3.227 \(\int \frac{x^7}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^8}{72 a^2 (a+b x)^8}+\frac{x^8}{9 a (a+b x)^9} \]

[Out]

x^8/(9*a*(a + b*x)^9) + x^8/(72*a^2*(a + b*x)^8)

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Rubi [A]  time = 0.0051056, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ \frac{x^8}{72 a^2 (a+b x)^8}+\frac{x^8}{9 a (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x)^10,x]

[Out]

x^8/(9*a*(a + b*x)^9) + x^8/(72*a^2*(a + b*x)^8)

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^7}{(a+b x)^{10}} \, dx &=\frac{x^8}{9 a (a+b x)^9}+\frac{\int \frac{x^7}{(a+b x)^9} \, dx}{9 a}\\ &=\frac{x^8}{9 a (a+b x)^9}+\frac{x^8}{72 a^2 (a+b x)^8}\\ \end{align*}

Mathematica [B]  time = 0.0198964, size = 86, normalized size = 2.46 \[ -\frac{36 a^5 b^2 x^2+84 a^4 b^3 x^3+126 a^3 b^4 x^4+126 a^2 b^5 x^5+9 a^6 b x+a^7+84 a b^6 x^6+36 b^7 x^7}{72 b^8 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x)^10,x]

[Out]

-(a^7 + 9*a^6*b*x + 36*a^5*b^2*x^2 + 84*a^4*b^3*x^3 + 126*a^3*b^4*x^4 + 126*a^2*b^5*x^5 + 84*a*b^6*x^6 + 36*b^
7*x^7)/(72*b^8*(a + b*x)^9)

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Maple [B]  time = 0.006, size = 117, normalized size = 3.3 \begin{align*} 7\,{\frac{{a}^{3}}{{b}^{8} \left ( bx+a \right ) ^{5}}}+{\frac{{a}^{7}}{9\,{b}^{8} \left ( bx+a \right ) ^{9}}}-{\frac{7\,{a}^{6}}{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{21\,{a}^{2}}{4\,{b}^{8} \left ( bx+a \right ) ^{4}}}+3\,{\frac{{a}^{5}}{{b}^{8} \left ( bx+a \right ) ^{7}}}-{\frac{1}{2\,{b}^{8} \left ( bx+a \right ) ^{2}}}-{\frac{35\,{a}^{4}}{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}+{\frac{7\,a}{3\,{b}^{8} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x+a)^10,x)

[Out]

7/b^8*a^3/(b*x+a)^5+1/9/b^8*a^7/(b*x+a)^9-7/8/b^8*a^6/(b*x+a)^8-21/4/b^8*a^2/(b*x+a)^4+3/b^8*a^5/(b*x+a)^7-1/2
/b^8/(b*x+a)^2-35/6/b^8*a^4/(b*x+a)^6+7/3/b^8*a/(b*x+a)^3

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Maxima [B]  time = 1.07251, size = 236, normalized size = 6.74 \begin{align*} -\frac{36 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 126 \, a^{3} b^{4} x^{4} + 84 \, a^{4} b^{3} x^{3} + 36 \, a^{5} b^{2} x^{2} + 9 \, a^{6} b x + a^{7}}{72 \,{\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/72*(36*b^7*x^7 + 84*a*b^6*x^6 + 126*a^2*b^5*x^5 + 126*a^3*b^4*x^4 + 84*a^4*b^3*x^3 + 36*a^5*b^2*x^2 + 9*a^6
*b*x + a^7)/(b^17*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^5 + 126*a^5*b^12*x^4
+ 84*a^6*b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a^9*b^8)

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Fricas [B]  time = 1.49776, size = 382, normalized size = 10.91 \begin{align*} -\frac{36 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 126 \, a^{3} b^{4} x^{4} + 84 \, a^{4} b^{3} x^{3} + 36 \, a^{5} b^{2} x^{2} + 9 \, a^{6} b x + a^{7}}{72 \,{\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/72*(36*b^7*x^7 + 84*a*b^6*x^6 + 126*a^2*b^5*x^5 + 126*a^3*b^4*x^4 + 84*a^4*b^3*x^3 + 36*a^5*b^2*x^2 + 9*a^6
*b*x + a^7)/(b^17*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^5 + 126*a^5*b^12*x^4
+ 84*a^6*b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a^9*b^8)

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Sympy [B]  time = 1.47864, size = 187, normalized size = 5.34 \begin{align*} - \frac{a^{7} + 9 a^{6} b x + 36 a^{5} b^{2} x^{2} + 84 a^{4} b^{3} x^{3} + 126 a^{3} b^{4} x^{4} + 126 a^{2} b^{5} x^{5} + 84 a b^{6} x^{6} + 36 b^{7} x^{7}}{72 a^{9} b^{8} + 648 a^{8} b^{9} x + 2592 a^{7} b^{10} x^{2} + 6048 a^{6} b^{11} x^{3} + 9072 a^{5} b^{12} x^{4} + 9072 a^{4} b^{13} x^{5} + 6048 a^{3} b^{14} x^{6} + 2592 a^{2} b^{15} x^{7} + 648 a b^{16} x^{8} + 72 b^{17} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x+a)**10,x)

[Out]

-(a**7 + 9*a**6*b*x + 36*a**5*b**2*x**2 + 84*a**4*b**3*x**3 + 126*a**3*b**4*x**4 + 126*a**2*b**5*x**5 + 84*a*b
**6*x**6 + 36*b**7*x**7)/(72*a**9*b**8 + 648*a**8*b**9*x + 2592*a**7*b**10*x**2 + 6048*a**6*b**11*x**3 + 9072*
a**5*b**12*x**4 + 9072*a**4*b**13*x**5 + 6048*a**3*b**14*x**6 + 2592*a**2*b**15*x**7 + 648*a*b**16*x**8 + 72*b
**17*x**9)

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Giac [B]  time = 1.21552, size = 113, normalized size = 3.23 \begin{align*} -\frac{36 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 126 \, a^{3} b^{4} x^{4} + 84 \, a^{4} b^{3} x^{3} + 36 \, a^{5} b^{2} x^{2} + 9 \, a^{6} b x + a^{7}}{72 \,{\left (b x + a\right )}^{9} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/72*(36*b^7*x^7 + 84*a*b^6*x^6 + 126*a^2*b^5*x^5 + 126*a^3*b^4*x^4 + 84*a^4*b^3*x^3 + 36*a^5*b^2*x^2 + 9*a^6
*b*x + a^7)/((b*x + a)^9*b^8)