3.3.9 \(\int \frac {x^8}{(a+b x)^7} \, dx\)

Optimal. Leaf size=128 \[ -\frac {a^8}{6 b^9 (a+b x)^6}+\frac {8 a^7}{5 b^9 (a+b x)^5}-\frac {7 a^6}{b^9 (a+b x)^4}+\frac {56 a^5}{3 b^9 (a+b x)^3}-\frac {35 a^4}{b^9 (a+b x)^2}+\frac {56 a^3}{b^9 (a+b x)}+\frac {28 a^2 \log (a+b x)}{b^9}-\frac {7 a x}{b^8}+\frac {x^2}{2 b^7} \]

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Rubi [A]  time = 0.09, antiderivative size = 128, 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^8}{6 b^9 (a+b x)^6}+\frac {8 a^7}{5 b^9 (a+b x)^5}-\frac {7 a^6}{b^9 (a+b x)^4}+\frac {56 a^5}{3 b^9 (a+b x)^3}-\frac {35 a^4}{b^9 (a+b x)^2}+\frac {56 a^3}{b^9 (a+b x)}+\frac {28 a^2 \log (a+b x)}{b^9}-\frac {7 a x}{b^8}+\frac {x^2}{2 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x)^7,x]

[Out]

(-7*a*x)/b^8 + x^2/(2*b^7) - a^8/(6*b^9*(a + b*x)^6) + (8*a^7)/(5*b^9*(a + b*x)^5) - (7*a^6)/(b^9*(a + b*x)^4)
+ (56*a^5)/(3*b^9*(a + b*x)^3) - (35*a^4)/(b^9*(a + b*x)^2) + (56*a^3)/(b^9*(a + b*x)) + (28*a^2*Log[a + b*x]
)/b^9

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^8}{(a+b x)^7} \, dx &=\int \left (-\frac {7 a}{b^8}+\frac {x}{b^7}+\frac {a^8}{b^8 (a+b x)^7}-\frac {8 a^7}{b^8 (a+b x)^6}+\frac {28 a^6}{b^8 (a+b x)^5}-\frac {56 a^5}{b^8 (a+b x)^4}+\frac {70 a^4}{b^8 (a+b x)^3}-\frac {56 a^3}{b^8 (a+b x)^2}+\frac {28 a^2}{b^8 (a+b x)}\right ) \, dx\\ &=-\frac {7 a x}{b^8}+\frac {x^2}{2 b^7}-\frac {a^8}{6 b^9 (a+b x)^6}+\frac {8 a^7}{5 b^9 (a+b x)^5}-\frac {7 a^6}{b^9 (a+b x)^4}+\frac {56 a^5}{3 b^9 (a+b x)^3}-\frac {35 a^4}{b^9 (a+b x)^2}+\frac {56 a^3}{b^9 (a+b x)}+\frac {28 a^2 \log (a+b x)}{b^9}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 104, normalized size = 0.81 \begin {gather*} \frac {-\frac {5 a^8}{(a+b x)^6}+\frac {48 a^7}{(a+b x)^5}-\frac {210 a^6}{(a+b x)^4}+\frac {560 a^5}{(a+b x)^3}-\frac {1050 a^4}{(a+b x)^2}+\frac {1680 a^3}{a+b x}+840 a^2 \log (a+b x)-210 a b x+15 b^2 x^2}{30 b^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x)^7,x]

[Out]

(-210*a*b*x + 15*b^2*x^2 - (5*a^8)/(a + b*x)^6 + (48*a^7)/(a + b*x)^5 - (210*a^6)/(a + b*x)^4 + (560*a^5)/(a +
b*x)^3 - (1050*a^4)/(a + b*x)^2 + (1680*a^3)/(a + b*x) + 840*a^2*Log[a + b*x])/(30*b^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8}{(a+b x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^8/(a + b*x)^7,x]

[Out]

IntegrateAlgebraic[x^8/(a + b*x)^7, x]

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fricas [A]  time = 0.93, size = 228, normalized size = 1.78 \begin {gather*} \frac {15 \, b^{8} x^{8} - 120 \, a b^{7} x^{7} - 1035 \, a^{2} b^{6} x^{6} - 1170 \, a^{3} b^{5} x^{5} + 3375 \, a^{4} b^{4} x^{4} + 10100 \, a^{5} b^{3} x^{3} + 10725 \, a^{6} b^{2} x^{2} + 5298 \, a^{7} b x + 1023 \, a^{8} + 840 \, {\left (a^{2} b^{6} x^{6} + 6 \, a^{3} b^{5} x^{5} + 15 \, a^{4} b^{4} x^{4} + 20 \, a^{5} b^{3} x^{3} + 15 \, a^{6} b^{2} x^{2} + 6 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{30 \, {\left (b^{15} x^{6} + 6 \, a b^{14} x^{5} + 15 \, a^{2} b^{13} x^{4} + 20 \, a^{3} b^{12} x^{3} + 15 \, a^{4} b^{11} x^{2} + 6 \, a^{5} b^{10} x + a^{6} b^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^7,x, algorithm="fricas")

[Out]

1/30*(15*b^8*x^8 - 120*a*b^7*x^7 - 1035*a^2*b^6*x^6 - 1170*a^3*b^5*x^5 + 3375*a^4*b^4*x^4 + 10100*a^5*b^3*x^3
+ 10725*a^6*b^2*x^2 + 5298*a^7*b*x + 1023*a^8 + 840*(a^2*b^6*x^6 + 6*a^3*b^5*x^5 + 15*a^4*b^4*x^4 + 20*a^5*b^3
*x^3 + 15*a^6*b^2*x^2 + 6*a^7*b*x + a^8)*log(b*x + a))/(b^15*x^6 + 6*a*b^14*x^5 + 15*a^2*b^13*x^4 + 20*a^3*b^1
2*x^3 + 15*a^4*b^11*x^2 + 6*a^5*b^10*x + a^6*b^9)

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giac [A]  time = 1.07, size = 105, normalized size = 0.82 \begin {gather*} \frac {28 \, a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{9}} + \frac {b^{7} x^{2} - 14 \, a b^{6} x}{2 \, b^{14}} + \frac {1680 \, a^{3} b^{5} x^{5} + 7350 \, a^{4} b^{4} x^{4} + 13160 \, a^{5} b^{3} x^{3} + 11970 \, a^{6} b^{2} x^{2} + 5508 \, a^{7} b x + 1023 \, a^{8}}{30 \, {\left (b x + a\right )}^{6} b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^7,x, algorithm="giac")

[Out]

28*a^2*log(abs(b*x + a))/b^9 + 1/2*(b^7*x^2 - 14*a*b^6*x)/b^14 + 1/30*(1680*a^3*b^5*x^5 + 7350*a^4*b^4*x^4 + 1
3160*a^5*b^3*x^3 + 11970*a^6*b^2*x^2 + 5508*a^7*b*x + 1023*a^8)/((b*x + a)^6*b^9)

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maple [A]  time = 0.01, size = 121, normalized size = 0.95 \begin {gather*} -\frac {a^{8}}{6 \left (b x +a \right )^{6} b^{9}}+\frac {8 a^{7}}{5 \left (b x +a \right )^{5} b^{9}}-\frac {7 a^{6}}{\left (b x +a \right )^{4} b^{9}}+\frac {56 a^{5}}{3 \left (b x +a \right )^{3} b^{9}}-\frac {35 a^{4}}{\left (b x +a \right )^{2} b^{9}}+\frac {x^{2}}{2 b^{7}}+\frac {56 a^{3}}{\left (b x +a \right ) b^{9}}+\frac {28 a^{2} \ln \left (b x +a \right )}{b^{9}}-\frac {7 a x}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x+a)^7,x)

[Out]

-7*a*x/b^8+1/2*x^2/b^7-1/6*a^8/b^9/(b*x+a)^6+8/5*a^7/b^9/(b*x+a)^5-7*a^6/b^9/(b*x+a)^4+56/3*a^5/b^9/(b*x+a)^3-
35*a^4/b^9/(b*x+a)^2+56*a^3/b^9/(b*x+a)+28*a^2*ln(b*x+a)/b^9

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maxima [A]  time = 1.56, size = 157, normalized size = 1.23 \begin {gather*} \frac {1680 \, a^{3} b^{5} x^{5} + 7350 \, a^{4} b^{4} x^{4} + 13160 \, a^{5} b^{3} x^{3} + 11970 \, a^{6} b^{2} x^{2} + 5508 \, a^{7} b x + 1023 \, a^{8}}{30 \, {\left (b^{15} x^{6} + 6 \, a b^{14} x^{5} + 15 \, a^{2} b^{13} x^{4} + 20 \, a^{3} b^{12} x^{3} + 15 \, a^{4} b^{11} x^{2} + 6 \, a^{5} b^{10} x + a^{6} b^{9}\right )}} + \frac {28 \, a^{2} \log \left (b x + a\right )}{b^{9}} + \frac {b x^{2} - 14 \, a x}{2 \, b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^7,x, algorithm="maxima")

[Out]

1/30*(1680*a^3*b^5*x^5 + 7350*a^4*b^4*x^4 + 13160*a^5*b^3*x^3 + 11970*a^6*b^2*x^2 + 5508*a^7*b*x + 1023*a^8)/(
b^15*x^6 + 6*a*b^14*x^5 + 15*a^2*b^13*x^4 + 20*a^3*b^12*x^3 + 15*a^4*b^11*x^2 + 6*a^5*b^10*x + a^6*b^9) + 28*a
^2*log(b*x + a)/b^9 + 1/2*(b*x^2 - 14*a*x)/b^8

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mupad [B]  time = 0.18, size = 102, normalized size = 0.80 \begin {gather*} \frac {\frac {{\left (a+b\,x\right )}^2}{2}+\frac {56\,a^3}{a+b\,x}-\frac {35\,a^4}{{\left (a+b\,x\right )}^2}+\frac {56\,a^5}{3\,{\left (a+b\,x\right )}^3}-\frac {7\,a^6}{{\left (a+b\,x\right )}^4}+\frac {8\,a^7}{5\,{\left (a+b\,x\right )}^5}-\frac {a^8}{6\,{\left (a+b\,x\right )}^6}+28\,a^2\,\ln \left (a+b\,x\right )-8\,a\,b\,x}{b^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x)^7,x)

[Out]

((a + b*x)^2/2 + (56*a^3)/(a + b*x) - (35*a^4)/(a + b*x)^2 + (56*a^5)/(3*(a + b*x)^3) - (7*a^6)/(a + b*x)^4 +
(8*a^7)/(5*(a + b*x)^5) - a^8/(6*(a + b*x)^6) + 28*a^2*log(a + b*x) - 8*a*b*x)/b^9

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sympy [A]  time = 0.84, size = 165, normalized size = 1.29 \begin {gather*} \frac {28 a^{2} \log {\left (a + b x \right )}}{b^{9}} - \frac {7 a x}{b^{8}} + \frac {1023 a^{8} + 5508 a^{7} b x + 11970 a^{6} b^{2} x^{2} + 13160 a^{5} b^{3} x^{3} + 7350 a^{4} b^{4} x^{4} + 1680 a^{3} b^{5} x^{5}}{30 a^{6} b^{9} + 180 a^{5} b^{10} x + 450 a^{4} b^{11} x^{2} + 600 a^{3} b^{12} x^{3} + 450 a^{2} b^{13} x^{4} + 180 a b^{14} x^{5} + 30 b^{15} x^{6}} + \frac {x^{2}}{2 b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x+a)**7,x)

[Out]

28*a**2*log(a + b*x)/b**9 - 7*a*x/b**8 + (1023*a**8 + 5508*a**7*b*x + 11970*a**6*b**2*x**2 + 13160*a**5*b**3*x
**3 + 7350*a**4*b**4*x**4 + 1680*a**3*b**5*x**5)/(30*a**6*b**9 + 180*a**5*b**10*x + 450*a**4*b**11*x**2 + 600*
a**3*b**12*x**3 + 450*a**2*b**13*x**4 + 180*a*b**14*x**5 + 30*b**15*x**6) + x**2/(2*b**7)

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