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

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

[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)

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Rubi [A]  time = 0.0155327, antiderivative size = 69, normalized size of antiderivative = 1., 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} \[ \frac{x^6}{504 a^4 (a+b x)^6}+\frac{x^6}{84 a^3 (a+b x)^7}+\frac{x^6}{24 a^2 (a+b x)^8}+\frac{x^6}{9 a (a+b x)^9} \]

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 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])

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]

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*}

Mathematica [A]  time = 0.0227998, size = 64, normalized size = 0.93 \[ -\frac{36 a^3 b^2 x^2+84 a^2 b^3 x^3+9 a^4 b x+a^5+126 a b^4 x^4+126 b^5 x^5}{504 b^6 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x)^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*b^6*(a + b*x)^9)

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Maple [A]  time = 0.005, size = 86, normalized size = 1.3 \begin{align*} -{\frac{5\,{a}^{4}}{8\,{b}^{6} \left ( bx+a \right ) ^{8}}}+{\frac{10\,{a}^{3}}{7\,{b}^{6} \left ( bx+a \right ) ^{7}}}+{\frac{{a}^{5}}{9\,{b}^{6} \left ( bx+a \right ) ^{9}}}-{\frac{1}{4\,{b}^{6} \left ( bx+a \right ) ^{4}}}-{\frac{5\,{a}^{2}}{3\,{b}^{6} \left ( bx+a \right ) ^{6}}}+{\frac{a}{{b}^{6} \left ( bx+a \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.1517, size = 207, normalized size = 3. \begin{align*} -\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{align*}

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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Fricas [B]  time = 1.54984, size = 335, normalized size = 4.86 \begin{align*} -\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{align*}

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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Sympy [B]  time = 1.32557, size = 163, normalized size = 2.36 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.18393, size = 84, normalized size = 1.22 \begin{align*} -\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{align*}

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)