3.7.11 \(\int \frac {(a+b x)^5}{(a^2-b^2 x^2)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac {2 a^2}{b (a-b x)^2}-\frac {4 a}{b (a-b x)}-\frac {\log (a-b x)}{b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 43, 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} \begin {gather*} \frac {2 a^2}{b (a-b x)^2}-\frac {4 a}{b (a-b x)}-\frac {\log (a-b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a^2 - b^2*x^2)^3,x]

[Out]

(2*a^2)/(b*(a - b*x)^2) - (4*a)/(b*(a - b*x)) - 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)^5}{\left (a^2-b^2 x^2\right )^3} \, dx &=\int \frac {(a+b x)^2}{(a-b x)^3} \, dx\\ &=\int \left (\frac {4 a^2}{(a-b x)^3}-\frac {4 a}{(a-b x)^2}+\frac {1}{a-b x}\right ) \, dx\\ &=\frac {2 a^2}{b (a-b x)^2}-\frac {4 a}{b (a-b x)}-\frac {\log (a-b x)}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 30, normalized size = 0.70 \begin {gather*} -\frac {\frac {2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a^2 - b^2*x^2)^3,x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{\left (a^2-b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/(a^2 - b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/(a^2 - b^2*x^2)^3, x]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 60, normalized size = 1.40 \begin {gather*} \frac {4 \, a b x - 2 \, a^{2} - {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(-b^2*x^2+a^2)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.21, size = 40, normalized size = 0.93 \begin {gather*} -\frac {\log \left ({\left | b x - a \right |}\right )}{b} + \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(-b^2*x^2+a^2)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 47, normalized size = 1.09 \begin {gather*} \frac {2 a^{2}}{\left (b x -a \right )^{2} b}+\frac {4 a}{\left (b x -a \right ) b}-\frac {\ln \left (b x -a \right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(-b^2*x^2+a^2)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 49, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} - \frac {\log \left (b x - a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(-b^2*x^2+a^2)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.40, size = 46, normalized size = 1.07 \begin {gather*} \frac {4\,a\,x-\frac {2\,a^2}{b}}{a^2-2\,a\,b\,x+b^2\,x^2}-\frac {\ln \left (b\,x-a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/(a^2 - b^2*x^2)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 41, normalized size = 0.95 \begin {gather*} - \frac {2 a^{2} - 4 a b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} - \frac {\log {\left (- a + b x \right )}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(-b**2*x**2+a**2)**3,x)

[Out]

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

________________________________________________________________________________________