3.757 \(\int \frac {(a+b x)^7}{(a^2-b^2 x^2)^2} \, dx\)

Optimal. Leaf size=70 \[ \frac {32 a^5}{b (a-b x)}+\frac {80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4} \]

[Out]

49*a^3*x+23/2*a^2*b*x^2+7/3*a*b^2*x^3+1/4*b^3*x^4+32*a^5/b/(-b*x+a)+80*a^4*ln(-b*x+a)/b

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Rubi [A]  time = 0.05, antiderivative size = 70, 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} \[ \frac {23}{2} a^2 b x^2+\frac {32 a^5}{b (a-b x)}+\frac {80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/(a^2 - b^2*x^2)^2,x]

[Out]

49*a^3*x + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3 + (b^3*x^4)/4 + (32*a^5)/(b*(a - b*x)) + (80*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)^7}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^5}{(a-b x)^2} \, dx\\ &=\int \left (49 a^3+23 a^2 b x+7 a b^2 x^2+b^3 x^3+\frac {32 a^5}{(a-b x)^2}-\frac {80 a^4}{a-b x}\right ) \, dx\\ &=49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4}+\frac {32 a^5}{b (a-b x)}+\frac {80 a^4 \log (a-b x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 71, normalized size = 1.01 \[ -\frac {32 a^5}{b (b x-a)}+\frac {80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/(a^2 - b^2*x^2)^2,x]

[Out]

49*a^3*x + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3 + (b^3*x^4)/4 - (32*a^5)/(b*(-a + b*x)) + (80*a^4*Log[a - b*x])/
b

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fricas [A]  time = 0.79, size = 88, normalized size = 1.26 \[ \frac {3 \, b^{5} x^{5} + 25 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 450 \, a^{3} b^{2} x^{2} - 588 \, a^{4} b x - 384 \, a^{5} + 960 \, {\left (a^{4} b x - a^{5}\right )} \log \left (b x - a\right )}{12 \, {\left (b^{2} x - a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(-b^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^5 + 25*a*b^4*x^4 + 110*a^2*b^3*x^3 + 450*a^3*b^2*x^2 - 588*a^4*b*x - 384*a^5 + 960*(a^4*b*x - a^
5)*log(b*x - a))/(b^2*x - a*b)

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giac [A]  time = 0.17, size = 78, normalized size = 1.11 \[ \frac {80 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {32 \, a^{5}}{{\left (b x - a\right )} b} + \frac {3 \, b^{11} x^{4} + 28 \, a b^{10} x^{3} + 138 \, a^{2} b^{9} x^{2} + 588 \, a^{3} b^{8} x}{12 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(-b^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

80*a^4*log(abs(b*x - a))/b - 32*a^5/((b*x - a)*b) + 1/12*(3*b^11*x^4 + 28*a*b^10*x^3 + 138*a^2*b^9*x^2 + 588*a
^3*b^8*x)/b^8

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maple [A]  time = 0.08, size = 67, normalized size = 0.96 \[ \frac {b^{3} x^{4}}{4}+\frac {7 a \,b^{2} x^{3}}{3}+\frac {23 a^{2} b \,x^{2}}{2}-\frac {32 a^{5}}{\left (b x -a \right ) b}+\frac {80 a^{4} \ln \left (b x -a \right )}{b}+49 a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/(-b^2*x^2+a^2)^2,x)

[Out]

1/4*b^3*x^4+7/3*a*b^2*x^3+23/2*a^2*b*x^2+49*a^3*x+80*a^4/b*ln(b*x-a)-32*a^5/b/(b*x-a)

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maxima [A]  time = 1.35, size = 66, normalized size = 0.94 \[ \frac {1}{4} \, b^{3} x^{4} + \frac {7}{3} \, a b^{2} x^{3} + \frac {23}{2} \, a^{2} b x^{2} - \frac {32 \, a^{5}}{b^{2} x - a b} + 49 \, a^{3} x + \frac {80 \, a^{4} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(-b^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

1/4*b^3*x^4 + 7/3*a*b^2*x^3 + 23/2*a^2*b*x^2 - 32*a^5/(b^2*x - a*b) + 49*a^3*x + 80*a^4*log(b*x - a)/b

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mupad [B]  time = 0.05, size = 64, normalized size = 0.91 \[ 49\,a^3\,x+\frac {b^3\,x^4}{4}+\frac {32\,a^5}{b\,\left (a-b\,x\right )}+\frac {80\,a^4\,\ln \left (a-b\,x\right )}{b}+\frac {23\,a^2\,b\,x^2}{2}+\frac {7\,a\,b^2\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^7/(a^2 - b^2*x^2)^2,x)

[Out]

49*a^3*x + (b^3*x^4)/4 + (32*a^5)/(b*(a - b*x)) + (80*a^4*log(a - b*x))/b + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3

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sympy [A]  time = 0.31, size = 65, normalized size = 0.93 \[ - \frac {32 a^{5}}{- a b + b^{2} x} + \frac {80 a^{4} \log {\left (- a + b x \right )}}{b} + 49 a^{3} x + \frac {23 a^{2} b x^{2}}{2} + \frac {7 a b^{2} x^{3}}{3} + \frac {b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/(-b**2*x**2+a**2)**2,x)

[Out]

-32*a**5/(-a*b + b**2*x) + 80*a**4*log(-a + b*x)/b + 49*a**3*x + 23*a**2*b*x**2/2 + 7*a*b**2*x**3/3 + b**3*x**
4/4

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