### 3.748 $$\int \frac{(a+b x)^5}{a^2-b^2 x^2} \, dx$$

Optimal. Leaf size=66 $-\frac{2 a^2 (a+b x)^2}{b}-\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b}$

[Out]

-8*a^3*x - (2*a^2*(a + b*x)^2)/b - (2*a*(a + b*x)^3)/(3*b) - (a + b*x)^4/(4*b) - (16*a^4*Log[a - b*x])/b

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Rubi [A]  time = 0.0253941, antiderivative size = 66, 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} $-\frac{2 a^2 (a+b x)^2}{b}-\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^5/(a^2 - b^2*x^2),x]

[Out]

-8*a^3*x - (2*a^2*(a + b*x)^2)/b - (2*a*(a + b*x)^3)/(3*b) - (a + b*x)^4/(4*b) - (16*a^4*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)^5}{a^2-b^2 x^2} \, dx &=\int \frac{(a+b x)^4}{a-b x} \, dx\\ &=\int \left (-8 a^3+\frac{16 a^4}{a-b x}-4 a^2 (a+b x)-2 a (a+b x)^2-(a+b x)^3\right ) \, dx\\ &=-8 a^3 x-\frac{2 a^2 (a+b x)^2}{b}-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b}-\frac{16 a^4 \log (a-b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0059495, size = 54, normalized size = 0.82 $-\frac{11}{2} a^2 b x^2-\frac{16 a^4 \log (a-b x)}{b}-15 a^3 x-\frac{5}{3} a b^2 x^3-\frac{b^3 x^4}{4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^5/(a^2 - b^2*x^2),x]

[Out]

-15*a^3*x - (11*a^2*b*x^2)/2 - (5*a*b^2*x^3)/3 - (b^3*x^4)/4 - (16*a^4*Log[a - b*x])/b

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Maple [A]  time = 0.041, size = 50, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}{x}^{4}}{4}}-{\frac{5\,a{b}^{2}{x}^{3}}{3}}-{\frac{11\,{a}^{2}b{x}^{2}}{2}}-15\,x{a}^{3}-16\,{\frac{{a}^{4}\ln \left ( bx-a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(-b^2*x^2+a^2),x)

[Out]

-1/4*b^3*x^4-5/3*a*b^2*x^3-11/2*a^2*b*x^2-15*x*a^3-16*a^4/b*ln(b*x-a)

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Maxima [A]  time = 1.20587, size = 66, normalized size = 1. \begin{align*} -\frac{1}{4} \, b^{3} x^{4} - \frac{5}{3} \, a b^{2} x^{3} - \frac{11}{2} \, a^{2} b x^{2} - 15 \, a^{3} x - \frac{16 \, a^{4} \log \left (b x - a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(-b^2*x^2+a^2),x, algorithm="maxima")

[Out]

-1/4*b^3*x^4 - 5/3*a*b^2*x^3 - 11/2*a^2*b*x^2 - 15*a^3*x - 16*a^4*log(b*x - a)/b

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Fricas [A]  time = 1.82077, size = 122, normalized size = 1.85 \begin{align*} -\frac{3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 66 \, a^{2} b^{2} x^{2} + 180 \, a^{3} b x + 192 \, a^{4} \log \left (b x - a\right )}{12 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(-b^2*x^2+a^2),x, algorithm="fricas")

[Out]

-1/12*(3*b^4*x^4 + 20*a*b^3*x^3 + 66*a^2*b^2*x^2 + 180*a^3*b*x + 192*a^4*log(b*x - a))/b

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Sympy [A]  time = 0.327504, size = 53, normalized size = 0.8 \begin{align*} - \frac{16 a^{4} \log{\left (- a + b x \right )}}{b} - 15 a^{3} x - \frac{11 a^{2} b x^{2}}{2} - \frac{5 a b^{2} x^{3}}{3} - \frac{b^{3} x^{4}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(-b**2*x**2+a**2),x)

[Out]

-16*a**4*log(-a + b*x)/b - 15*a**3*x - 11*a**2*b*x**2/2 - 5*a*b**2*x**3/3 - b**3*x**4/4

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Giac [A]  time = 1.17887, size = 82, normalized size = 1.24 \begin{align*} -\frac{16 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{3 \, b^{7} x^{4} + 20 \, a b^{6} x^{3} + 66 \, a^{2} b^{5} x^{2} + 180 \, a^{3} b^{4} x}{12 \, b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(-b^2*x^2+a^2),x, algorithm="giac")

[Out]

-16*a^4*log(abs(b*x - a))/b - 1/12*(3*b^7*x^4 + 20*a*b^6*x^3 + 66*a^2*b^5*x^2 + 180*a^3*b^4*x)/b^4