3.228 \(\int \frac{x^6}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=52 \[ \frac{x^7}{252 a^3 (a+b x)^7}+\frac{x^7}{36 a^2 (a+b x)^8}+\frac{x^7}{9 a (a+b x)^9} \]

[Out]

x^7/(9*a*(a + b*x)^9) + x^7/(36*a^2*(a + b*x)^8) + x^7/(252*a^3*(a + b*x)^7)

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Rubi [A]  time = 0.0096319, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{x^7}{252 a^3 (a+b x)^7}+\frac{x^7}{36 a^2 (a+b x)^8}+\frac{x^7}{9 a (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x)^10,x]

[Out]

x^7/(9*a*(a + b*x)^9) + x^7/(36*a^2*(a + b*x)^8) + x^7/(252*a^3*(a + b*x)^7)

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^6}{(a+b x)^{10}} \, dx &=\frac{x^7}{9 a (a+b x)^9}+\frac{2 \int \frac{x^6}{(a+b x)^9} \, dx}{9 a}\\ &=\frac{x^7}{9 a (a+b x)^9}+\frac{x^7}{36 a^2 (a+b x)^8}+\frac{\int \frac{x^6}{(a+b x)^8} \, dx}{36 a^2}\\ &=\frac{x^7}{9 a (a+b x)^9}+\frac{x^7}{36 a^2 (a+b x)^8}+\frac{x^7}{252 a^3 (a+b x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0254138, size = 75, normalized size = 1.44 \[ -\frac{36 a^4 b^2 x^2+84 a^3 b^3 x^3+126 a^2 b^4 x^4+9 a^5 b x+a^6+126 a b^5 x^5+84 b^6 x^6}{252 b^7 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x)^10,x]

[Out]

-(a^6 + 9*a^5*b*x + 36*a^4*b^2*x^2 + 84*a^3*b^3*x^3 + 126*a^2*b^4*x^4 + 126*a*b^5*x^5 + 84*b^6*x^6)/(252*b^7*(
a + b*x)^9)

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Maple [B]  time = 0.006, size = 102, normalized size = 2. \begin{align*} -{\frac{15\,{a}^{4}}{7\,{b}^{7} \left ( bx+a \right ) ^{7}}}-{\frac{{a}^{6}}{9\,{b}^{7} \left ( bx+a \right ) ^{9}}}+{\frac{3\,{a}^{5}}{4\,{b}^{7} \left ( bx+a \right ) ^{8}}}+{\frac{3\,a}{2\,{b}^{7} \left ( bx+a \right ) ^{4}}}+{\frac{10\,{a}^{3}}{3\,{b}^{7} \left ( bx+a \right ) ^{6}}}-{\frac{1}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}}-3\,{\frac{{a}^{2}}{{b}^{7} \left ( bx+a \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x+a)^10,x)

[Out]

-15/7/b^7*a^4/(b*x+a)^7-1/9/b^7*a^6/(b*x+a)^9+3/4/b^7*a^5/(b*x+a)^8+3/2/b^7*a/(b*x+a)^4+10/3/b^7*a^3/(b*x+a)^6
-1/3/b^7/(b*x+a)^3-3/b^7*a^2/(b*x+a)^5

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Maxima [B]  time = 1.11422, size = 221, normalized size = 4.25 \begin{align*} -\frac{84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \,{\left (b^{16} x^{9} + 9 \, a b^{15} x^{8} + 36 \, a^{2} b^{14} x^{7} + 84 \, a^{3} b^{13} x^{6} + 126 \, a^{4} b^{12} x^{5} + 126 \, a^{5} b^{11} x^{4} + 84 \, a^{6} b^{10} x^{3} + 36 \, a^{7} b^{9} x^{2} + 9 \, a^{8} b^{8} x + a^{9} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/252*(84*b^6*x^6 + 126*a*b^5*x^5 + 126*a^2*b^4*x^4 + 84*a^3*b^3*x^3 + 36*a^4*b^2*x^2 + 9*a^5*b*x + a^6)/(b^1
6*x^9 + 9*a*b^15*x^8 + 36*a^2*b^14*x^7 + 84*a^3*b^13*x^6 + 126*a^4*b^12*x^5 + 126*a^5*b^11*x^4 + 84*a^6*b^10*x
^3 + 36*a^7*b^9*x^2 + 9*a^8*b^8*x + a^9*b^7)

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Fricas [B]  time = 1.53719, size = 359, normalized size = 6.9 \begin{align*} -\frac{84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \,{\left (b^{16} x^{9} + 9 \, a b^{15} x^{8} + 36 \, a^{2} b^{14} x^{7} + 84 \, a^{3} b^{13} x^{6} + 126 \, a^{4} b^{12} x^{5} + 126 \, a^{5} b^{11} x^{4} + 84 \, a^{6} b^{10} x^{3} + 36 \, a^{7} b^{9} x^{2} + 9 \, a^{8} b^{8} x + a^{9} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*x^6 + 126*a*b^5*x^5 + 126*a^2*b^4*x^4 + 84*a^3*b^3*x^3 + 36*a^4*b^2*x^2 + 9*a^5*b*x + a^6)/(b^1
6*x^9 + 9*a*b^15*x^8 + 36*a^2*b^14*x^7 + 84*a^3*b^13*x^6 + 126*a^4*b^12*x^5 + 126*a^5*b^11*x^4 + 84*a^6*b^10*x
^3 + 36*a^7*b^9*x^2 + 9*a^8*b^8*x + a^9*b^7)

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Sympy [B]  time = 1.41402, size = 175, normalized size = 3.37 \begin{align*} - \frac{a^{6} + 9 a^{5} b x + 36 a^{4} b^{2} x^{2} + 84 a^{3} b^{3} x^{3} + 126 a^{2} b^{4} x^{4} + 126 a b^{5} x^{5} + 84 b^{6} x^{6}}{252 a^{9} b^{7} + 2268 a^{8} b^{8} x + 9072 a^{7} b^{9} x^{2} + 21168 a^{6} b^{10} x^{3} + 31752 a^{5} b^{11} x^{4} + 31752 a^{4} b^{12} x^{5} + 21168 a^{3} b^{13} x^{6} + 9072 a^{2} b^{14} x^{7} + 2268 a b^{15} x^{8} + 252 b^{16} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**10,x)

[Out]

-(a**6 + 9*a**5*b*x + 36*a**4*b**2*x**2 + 84*a**3*b**3*x**3 + 126*a**2*b**4*x**4 + 126*a*b**5*x**5 + 84*b**6*x
**6)/(252*a**9*b**7 + 2268*a**8*b**8*x + 9072*a**7*b**9*x**2 + 21168*a**6*b**10*x**3 + 31752*a**5*b**11*x**4 +
31752*a**4*b**12*x**5 + 21168*a**3*b**13*x**6 + 9072*a**2*b**14*x**7 + 2268*a*b**15*x**8 + 252*b**16*x**9)

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Giac [A]  time = 1.21344, size = 99, normalized size = 1.9 \begin{align*} -\frac{84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \,{\left (b x + a\right )}^{9} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6 + 126*a*b^5*x^5 + 126*a^2*b^4*x^4 + 84*a^3*b^3*x^3 + 36*a^4*b^2*x^2 + 9*a^5*b*x + a^6)/((b*
x + a)^9*b^7)