### 3.768 $$\int \frac{(a+b x)^7}{(a^2-b^2 x^2)^3} \, dx$$

Optimal. Leaf size=60 $\frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2}$

[Out]

-7*a*x - (b*x^2)/2 + (8*a^4)/(b*(a - b*x)^2) - (32*a^3)/(b*(a - b*x)) - (24*a^2*Log[a - b*x])/b

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Rubi [A]  time = 0.0406074, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {627, 43} $\frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^7/(a^2 - b^2*x^2)^3,x]

[Out]

-7*a*x - (b*x^2)/2 + (8*a^4)/(b*(a - b*x)^2) - (32*a^3)/(b*(a - b*x)) - (24*a^2*Log[a - b*x])/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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^7}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac{(a+b x)^4}{(a-b x)^3} \, dx\\ &=\int \left (-7 a-b x+\frac{16 a^4}{(a-b x)^3}-\frac{32 a^3}{(a-b x)^2}+\frac{24 a^2}{a-b x}\right ) \, dx\\ &=-7 a x-\frac{b x^2}{2}+\frac{8 a^4}{b (a-b x)^2}-\frac{32 a^3}{b (a-b x)}-\frac{24 a^2 \log (a-b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0280393, size = 62, normalized size = 1.03 $\frac{8 a^4}{b (b x-a)^2}+\frac{32 a^3}{b (b x-a)}-\frac{24 a^2 \log (a-b x)}{b}-7 a x-\frac{b x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^7/(a^2 - b^2*x^2)^3,x]

[Out]

-7*a*x - (b*x^2)/2 + (8*a^4)/(b*(-a + b*x)^2) + (32*a^3)/(b*(-a + b*x)) - (24*a^2*Log[a - b*x])/b

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Maple [A]  time = 0.047, size = 62, normalized size = 1. \begin{align*} -{\frac{b{x}^{2}}{2}}-7\,ax+8\,{\frac{{a}^{4}}{b \left ( bx-a \right ) ^{2}}}+32\,{\frac{{a}^{3}}{b \left ( bx-a \right ) }}-24\,{\frac{{a}^{2}\ln \left ( bx-a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/(-b^2*x^2+a^2)^3,x)

[Out]

-1/2*b*x^2-7*a*x+8*a^4/b/(b*x-a)^2+32/b*a^3/(b*x-a)-24/b*a^2*ln(b*x-a)

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Maxima [A]  time = 1.08461, size = 86, normalized size = 1.43 \begin{align*} -\frac{1}{2} \, b x^{2} - 7 \, a x - \frac{24 \, a^{2} \log \left (b x - a\right )}{b} + \frac{8 \,{\left (4 \, a^{3} b x - 3 \, a^{4}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(-b^2*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/2*b*x^2 - 7*a*x - 24*a^2*log(b*x - a)/b + 8*(4*a^3*b*x - 3*a^4)/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Fricas [A]  time = 1.62566, size = 203, normalized size = 3.38 \begin{align*} -\frac{b^{4} x^{4} + 12 \, a b^{3} x^{3} - 27 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x + 48 \, a^{4} + 48 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )} \log \left (b x - a\right )}{2 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(-b^2*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/2*(b^4*x^4 + 12*a*b^3*x^3 - 27*a^2*b^2*x^2 - 50*a^3*b*x + 48*a^4 + 48*(a^2*b^2*x^2 - 2*a^3*b*x + a^4)*log(b
*x - a))/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Sympy [A]  time = 0.495016, size = 58, normalized size = 0.97 \begin{align*} - \frac{24 a^{2} \log{\left (- a + b x \right )}}{b} - 7 a x - \frac{b x^{2}}{2} + \frac{- 24 a^{4} + 32 a^{3} b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/(-b**2*x**2+a**2)**3,x)

[Out]

-24*a**2*log(-a + b*x)/b - 7*a*x - b*x**2/2 + (-24*a**4 + 32*a**3*b*x)/(a**2*b - 2*a*b**2*x + b**3*x**2)

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Giac [A]  time = 1.225, size = 88, normalized size = 1.47 \begin{align*} -\frac{24 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} + \frac{8 \,{\left (4 \, a^{3} b x - 3 \, a^{4}\right )}}{{\left (b x - a\right )}^{2} b} - \frac{b^{7} x^{2} + 14 \, a b^{6} x}{2 \, b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(-b^2*x^2+a^2)^3,x, algorithm="giac")

[Out]

-24*a^2*log(abs(b*x - a))/b + 8*(4*a^3*b*x - 3*a^4)/((b*x - a)^2*b) - 1/2*(b^7*x^2 + 14*a*b^6*x)/b^6