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

Optimal. Leaf size=66 \[ -\frac {16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac {2 a^2 (a+b x)^2}{b}-\frac {2 a (a+b x)^3}{3 b}-\frac {(a+b x)^4}{4 b} \]

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Rubi [A]  time = 0.03, antiderivative size = 66, 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 (a+b x)^2}{b}-\frac {16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac {2 a (a+b x)^3}{3 b}-\frac {(a+b x)^4}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a^2 - b^2*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 54, normalized size = 0.82 \begin {gather*} -\frac {16 a^4 \log (a-b x)}{b}-15 a^3 x-\frac {11}{2} a^2 b x^2-\frac {5}{3} a b^2 x^3-\frac {b^3 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-15*a^3*x - (11*a^2*b*x^2)/2 - (5*a*b^2*x^3)/3 - (b^3*x^4)/4 - (16*a^4*Log[a - b*x])/b

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 54, normalized size = 0.82 \begin {gather*} -\frac {3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 66 \, a^{2} b^{2} x^{2} + 180 \, a^{3} b x + 192 \, a^{4} \log \left (b x - a\right )}{12 \, 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),x, algorithm="fricas")

[Out]

-1/12*(3*b^4*x^4 + 20*a*b^3*x^3 + 66*a^2*b^2*x^2 + 180*a^3*b*x + 192*a^4*log(b*x - a))/b

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giac [A]  time = 0.16, size = 61, normalized size = 0.92 \begin {gather*} -\frac {16 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {3 \, b^{7} x^{4} + 20 \, a b^{6} x^{3} + 66 \, a^{2} b^{5} x^{2} + 180 \, a^{3} b^{4} x}{12 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a^4*log(abs(b*x - a))/b - 1/12*(3*b^7*x^4 + 20*a*b^6*x^3 + 66*a^2*b^5*x^2 + 180*a^3*b^4*x)/b^4

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maple [A]  time = 0.05, size = 50, normalized size = 0.76 \begin {gather*} -\frac {b^{3} x^{4}}{4}-\frac {5 a \,b^{2} x^{3}}{3}-\frac {11 a^{2} b \,x^{2}}{2}-\frac {16 a^{4} \ln \left (b x -a \right )}{b}-15 a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^3*x^4-5/3*a*b^2*x^3-11/2*a^2*b*x^2-15*a^3*x-16*a^4/b*ln(b*x-a)

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maxima [A]  time = 1.33, size = 49, normalized size = 0.74 \begin {gather*} -\frac {1}{4} \, b^{3} x^{4} - \frac {5}{3} \, a b^{2} x^{3} - \frac {11}{2} \, a^{2} b x^{2} - 15 \, a^{3} x - \frac {16 \, a^{4} \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),x, algorithm="maxima")

[Out]

-1/4*b^3*x^4 - 5/3*a*b^2*x^3 - 11/2*a^2*b*x^2 - 15*a^3*x - 16*a^4*log(b*x - a)/b

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mupad [B]  time = 0.38, size = 49, normalized size = 0.74 \begin {gather*} -15\,a^3\,x-\frac {b^3\,x^4}{4}-\frac {11\,a^2\,b\,x^2}{2}-\frac {5\,a\,b^2\,x^3}{3}-\frac {16\,a^4\,\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),x)

[Out]

- 15*a^3*x - (b^3*x^4)/4 - (11*a^2*b*x^2)/2 - (5*a*b^2*x^3)/3 - (16*a^4*log(b*x - a))/b

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sympy [A]  time = 0.19, size = 53, normalized size = 0.80 \begin {gather*} - \frac {16 a^{4} \log {\left (- a + b x \right )}}{b} - 15 a^{3} x - \frac {11 a^{2} b x^{2}}{2} - \frac {5 a b^{2} x^{3}}{3} - \frac {b^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a**4*log(-a + b*x)/b - 15*a**3*x - 11*a**2*b*x**2/2 - 5*a*b**2*x**3/3 - b**3*x**4/4

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