3.230 \(\int \frac{x^4}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{a^4}{9 b^5 (a+b x)^9}+\frac{a^3}{2 b^5 (a+b x)^8}-\frac{6 a^2}{7 b^5 (a+b x)^7}+\frac{2 a}{3 b^5 (a+b x)^6}-\frac{1}{5 b^5 (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0401836, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{a^4}{9 b^5 (a+b x)^9}+\frac{a^3}{2 b^5 (a+b x)^8}-\frac{6 a^2}{7 b^5 (a+b x)^7}+\frac{2 a}{3 b^5 (a+b x)^6}-\frac{1}{5 b^5 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^10,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^4}{(a+b x)^{10}} \, dx &=\int \left (\frac{a^4}{b^4 (a+b x)^{10}}-\frac{4 a^3}{b^4 (a+b x)^9}+\frac{6 a^2}{b^4 (a+b x)^8}-\frac{4 a}{b^4 (a+b x)^7}+\frac{1}{b^4 (a+b x)^6}\right ) \, dx\\ &=-\frac{a^4}{9 b^5 (a+b x)^9}+\frac{a^3}{2 b^5 (a+b x)^8}-\frac{6 a^2}{7 b^5 (a+b x)^7}+\frac{2 a}{3 b^5 (a+b x)^6}-\frac{1}{5 b^5 (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0235085, size = 53, normalized size = 0.65 \[ -\frac{36 a^2 b^2 x^2+9 a^3 b x+a^4+84 a b^3 x^3+126 b^4 x^4}{630 b^5 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^10,x]

[Out]

-(a^4 + 9*a^3*b*x + 36*a^2*b^2*x^2 + 84*a*b^3*x^3 + 126*b^4*x^4)/(630*b^5*(a + b*x)^9)

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Maple [A]  time = 0.005, size = 72, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{9\,{b}^{5} \left ( bx+a \right ) ^{9}}}+{\frac{{a}^{3}}{2\,{b}^{5} \left ( bx+a \right ) ^{8}}}-{\frac{6\,{a}^{2}}{7\,{b}^{5} \left ( bx+a \right ) ^{7}}}+{\frac{2\,a}{3\,{b}^{5} \left ( bx+a \right ) ^{6}}}-{\frac{1}{5\,{b}^{5} \left ( bx+a \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^10,x)

[Out]

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

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Maxima [A]  time = 1.15553, size = 192, normalized size = 2.37 \begin{align*} -\frac{126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \,{\left (b^{14} x^{9} + 9 \, a b^{13} x^{8} + 36 \, a^{2} b^{12} x^{7} + 84 \, a^{3} b^{11} x^{6} + 126 \, a^{4} b^{10} x^{5} + 126 \, a^{5} b^{9} x^{4} + 84 \, a^{6} b^{8} x^{3} + 36 \, a^{7} b^{7} x^{2} + 9 \, a^{8} b^{6} x + a^{9} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/(b^14*x^9 + 9*a*b^13*x^8 + 36*a^2*b^12*
x^7 + 84*a^3*b^11*x^6 + 126*a^4*b^10*x^5 + 126*a^5*b^9*x^4 + 84*a^6*b^8*x^3 + 36*a^7*b^7*x^2 + 9*a^8*b^6*x + a
^9*b^5)

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Fricas [A]  time = 1.4368, size = 309, normalized size = 3.81 \begin{align*} -\frac{126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \,{\left (b^{14} x^{9} + 9 \, a b^{13} x^{8} + 36 \, a^{2} b^{12} x^{7} + 84 \, a^{3} b^{11} x^{6} + 126 \, a^{4} b^{10} x^{5} + 126 \, a^{5} b^{9} x^{4} + 84 \, a^{6} b^{8} x^{3} + 36 \, a^{7} b^{7} x^{2} + 9 \, a^{8} b^{6} x + a^{9} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/(b^14*x^9 + 9*a*b^13*x^8 + 36*a^2*b^12*
x^7 + 84*a^3*b^11*x^6 + 126*a^4*b^10*x^5 + 126*a^5*b^9*x^4 + 84*a^6*b^8*x^3 + 36*a^7*b^7*x^2 + 9*a^8*b^6*x + a
^9*b^5)

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Sympy [B]  time = 1.24915, size = 151, normalized size = 1.86 \begin{align*} - \frac{a^{4} + 9 a^{3} b x + 36 a^{2} b^{2} x^{2} + 84 a b^{3} x^{3} + 126 b^{4} x^{4}}{630 a^{9} b^{5} + 5670 a^{8} b^{6} x + 22680 a^{7} b^{7} x^{2} + 52920 a^{6} b^{8} x^{3} + 79380 a^{5} b^{9} x^{4} + 79380 a^{4} b^{10} x^{5} + 52920 a^{3} b^{11} x^{6} + 22680 a^{2} b^{12} x^{7} + 5670 a b^{13} x^{8} + 630 b^{14} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**10,x)

[Out]

-(a**4 + 9*a**3*b*x + 36*a**2*b**2*x**2 + 84*a*b**3*x**3 + 126*b**4*x**4)/(630*a**9*b**5 + 5670*a**8*b**6*x +
22680*a**7*b**7*x**2 + 52920*a**6*b**8*x**3 + 79380*a**5*b**9*x**4 + 79380*a**4*b**10*x**5 + 52920*a**3*b**11*
x**6 + 22680*a**2*b**12*x**7 + 5670*a*b**13*x**8 + 630*b**14*x**9)

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Giac [A]  time = 1.1815, size = 69, normalized size = 0.85 \begin{align*} -\frac{126 \, b^{4} x^{4} + 84 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x + a^{4}}{630 \,{\left (b x + a\right )}^{9} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 84*a*b^3*x^3 + 36*a^2*b^2*x^2 + 9*a^3*b*x + a^4)/((b*x + a)^9*b^5)