3.3.29 \(\int \frac {x^5}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=69 \[ \frac {x^6}{504 a^4 (a+b x)^6}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{9 a (a+b x)^9} \]

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Rubi [A]  time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {45, 37} \begin {gather*} \frac {x^6}{504 a^4 (a+b x)^6}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{9 a (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x)^10,x]

[Out]

x^6/(9*a*(a + b*x)^9) + x^6/(24*a^2*(a + b*x)^8) + x^6/(84*a^3*(a + b*x)^7) + x^6/(504*a^4*(a + b*x)^6)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {x^5}{(a+b x)^{10}} \, dx &=\frac {x^6}{9 a (a+b x)^9}+\frac {\int \frac {x^5}{(a+b x)^9} \, dx}{3 a}\\ &=\frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {\int \frac {x^5}{(a+b x)^8} \, dx}{12 a^2}\\ &=\frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {\int \frac {x^5}{(a+b x)^7} \, dx}{84 a^3}\\ &=\frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {x^6}{504 a^4 (a+b x)^6}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 64, normalized size = 0.93 \begin {gather*} -\frac {a^5+9 a^4 b x+36 a^3 b^2 x^2+84 a^2 b^3 x^3+126 a b^4 x^4+126 b^5 x^5}{504 b^6 (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x)^10,x]

[Out]

-1/504*(a^5 + 9*a^4*b*x + 36*a^3*b^2*x^2 + 84*a^2*b^3*x^3 + 126*a*b^4*x^4 + 126*b^5*x^5)/(b^6*(a + b*x)^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{(a+b x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5/(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[x^5/(a + b*x)^10, x]

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fricas [B]  time = 0.98, size = 153, normalized size = 2.22 \begin {gather*} -\frac {126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \, {\left (b^{15} x^{9} + 9 \, a b^{14} x^{8} + 36 \, a^{2} b^{13} x^{7} + 84 \, a^{3} b^{12} x^{6} + 126 \, a^{4} b^{11} x^{5} + 126 \, a^{5} b^{10} x^{4} + 84 \, a^{6} b^{9} x^{3} + 36 \, a^{7} b^{8} x^{2} + 9 \, a^{8} b^{7} x + a^{9} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*x + a^5)/(b^15*x^9 + 9*a*b^14*
x^8 + 36*a^2*b^13*x^7 + 84*a^3*b^12*x^6 + 126*a^4*b^11*x^5 + 126*a^5*b^10*x^4 + 84*a^6*b^9*x^3 + 36*a^7*b^8*x^
2 + 9*a^8*b^7*x + a^9*b^6)

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giac [A]  time = 0.90, size = 62, normalized size = 0.90 \begin {gather*} -\frac {126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \, {\left (b x + a\right )}^{9} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*x + a^5)/((b*x + a)^9*b^6)

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maple [A]  time = 0.00, size = 86, normalized size = 1.25 \begin {gather*} \frac {a^{5}}{9 \left (b x +a \right )^{9} b^{6}}-\frac {5 a^{4}}{8 \left (b x +a \right )^{8} b^{6}}+\frac {10 a^{3}}{7 \left (b x +a \right )^{7} b^{6}}-\frac {5 a^{2}}{3 \left (b x +a \right )^{6} b^{6}}+\frac {a}{\left (b x +a \right )^{5} b^{6}}-\frac {1}{4 \left (b x +a \right )^{4} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^10,x)

[Out]

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

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maxima [B]  time = 1.54, size = 153, normalized size = 2.22 \begin {gather*} -\frac {126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \, {\left (b^{15} x^{9} + 9 \, a b^{14} x^{8} + 36 \, a^{2} b^{13} x^{7} + 84 \, a^{3} b^{12} x^{6} + 126 \, a^{4} b^{11} x^{5} + 126 \, a^{5} b^{10} x^{4} + 84 \, a^{6} b^{9} x^{3} + 36 \, a^{7} b^{8} x^{2} + 9 \, a^{8} b^{7} x + a^{9} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*x + a^5)/(b^15*x^9 + 9*a*b^14*
x^8 + 36*a^2*b^13*x^7 + 84*a^3*b^12*x^6 + 126*a^4*b^11*x^5 + 126*a^5*b^10*x^4 + 84*a^6*b^9*x^3 + 36*a^7*b^8*x^
2 + 9*a^8*b^7*x + a^9*b^6)

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mupad [B]  time = 0.08, size = 71, normalized size = 1.03 \begin {gather*} \frac {\frac {a}{{\left (a+b\,x\right )}^5}-\frac {1}{4\,{\left (a+b\,x\right )}^4}-\frac {5\,a^2}{3\,{\left (a+b\,x\right )}^6}+\frac {10\,a^3}{7\,{\left (a+b\,x\right )}^7}-\frac {5\,a^4}{8\,{\left (a+b\,x\right )}^8}+\frac {a^5}{9\,{\left (a+b\,x\right )}^9}}{b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b*x)^10,x)

[Out]

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

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sympy [B]  time = 0.84, size = 163, normalized size = 2.36 \begin {gather*} \frac {- a^{5} - 9 a^{4} b x - 36 a^{3} b^{2} x^{2} - 84 a^{2} b^{3} x^{3} - 126 a b^{4} x^{4} - 126 b^{5} x^{5}}{504 a^{9} b^{6} + 4536 a^{8} b^{7} x + 18144 a^{7} b^{8} x^{2} + 42336 a^{6} b^{9} x^{3} + 63504 a^{5} b^{10} x^{4} + 63504 a^{4} b^{11} x^{5} + 42336 a^{3} b^{12} x^{6} + 18144 a^{2} b^{13} x^{7} + 4536 a b^{14} x^{8} + 504 b^{15} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**10,x)

[Out]

(-a**5 - 9*a**4*b*x - 36*a**3*b**2*x**2 - 84*a**2*b**3*x**3 - 126*a*b**4*x**4 - 126*b**5*x**5)/(504*a**9*b**6
+ 4536*a**8*b**7*x + 18144*a**7*b**8*x**2 + 42336*a**6*b**9*x**3 + 63504*a**5*b**10*x**4 + 63504*a**4*b**11*x*
*5 + 42336*a**3*b**12*x**6 + 18144*a**2*b**13*x**7 + 4536*a*b**14*x**8 + 504*b**15*x**9)

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