3.2.93 \(\int \frac {x^8}{(a+b x)^4} \, dx\)

Optimal. Leaf size=114 \[ -\frac {a^8}{3 b^9 (a+b x)^3}+\frac {4 a^7}{b^9 (a+b x)^2}-\frac {28 a^6}{b^9 (a+b x)}-\frac {56 a^5 \log (a+b x)}{b^9}+\frac {35 a^4 x}{b^8}-\frac {10 a^3 x^2}{b^7}+\frac {10 a^2 x^3}{3 b^6}-\frac {a x^4}{b^5}+\frac {x^5}{5 b^4} \]

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Rubi [A]  time = 0.09, antiderivative size = 114, 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 {10 a^3 x^2}{b^7}+\frac {10 a^2 x^3}{3 b^6}-\frac {a^8}{3 b^9 (a+b x)^3}+\frac {4 a^7}{b^9 (a+b x)^2}-\frac {28 a^6}{b^9 (a+b x)}+\frac {35 a^4 x}{b^8}-\frac {56 a^5 \log (a+b x)}{b^9}-\frac {a x^4}{b^5}+\frac {x^5}{5 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x)^4,x]

[Out]

(35*a^4*x)/b^8 - (10*a^3*x^2)/b^7 + (10*a^2*x^3)/(3*b^6) - (a*x^4)/b^5 + x^5/(5*b^4) - a^8/(3*b^9*(a + b*x)^3)
+ (4*a^7)/(b^9*(a + b*x)^2) - (28*a^6)/(b^9*(a + b*x)) - (56*a^5*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)^4} \, dx &=\int \left (\frac {35 a^4}{b^8}-\frac {20 a^3 x}{b^7}+\frac {10 a^2 x^2}{b^6}-\frac {4 a x^3}{b^5}+\frac {x^4}{b^4}+\frac {a^8}{b^8 (a+b x)^4}-\frac {8 a^7}{b^8 (a+b x)^3}+\frac {28 a^6}{b^8 (a+b x)^2}-\frac {56 a^5}{b^8 (a+b x)}\right ) \, dx\\ &=\frac {35 a^4 x}{b^8}-\frac {10 a^3 x^2}{b^7}+\frac {10 a^2 x^3}{3 b^6}-\frac {a x^4}{b^5}+\frac {x^5}{5 b^4}-\frac {a^8}{3 b^9 (a+b x)^3}+\frac {4 a^7}{b^9 (a+b x)^2}-\frac {28 a^6}{b^9 (a+b x)}-\frac {56 a^5 \log (a+b x)}{b^9}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 101, normalized size = 0.89 \begin {gather*} \frac {-\frac {5 a^8}{(a+b x)^3}+\frac {60 a^7}{(a+b x)^2}-\frac {420 a^6}{a+b x}-840 a^5 \log (a+b x)+525 a^4 b x-150 a^3 b^2 x^2+50 a^2 b^3 x^3-15 a b^4 x^4+3 b^5 x^5}{15 b^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x)^4,x]

[Out]

(525*a^4*b*x - 150*a^3*b^2*x^2 + 50*a^2*b^3*x^3 - 15*a*b^4*x^4 + 3*b^5*x^5 - (5*a^8)/(a + b*x)^3 + (60*a^7)/(a
+ b*x)^2 - (420*a^6)/(a + b*x) - 840*a^5*Log[a + b*x])/(15*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)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^8/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[x^8/(a + b*x)^4, x]

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fricas [A]  time = 0.86, size = 162, normalized size = 1.42 \begin {gather*} \frac {3 \, b^{8} x^{8} - 6 \, a b^{7} x^{7} + 14 \, a^{2} b^{6} x^{6} - 42 \, a^{3} b^{5} x^{5} + 210 \, a^{4} b^{4} x^{4} + 1175 \, a^{5} b^{3} x^{3} + 1005 \, a^{6} b^{2} x^{2} - 255 \, a^{7} b x - 365 \, a^{8} - 840 \, {\left (a^{5} b^{3} x^{3} + 3 \, a^{6} b^{2} x^{2} + 3 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{15 \, {\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^4,x, algorithm="fricas")

[Out]

1/15*(3*b^8*x^8 - 6*a*b^7*x^7 + 14*a^2*b^6*x^6 - 42*a^3*b^5*x^5 + 210*a^4*b^4*x^4 + 1175*a^5*b^3*x^3 + 1005*a^
6*b^2*x^2 - 255*a^7*b*x - 365*a^8 - 840*(a^5*b^3*x^3 + 3*a^6*b^2*x^2 + 3*a^7*b*x + a^8)*log(b*x + a))/(b^12*x^
3 + 3*a*b^11*x^2 + 3*a^2*b^10*x + a^3*b^9)

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giac [A]  time = 1.11, size = 106, normalized size = 0.93 \begin {gather*} -\frac {56 \, a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{9}} - \frac {84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \, {\left (b x + a\right )}^{3} b^{9}} + \frac {3 \, b^{16} x^{5} - 15 \, a b^{15} x^{4} + 50 \, a^{2} b^{14} x^{3} - 150 \, a^{3} b^{13} x^{2} + 525 \, a^{4} b^{12} x}{15 \, b^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^4,x, algorithm="giac")

[Out]

-56*a^5*log(abs(b*x + a))/b^9 - 1/3*(84*a^6*b^2*x^2 + 156*a^7*b*x + 73*a^8)/((b*x + a)^3*b^9) + 1/15*(3*b^16*x
^5 - 15*a*b^15*x^4 + 50*a^2*b^14*x^3 - 150*a^3*b^13*x^2 + 525*a^4*b^12*x)/b^20

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maple [A]  time = 0.01, size = 109, normalized size = 0.96 \begin {gather*} \frac {x^{5}}{5 b^{4}}-\frac {a \,x^{4}}{b^{5}}-\frac {a^{8}}{3 \left (b x +a \right )^{3} b^{9}}+\frac {10 a^{2} x^{3}}{3 b^{6}}+\frac {4 a^{7}}{\left (b x +a \right )^{2} b^{9}}-\frac {10 a^{3} x^{2}}{b^{7}}-\frac {28 a^{6}}{\left (b x +a \right ) b^{9}}-\frac {56 a^{5} \ln \left (b x +a \right )}{b^{9}}+\frac {35 a^{4} x}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x+a)^4,x)

[Out]

35*a^4*x/b^8-10*a^3*x^2/b^7+10/3*a^2*x^3/b^6-a*x^4/b^5+1/5*x^5/b^4-1/3*a^8/b^9/(b*x+a)^3+4*a^7/b^9/(b*x+a)^2-2
8*a^6/b^9/(b*x+a)-56*a^5*ln(b*x+a)/b^9

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maxima [A]  time = 1.44, size = 125, normalized size = 1.10 \begin {gather*} -\frac {84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \, {\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} - \frac {56 \, a^{5} \log \left (b x + a\right )}{b^{9}} + \frac {3 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 50 \, a^{2} b^{2} x^{3} - 150 \, a^{3} b x^{2} + 525 \, a^{4} x}{15 \, b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/3*(84*a^6*b^2*x^2 + 156*a^7*b*x + 73*a^8)/(b^12*x^3 + 3*a*b^11*x^2 + 3*a^2*b^10*x + a^3*b^9) - 56*a^5*log(b
*x + a)/b^9 + 1/15*(3*b^4*x^5 - 15*a*b^3*x^4 + 50*a^2*b^2*x^3 - 150*a^3*b*x^2 + 525*a^4*x)/b^8

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mupad [B]  time = 0.37, size = 103, normalized size = 0.90 \begin {gather*} -\frac {2\,a\,{\left (a+b\,x\right )}^4-\frac {{\left (a+b\,x\right )}^5}{5}-\frac {28\,a^2\,{\left (a+b\,x\right )}^3}{3}+28\,a^3\,{\left (a+b\,x\right )}^2+\frac {28\,a^6}{a+b\,x}-\frac {4\,a^7}{{\left (a+b\,x\right )}^2}+\frac {a^8}{3\,{\left (a+b\,x\right )}^3}+56\,a^5\,\ln \left (a+b\,x\right )-70\,a^4\,b\,x}{b^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x)^4,x)

[Out]

-(2*a*(a + b*x)^4 - (a + b*x)^5/5 - (28*a^2*(a + b*x)^3)/3 + 28*a^3*(a + b*x)^2 + (28*a^6)/(a + b*x) - (4*a^7)
/(a + b*x)^2 + a^8/(3*(a + b*x)^3) + 56*a^5*log(a + b*x) - 70*a^4*b*x)/b^9

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sympy [A]  time = 0.52, size = 131, normalized size = 1.15 \begin {gather*} - \frac {56 a^{5} \log {\left (a + b x \right )}}{b^{9}} + \frac {35 a^{4} x}{b^{8}} - \frac {10 a^{3} x^{2}}{b^{7}} + \frac {10 a^{2} x^{3}}{3 b^{6}} - \frac {a x^{4}}{b^{5}} + \frac {- 73 a^{8} - 156 a^{7} b x - 84 a^{6} b^{2} x^{2}}{3 a^{3} b^{9} + 9 a^{2} b^{10} x + 9 a b^{11} x^{2} + 3 b^{12} x^{3}} + \frac {x^{5}}{5 b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x+a)**4,x)

[Out]

-56*a**5*log(a + b*x)/b**9 + 35*a**4*x/b**8 - 10*a**3*x**2/b**7 + 10*a**2*x**3/(3*b**6) - a*x**4/b**5 + (-73*a
**8 - 156*a**7*b*x - 84*a**6*b**2*x**2)/(3*a**3*b**9 + 9*a**2*b**10*x + 9*a*b**11*x**2 + 3*b**12*x**3) + x**5/
(5*b**4)

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