### 3.758 $$\int \frac{(a+b x)^6}{(a^2-b^2 x^2)^2} \, dx$$

Optimal. Leaf size=55 $\frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}$

[Out]

17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 + (16*a^4)/(b*(a - b*x)) + (32*a^3*Log[a - b*x])/b

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Rubi [A]  time = 0.0410961, antiderivative size = 55, 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} $\frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^6/(a^2 - b^2*x^2)^2,x]

[Out]

17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 + (16*a^4)/(b*(a - b*x)) + (32*a^3*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)^6}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{(a+b x)^4}{(a-b x)^2} \, dx\\ &=\int \left (17 a^2+6 a b x+b^2 x^2+\frac{16 a^4}{(a-b x)^2}-\frac{32 a^3}{a-b x}\right ) \, dx\\ &=17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}+\frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0280465, size = 56, normalized size = 1.02 $-\frac{16 a^4}{b (b x-a)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^6/(a^2 - b^2*x^2)^2,x]

[Out]

17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 - (16*a^4)/(b*(-a + b*x)) + (32*a^3*Log[a - b*x])/b

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Maple [A]  time = 0.046, size = 56, normalized size = 1. \begin{align*}{\frac{{b}^{2}{x}^{3}}{3}}+3\,ab{x}^{2}+17\,{a}^{2}x-16\,{\frac{{a}^{4}}{b \left ( bx-a \right ) }}+32\,{\frac{{a}^{3}\ln \left ( bx-a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/(-b^2*x^2+a^2)^2,x)

[Out]

1/3*b^2*x^3+3*a*b*x^2+17*a^2*x-16*a^4/b/(b*x-a)+32/b*a^3*ln(b*x-a)

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Maxima [A]  time = 1.11841, size = 74, normalized size = 1.35 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + 3 \, a b x^{2} - \frac{16 \, a^{4}}{b^{2} x - a b} + 17 \, a^{2} x + \frac{32 \, a^{3} \log \left (b x - a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(-b^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

1/3*b^2*x^3 + 3*a*b*x^2 - 16*a^4/(b^2*x - a*b) + 17*a^2*x + 32*a^3*log(b*x - a)/b

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Fricas [A]  time = 1.66251, size = 157, normalized size = 2.85 \begin{align*} \frac{b^{4} x^{4} + 8 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} - 51 \, a^{3} b x - 48 \, a^{4} + 96 \,{\left (a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{3 \,{\left (b^{2} x - a b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(-b^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

1/3*(b^4*x^4 + 8*a*b^3*x^3 + 42*a^2*b^2*x^2 - 51*a^3*b*x - 48*a^4 + 96*(a^3*b*x - a^4)*log(b*x - a))/(b^2*x -
a*b)

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Sympy [A]  time = 0.397923, size = 49, normalized size = 0.89 \begin{align*} - \frac{16 a^{4}}{- a b + b^{2} x} + \frac{32 a^{3} \log{\left (- a + b x \right )}}{b} + 17 a^{2} x + 3 a b x^{2} + \frac{b^{2} x^{3}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/(-b**2*x**2+a**2)**2,x)

[Out]

-16*a**4/(-a*b + b**2*x) + 32*a**3*log(-a + b*x)/b + 17*a**2*x + 3*a*b*x**2 + b**2*x**3/3

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Giac [A]  time = 1.21901, size = 89, normalized size = 1.62 \begin{align*} \frac{32 \, a^{3} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{16 \, a^{4}}{{\left (b x - a\right )} b} + \frac{b^{8} x^{3} + 9 \, a b^{7} x^{2} + 51 \, a^{2} b^{6} x}{3 \, b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(-b^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

32*a^3*log(abs(b*x - a))/b - 16*a^4/((b*x - a)*b) + 1/3*(b^8*x^3 + 9*a*b^7*x^2 + 51*a^2*b^6*x)/b^6