### 3.750 $$\int \frac{(a+b x)^3}{a^2-b^2 x^2} \, dx$$

Optimal. Leaf size=28 $-\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2}$

[Out]

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

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Rubi [A]  time = 0.0146225, antiderivative size = 28, 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} $-\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^3/(a^2 - b^2*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0048804, size = 28, normalized size = 1. $-\frac{4 a^2 \log (a-b x)}{b}-3 a x-\frac{b x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 28, normalized size = 1. \begin{align*} -{\frac{b{x}^{2}}{2}}-3\,ax-4\,{\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)^3/(-b^2*x^2+a^2),x)

[Out]

-1/2*b*x^2-3*a*x-4/b*a^2*ln(b*x-a)

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Maxima [A]  time = 1.23439, size = 36, normalized size = 1.29 \begin{align*} -\frac{1}{2} \, b x^{2} - 3 \, a x - \frac{4 \, a^{2} \log \left (b x - a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*x^2 - 3*a*x - 4*a^2*log(b*x - a)/b

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Fricas [A]  time = 1.62271, size = 66, normalized size = 2.36 \begin{align*} -\frac{b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^2*x^2 + 6*a*b*x + 8*a^2*log(b*x - a))/b

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Sympy [A]  time = 0.291171, size = 26, normalized size = 0.93 \begin{align*} - \frac{4 a^{2} \log{\left (- a + b x \right )}}{b} - 3 a x - \frac{b x^{2}}{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.23165, size = 51, normalized size = 1.82 \begin{align*} -\frac{4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac{b^{3} x^{2} + 6 \, a b^{2} x}{2 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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