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

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^8}{72 a^2 (a+b x)^8}+\frac {x^8}{9 a (a+b x)^9} \end {gather*}

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 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^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*}

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Mathematica [B]  time = 0.01, size = 86, normalized size = 2.46 \begin {gather*} -\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 b^8 (a+b x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x)^10,x]

[Out]

-1/72*(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)/(b^8*(a + b*x)^9)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^7}{(a+b x)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7/(a + b*x)^10,x]

[Out]

IntegrateAlgebraic[x^7/(a + b*x)^10, x]

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fricas [B]  time = 0.88, size = 175, normalized size = 5.00 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 0.96, size = 84, normalized size = 2.40 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.01, size = 117, normalized size = 3.34 \begin {gather*} \frac {a^{7}}{9 \left (b x +a \right )^{9} b^{8}}-\frac {7 a^{6}}{8 \left (b x +a \right )^{8} b^{8}}+\frac {3 a^{5}}{\left (b x +a \right )^{7} b^{8}}-\frac {35 a^{4}}{6 \left (b x +a \right )^{6} b^{8}}+\frac {7 a^{3}}{\left (b x +a \right )^{5} b^{8}}-\frac {21 a^{2}}{4 \left (b x +a \right )^{4} b^{8}}+\frac {7 a}{3 \left (b x +a \right )^{3} b^{8}}-\frac {1}{2 \left (b x +a \right )^{2} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.59, size = 175, normalized size = 5.00 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.13, size = 22, normalized size = 0.63 \begin {gather*} \frac {x^8\,\left (9\,a+b\,x\right )}{72\,a^2\,{\left (a+b\,x\right )}^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x)^10,x)

[Out]

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

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sympy [B]  time = 1.00, size = 187, normalized size = 5.34 \begin {gather*} \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 {gather*}

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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