b**3*x**3 + 15*b**4*x**4)/(48*a**10*b + 48*a**9*b**2*x - 96*
a**8*b**3*x**2 - 96*a**7*b**4*x**3 + 48*a**6*b**5*x**4 + 48*a**5*b**6*x**5) - (5*log(-a/b + x)/32 - 5*log(a/b
+ x)/32)/(a**6*b)

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Giac [A]  time = 1.17042, size = 136, normalized size = 1.3 \begin{align*} \frac{5 \, \log \left ({\left | b x + a \right |}\right )}{32 \, a^{6} b} - \frac{5 \, \log \left ({\left | b x - a \right |}\right )}{32 \, a^{6} b} - \frac{15 \, a b^{4} x^{4} + 15 \, a^{2} b^{3} x^{3} - 25 \, a^{3} b^{2} x^{2} - 25 \, a^{4} b x + 8 \, a^{5}}{48 \,{\left (b x + a\right )}^{3}{\left (b x - a\right )}^{2} a^{6} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2)^3,x, algorithm="giac")

[Out]

5/32*log(abs(b*x + a))/(a^6*b) - 5/32*log(abs(b*x - a))/(a^6*b) - 1/48*(15*a*b^4*x^4 + 15*a^2*b^3*x^3 - 25*a^3
*b^2*x^2 - 25*a^4*b*x + 8*a^5)/((b*x + a)^3*(b*x - a)^2*a^6*b)