### 3.769 $$\int \frac{(a+b x)^6}{(a^2-b^2 x^2)^3} \, dx$$

Optimal. Leaf size=49 $\frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x$

[Out]

-x + (4*a^3)/(b*(a - b*x)^2) - (12*a^2)/(b*(a - b*x)) - (6*a*Log[a - b*x])/b

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Rubi [A]  time = 0.0295678, antiderivative size = 49, 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} $\frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^6/(a^2 - b^2*x^2)^3,x]

[Out]

-x + (4*a^3)/(b*(a - b*x)^2) - (12*a^2)/(b*(a - b*x)) - (6*a*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 )^3} \, dx &=\int \frac{(a+b x)^3}{(a-b x)^3} \, dx\\ &=\int \left (-1+\frac{8 a^3}{(a-b x)^3}-\frac{12 a^2}{(a-b x)^2}+\frac{6 a}{a-b x}\right ) \, dx\\ &=-x+\frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.044352, size = 41, normalized size = 0.84 $\frac{4 a^2 (3 b x-2 a)}{b (a-b x)^2}-\frac{6 a \log (a-b x)}{b}-x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^6/(a^2 - b^2*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.045, size = 53, normalized size = 1.1 \begin{align*} -x+4\,{\frac{{a}^{3}}{b \left ( bx-a \right ) ^{2}}}+12\,{\frac{{a}^{2}}{b \left ( bx-a \right ) }}-6\,{\frac{a\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)^3,x)

[Out]

-x+4/b*a^3/(b*x-a)^2+12/b*a^2/(b*x-a)-6/b*a*ln(b*x-a)

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Maxima [A]  time = 1.03819, size = 74, normalized size = 1.51 \begin{align*} -x - \frac{6 \, a \log \left (b x - a\right )}{b} + \frac{4 \,{\left (3 \, a^{2} b x - 2 \, a^{3}\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)^6/(-b^2*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-x - 6*a*log(b*x - a)/b + 4*(3*a^2*b*x - 2*a^3)/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Fricas [A]  time = 1.74147, size = 167, normalized size = 3.41 \begin{align*} -\frac{b^{3} x^{3} - 2 \, a b^{2} x^{2} - 11 \, a^{2} b x + 8 \, a^{3} + 6 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\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)^6/(-b^2*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-(b^3*x^3 - 2*a*b^2*x^2 - 11*a^2*b*x + 8*a^3 + 6*(a*b^2*x^2 - 2*a^2*b*x + a^3)*log(b*x - a))/(b^3*x^2 - 2*a*b^
2*x + a^2*b)

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Sympy [A]  time = 0.426304, size = 46, normalized size = 0.94 \begin{align*} - \frac{6 a \log{\left (- a + b x \right )}}{b} - x + \frac{- 8 a^{3} + 12 a^{2} 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)**6/(-b**2*x**2+a**2)**3,x)

[Out]

-6*a*log(-a + b*x)/b - x + (-8*a**3 + 12*a**2*b*x)/(a**2*b - 2*a*b**2*x + b**3*x**2)

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Giac [A]  time = 1.17542, size = 62, normalized size = 1.27 \begin{align*} -x - \frac{6 \, a \log \left ({\left | b x - a \right |}\right )}{b} + \frac{4 \,{\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{{\left (b x - a\right )}^{2} 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)^3,x, algorithm="giac")

[Out]

-x - 6*a*log(abs(b*x - a))/b + 4*(3*a^2*b*x - 2*a^3)/((b*x - a)^2*b)