3.231 \(\int \frac{x^3}{(a+b x)^{10}} \, dx\)

Optimal. Leaf size=64 \[ \frac{a^3}{9 b^4 (a+b x)^9}-\frac{3 a^2}{8 b^4 (a+b x)^8}+\frac{3 a}{7 b^4 (a+b x)^7}-\frac{1}{6 b^4 (a+b x)^6} \]

[Out]

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

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Rubi [A]  time = 0.0297847, antiderivative size = 64, 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} \[ \frac{a^3}{9 b^4 (a+b x)^9}-\frac{3 a^2}{8 b^4 (a+b x)^8}+\frac{3 a}{7 b^4 (a+b x)^7}-\frac{1}{6 b^4 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x)^10,x]

[Out]

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

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

Mathematica [A]  time = 0.0178355, size = 42, normalized size = 0.66 \[ -\frac{9 a^2 b x+a^3+36 a b^2 x^2+84 b^3 x^3}{504 b^4 (a+b x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x)^10,x]

[Out]

-(a^3 + 9*a^2*b*x + 36*a*b^2*x^2 + 84*b^3*x^3)/(504*b^4*(a + b*x)^9)

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Maple [A]  time = 0.006, size = 57, normalized size = 0.9 \begin{align*}{\frac{{a}^{3}}{9\,{b}^{4} \left ( bx+a \right ) ^{9}}}-{\frac{3\,{a}^{2}}{8\,{b}^{4} \left ( bx+a \right ) ^{8}}}+{\frac{3\,a}{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{1}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x+a)^10,x)

[Out]

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

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Maxima [B]  time = 1.09933, size = 177, normalized size = 2.77 \begin{align*} -\frac{84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \,{\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/504*(84*b^3*x^3 + 36*a*b^2*x^2 + 9*a^2*b*x + a^3)/(b^13*x^9 + 9*a*b^12*x^8 + 36*a^2*b^11*x^7 + 84*a^3*b^10*
x^6 + 126*a^4*b^9*x^5 + 126*a^5*b^8*x^4 + 84*a^6*b^7*x^3 + 36*a^7*b^6*x^2 + 9*a^8*b^5*x + a^9*b^4)

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Fricas [B]  time = 1.47435, size = 284, normalized size = 4.44 \begin{align*} -\frac{84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \,{\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/504*(84*b^3*x^3 + 36*a*b^2*x^2 + 9*a^2*b*x + a^3)/(b^13*x^9 + 9*a*b^12*x^8 + 36*a^2*b^11*x^7 + 84*a^3*b^10*
x^6 + 126*a^4*b^9*x^5 + 126*a^5*b^8*x^4 + 84*a^6*b^7*x^3 + 36*a^7*b^6*x^2 + 9*a^8*b^5*x + a^9*b^4)

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Sympy [B]  time = 1.10376, size = 139, normalized size = 2.17 \begin{align*} - \frac{a^{3} + 9 a^{2} b x + 36 a b^{2} x^{2} + 84 b^{3} x^{3}}{504 a^{9} b^{4} + 4536 a^{8} b^{5} x + 18144 a^{7} b^{6} x^{2} + 42336 a^{6} b^{7} x^{3} + 63504 a^{5} b^{8} x^{4} + 63504 a^{4} b^{9} x^{5} + 42336 a^{3} b^{10} x^{6} + 18144 a^{2} b^{11} x^{7} + 4536 a b^{12} x^{8} + 504 b^{13} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x+a)**10,x)

[Out]

-(a**3 + 9*a**2*b*x + 36*a*b**2*x**2 + 84*b**3*x**3)/(504*a**9*b**4 + 4536*a**8*b**5*x + 18144*a**7*b**6*x**2
+ 42336*a**6*b**7*x**3 + 63504*a**5*b**8*x**4 + 63504*a**4*b**9*x**5 + 42336*a**3*b**10*x**6 + 18144*a**2*b**1
1*x**7 + 4536*a*b**12*x**8 + 504*b**13*x**9)

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Giac [A]  time = 1.21656, size = 54, normalized size = 0.84 \begin{align*} -\frac{84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \,{\left (b x + a\right )}^{9} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/504*(84*b^3*x^3 + 36*a*b^2*x^2 + 9*a^2*b*x + a^3)/((b*x + a)^9*b^4)