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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 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])

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]))

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 = 0.01, size = 28, normalized size = 1.00 \[ -\frac {4 a^2 \log (a-b x)}{b}-3 a x-\frac {b x^2}{2} \]

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

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 31, normalized size = 1.11 \[ -\frac {b^{2} x^{2} + 6 \, a b x + 8 \, a^{2} \log \left (b x - a\right )}{2 \, b} \]

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

________________________________________________________________________________________

giac [A]  time = 0.18, size = 38, normalized size = 1.36 \[ -\frac {4 \, a^{2} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {b^{3} x^{2} + 6 \, a b^{2} x}{2 \, b^{2}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 28, normalized size = 1.00 \[ -\frac {b \,x^{2}}{2}-\frac {4 a^{2} \ln \left (b x -a \right )}{b}-3 a x \]

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*a^2/b*ln(b*x-a)

________________________________________________________________________________________

maxima [A]  time = 1.37, size = 27, normalized size = 0.96 \[ -\frac {1}{2} \, b x^{2} - 3 \, a x - \frac {4 \, a^{2} \log \left (b x - a\right )}{b} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.41, size = 27, normalized size = 0.96 \[ -3\,a\,x-\frac {b\,x^2}{2}-\frac {4\,a^2\,\ln \left (b\,x-a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[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(b*x - a))/b

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 26, normalized size = 0.93 \[ - \frac {4 a^{2} \log {\left (- a + b x \right )}}{b} - 3 a x - \frac {b x^{2}}{2} \]

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

________________________________________________________________________________________