3.747 \(\int \frac {(a+b x)^6}{a^2-b^2 x^2} \, dx\)

Optimal. Leaf size=83 \[ -\frac {32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac {4 a^3 (a+b x)^2}{b}-\frac {4 a^2 (a+b x)^3}{3 b}-\frac {a (a+b x)^4}{2 b}-\frac {(a+b x)^5}{5 b} \]

[Out]

-16*a^4*x-4*a^3*(b*x+a)^2/b-4/3*a^2*(b*x+a)^3/b-1/2*a*(b*x+a)^4/b-1/5*(b*x+a)^5/b-32*a^5*ln(-b*x+a)/b

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Rubi [A]  time = 0.03, antiderivative size = 83, 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} \[ -\frac {4 a^3 (a+b x)^2}{b}-\frac {4 a^2 (a+b x)^3}{3 b}-\frac {32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac {a (a+b x)^4}{2 b}-\frac {(a+b x)^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^6/(a^2 - b^2*x^2),x]

[Out]

-16*a^4*x - (4*a^3*(a + b*x)^2)/b - (4*a^2*(a + b*x)^3)/(3*b) - (a*(a + b*x)^4)/(2*b) - (a + b*x)^5/(5*b) - (3
2*a^5*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)^6}{a^2-b^2 x^2} \, dx &=\int \frac {(a+b x)^5}{a-b x} \, dx\\ &=\int \left (-16 a^4+\frac {32 a^5}{a-b x}-8 a^3 (a+b x)-4 a^2 (a+b x)^2-2 a (a+b x)^3-(a+b x)^4\right ) \, dx\\ &=-16 a^4 x-\frac {4 a^3 (a+b x)^2}{b}-\frac {4 a^2 (a+b x)^3}{3 b}-\frac {a (a+b x)^4}{2 b}-\frac {(a+b x)^5}{5 b}-\frac {32 a^5 \log (a-b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 65, normalized size = 0.78 \[ -\frac {32 a^5 \log (a-b x)}{b}-31 a^4 x-13 a^3 b x^2-\frac {16}{3} a^2 b^2 x^3-\frac {3}{2} a b^3 x^4-\frac {b^4 x^5}{5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-31*a^4*x - 13*a^3*b*x^2 - (16*a^2*b^2*x^3)/3 - (3*a*b^3*x^4)/2 - (b^4*x^5)/5 - (32*a^5*Log[a - b*x])/b

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fricas [A]  time = 0.79, size = 65, normalized size = 0.78 \[ -\frac {6 \, b^{5} x^{5} + 45 \, a b^{4} x^{4} + 160 \, a^{2} b^{3} x^{3} + 390 \, a^{3} b^{2} x^{2} + 930 \, a^{4} b x + 960 \, a^{5} \log \left (b x - a\right )}{30 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(-b^2*x^2+a^2),x, algorithm="fricas")

[Out]

-1/30*(6*b^5*x^5 + 45*a*b^4*x^4 + 160*a^2*b^3*x^3 + 390*a^3*b^2*x^2 + 930*a^4*b*x + 960*a^5*log(b*x - a))/b

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giac [A]  time = 0.17, size = 72, normalized size = 0.87 \[ -\frac {32 \, a^{5} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {6 \, b^{9} x^{5} + 45 \, a b^{8} x^{4} + 160 \, a^{2} b^{7} x^{3} + 390 \, a^{3} b^{6} x^{2} + 930 \, a^{4} b^{5} x}{30 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(-b^2*x^2+a^2),x, algorithm="giac")

[Out]

-32*a^5*log(abs(b*x - a))/b - 1/30*(6*b^9*x^5 + 45*a*b^8*x^4 + 160*a^2*b^7*x^3 + 390*a^3*b^6*x^2 + 930*a^4*b^5
*x)/b^5

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maple [A]  time = 0.04, size = 61, normalized size = 0.73 \[ -\frac {b^{4} x^{5}}{5}-\frac {3 a \,b^{3} x^{4}}{2}-\frac {16 a^{2} b^{2} x^{3}}{3}-13 a^{3} b \,x^{2}-\frac {32 a^{5} \ln \left (b x -a \right )}{b}-31 a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6/(-b^2*x^2+a^2),x)

[Out]

-1/5*b^4*x^5-3/2*x^4*a*b^3-16/3*b^2*a^2*x^3-13*a^3*b*x^2-31*a^4*x-32*a^5/b*ln(b*x-a)

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maxima [A]  time = 1.34, size = 60, normalized size = 0.72 \[ -\frac {1}{5} \, b^{4} x^{5} - \frac {3}{2} \, a b^{3} x^{4} - \frac {16}{3} \, a^{2} b^{2} x^{3} - 13 \, a^{3} b x^{2} - 31 \, a^{4} x - \frac {32 \, a^{5} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6/(-b^2*x^2+a^2),x, algorithm="maxima")

[Out]

-1/5*b^4*x^5 - 3/2*a*b^3*x^4 - 16/3*a^2*b^2*x^3 - 13*a^3*b*x^2 - 31*a^4*x - 32*a^5*log(b*x - a)/b

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mupad [B]  time = 0.40, size = 60, normalized size = 0.72 \[ -31\,a^4\,x-\frac {b^4\,x^5}{5}-13\,a^3\,b\,x^2-\frac {3\,a\,b^3\,x^4}{2}-\frac {32\,a^5\,\ln \left (b\,x-a\right )}{b}-\frac {16\,a^2\,b^2\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^6/(a^2 - b^2*x^2),x)

[Out]

- 31*a^4*x - (b^4*x^5)/5 - 13*a^3*b*x^2 - (3*a*b^3*x^4)/2 - (32*a^5*log(b*x - a))/b - (16*a^2*b^2*x^3)/3

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sympy [A]  time = 0.22, size = 65, normalized size = 0.78 \[ - \frac {32 a^{5} \log {\left (- a + b x \right )}}{b} - 31 a^{4} x - 13 a^{3} b x^{2} - \frac {16 a^{2} b^{2} x^{3}}{3} - \frac {3 a b^{3} x^{4}}{2} - \frac {b^{4} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6/(-b**2*x**2+a**2),x)

[Out]

-32*a**5*log(-a + b*x)/b - 31*a**4*x - 13*a**3*b*x**2 - 16*a**2*b**2*x**3/3 - 3*a*b**3*x**4/2 - b**4*x**5/5

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