3.3.10 \(\int \frac {x^7}{(a+b x)^7} \, dx\)

Optimal. Leaf size=118 \[ \frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8}+\frac {x}{b^7} \]

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Rubi [A]  time = 0.07, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} \frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8}+\frac {x}{b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x)^7,x]

[Out]

x/b^7 + a^7/(6*b^8*(a + b*x)^6) - (7*a^6)/(5*b^8*(a + b*x)^5) + (21*a^5)/(4*b^8*(a + b*x)^4) - (35*a^4)/(3*b^8
*(a + b*x)^3) + (35*a^3)/(2*b^8*(a + b*x)^2) - (21*a^2)/(b^8*(a + b*x)) - (7*a*Log[a + b*x])/b^8

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])

Rubi steps

\begin {align*} \int \frac {x^7}{(a+b x)^7} \, dx &=\int \left (\frac {1}{b^7}-\frac {a^7}{b^7 (a+b x)^7}+\frac {7 a^6}{b^7 (a+b x)^6}-\frac {21 a^5}{b^7 (a+b x)^5}+\frac {35 a^4}{b^7 (a+b x)^4}-\frac {35 a^3}{b^7 (a+b x)^3}+\frac {21 a^2}{b^7 (a+b x)^2}-\frac {7 a}{b^7 (a+b x)}\right ) \, dx\\ &=\frac {x}{b^7}+\frac {a^7}{6 b^8 (a+b x)^6}-\frac {7 a^6}{5 b^8 (a+b x)^5}+\frac {21 a^5}{4 b^8 (a+b x)^4}-\frac {35 a^4}{3 b^8 (a+b x)^3}+\frac {35 a^3}{2 b^8 (a+b x)^2}-\frac {21 a^2}{b^8 (a+b x)}-\frac {7 a \log (a+b x)}{b^8}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 104, normalized size = 0.88 \begin {gather*} -\frac {669 a^7+3594 a^6 b x+7725 a^5 b^2 x^2+8200 a^4 b^3 x^3+4050 a^3 b^4 x^4+360 a^2 b^5 x^5-360 a b^6 x^6+420 a (a+b x)^6 \log (a+b x)-60 b^7 x^7}{60 b^8 (a+b x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x)^7,x]

[Out]

-1/60*(669*a^7 + 3594*a^6*b*x + 7725*a^5*b^2*x^2 + 8200*a^4*b^3*x^3 + 4050*a^3*b^4*x^4 + 360*a^2*b^5*x^5 - 360
*a*b^6*x^6 - 60*b^7*x^7 + 420*a*(a + b*x)^6*Log[a + b*x])/(b^8*(a + b*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7}{(a+b x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7/(a + b*x)^7,x]

[Out]

IntegrateAlgebraic[x^7/(a + b*x)^7, x]

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fricas [A]  time = 1.03, size = 215, normalized size = 1.82 \begin {gather*} \frac {60 \, b^{7} x^{7} + 360 \, a b^{6} x^{6} - 360 \, a^{2} b^{5} x^{5} - 4050 \, a^{3} b^{4} x^{4} - 8200 \, a^{4} b^{3} x^{3} - 7725 \, a^{5} b^{2} x^{2} - 3594 \, a^{6} b x - 669 \, a^{7} - 420 \, {\left (a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{5} + 15 \, a^{3} b^{4} x^{4} + 20 \, a^{4} b^{3} x^{3} + 15 \, a^{5} b^{2} x^{2} + 6 \, a^{6} b x + a^{7}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(60*b^7*x^7 + 360*a*b^6*x^6 - 360*a^2*b^5*x^5 - 4050*a^3*b^4*x^4 - 8200*a^4*b^3*x^3 - 7725*a^5*b^2*x^2 -
3594*a^6*b*x - 669*a^7 - 420*(a*b^6*x^6 + 6*a^2*b^5*x^5 + 15*a^3*b^4*x^4 + 20*a^4*b^3*x^3 + 15*a^5*b^2*x^2 + 6
*a^6*b*x + a^7)*log(b*x + a))/(b^14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 +
6*a^5*b^9*x + a^6*b^8)

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giac [A]  time = 0.92, size = 88, normalized size = 0.75 \begin {gather*} \frac {x}{b^{7}} - \frac {7 \, a \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \, {\left (b x + a\right )}^{6} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b^7 - 7*a*log(abs(b*x + a))/b^8 - 1/60*(1260*a^2*b^5*x^5 + 5250*a^3*b^4*x^4 + 9100*a^4*b^3*x^3 + 8085*a^5*b^
2*x^2 + 3654*a^6*b*x + 669*a^7)/((b*x + a)^6*b^8)

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maple [A]  time = 0.01, size = 109, normalized size = 0.92 \begin {gather*} \frac {a^{7}}{6 \left (b x +a \right )^{6} b^{8}}-\frac {7 a^{6}}{5 \left (b x +a \right )^{5} b^{8}}+\frac {21 a^{5}}{4 \left (b x +a \right )^{4} b^{8}}-\frac {35 a^{4}}{3 \left (b x +a \right )^{3} b^{8}}+\frac {35 a^{3}}{2 \left (b x +a \right )^{2} b^{8}}-\frac {21 a^{2}}{\left (b x +a \right ) b^{8}}-\frac {7 a \ln \left (b x +a \right )}{b^{8}}+\frac {x}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b^7+1/6*a^7/b^8/(b*x+a)^6-7/5*a^6/b^8/(b*x+a)^5+21/4*a^5/b^8/(b*x+a)^4-35/3*a^4/b^8/(b*x+a)^3+35/2*a^3/b^8/(
b*x+a)^2-21*a^2/b^8/(b*x+a)-7*a*ln(b*x+a)/b^8

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maxima [A]  time = 1.49, size = 145, normalized size = 1.23 \begin {gather*} -\frac {1260 \, a^{2} b^{5} x^{5} + 5250 \, a^{3} b^{4} x^{4} + 9100 \, a^{4} b^{3} x^{3} + 8085 \, a^{5} b^{2} x^{2} + 3654 \, a^{6} b x + 669 \, a^{7}}{60 \, {\left (b^{14} x^{6} + 6 \, a b^{13} x^{5} + 15 \, a^{2} b^{12} x^{4} + 20 \, a^{3} b^{11} x^{3} + 15 \, a^{4} b^{10} x^{2} + 6 \, a^{5} b^{9} x + a^{6} b^{8}\right )}} + \frac {x}{b^{7}} - \frac {7 \, a \log \left (b x + a\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(1260*a^2*b^5*x^5 + 5250*a^3*b^4*x^4 + 9100*a^4*b^3*x^3 + 8085*a^5*b^2*x^2 + 3654*a^6*b*x + 669*a^7)/(b^
14*x^6 + 6*a*b^13*x^5 + 15*a^2*b^12*x^4 + 20*a^3*b^11*x^3 + 15*a^4*b^10*x^2 + 6*a^5*b^9*x + a^6*b^8) + x/b^7 -
7*a*log(b*x + a)/b^8

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mupad [B]  time = 0.34, size = 91, normalized size = 0.77 \begin {gather*} -\frac {7\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {21\,a^2}{a+b\,x}-\frac {35\,a^3}{2\,{\left (a+b\,x\right )}^2}+\frac {35\,a^4}{3\,{\left (a+b\,x\right )}^3}-\frac {21\,a^5}{4\,{\left (a+b\,x\right )}^4}+\frac {7\,a^6}{5\,{\left (a+b\,x\right )}^5}-\frac {a^7}{6\,{\left (a+b\,x\right )}^6}}{b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x)^7,x)

[Out]

-(7*a*log(a + b*x) - b*x + (21*a^2)/(a + b*x) - (35*a^3)/(2*(a + b*x)^2) + (35*a^4)/(3*(a + b*x)^3) - (21*a^5)
/(4*(a + b*x)^4) + (7*a^6)/(5*(a + b*x)^5) - a^7/(6*(a + b*x)^6))/b^8

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sympy [A]  time = 0.82, size = 153, normalized size = 1.30 \begin {gather*} - \frac {7 a \log {\left (a + b x \right )}}{b^{8}} + \frac {- 669 a^{7} - 3654 a^{6} b x - 8085 a^{5} b^{2} x^{2} - 9100 a^{4} b^{3} x^{3} - 5250 a^{3} b^{4} x^{4} - 1260 a^{2} b^{5} x^{5}}{60 a^{6} b^{8} + 360 a^{5} b^{9} x + 900 a^{4} b^{10} x^{2} + 1200 a^{3} b^{11} x^{3} + 900 a^{2} b^{12} x^{4} + 360 a b^{13} x^{5} + 60 b^{14} x^{6}} + \frac {x}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*a*log(a + b*x)/b**8 + (-669*a**7 - 3654*a**6*b*x - 8085*a**5*b**2*x**2 - 9100*a**4*b**3*x**3 - 5250*a**3*b*
*4*x**4 - 1260*a**2*b**5*x**5)/(60*a**6*b**8 + 360*a**5*b**9*x + 900*a**4*b**10*x**2 + 1200*a**3*b**11*x**3 +
900*a**2*b**12*x**4 + 360*a*b**13*x**5 + 60*b**14*x**6) + x/b**7

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