\(\int \frac {x^3}{(a+b x)^{10}} \, dx\) [231]

Optimal result

Integrand size = 11, antiderivative size = 64 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=\frac {a^3}{9 b^4 (a+b x)^9}-\frac {3 a^2}{8 b^4 (a+b x)^8}+\frac {3 a}{7 b^4 (a+b x)^7}-\frac {1}{6 b^4 (a+b x)^6} \]

[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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=\frac {a^3}{9 b^4 (a+b x)^9}-\frac {3 a^2}{8 b^4 (a+b x)^8}+\frac {3 a}{7 b^4 (a+b x)^7}-\frac {1}{6 b^4 (a+b x)^6} \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.66 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=-\frac {a^3+9 a^2 b x+36 a b^2 x^2+84 b^3 x^3}{504 b^4 (a+b x)^9} \]

[In]

Integrate[x^3/(a + b*x)^10,x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {84 b^{3} x^{3}+36 a \,b^{2} x^{2}+9 a^{2} b x +a^{3}}{504 \left (b x +a \right )^{9} b^{4}}\)	\(41\)
norman	\(\frac {-\frac {x^{3}}{6 b}-\frac {a \,x^{2}}{14 b^{2}}-\frac {a^{2} x}{56 b^{3}}-\frac {a^{3}}{504 b^{4}}}{\left (b x +a \right )^{9}}\)	\(44\)
risch	\(\frac {-\frac {x^{3}}{6 b}-\frac {a \,x^{2}}{14 b^{2}}-\frac {a^{2} x}{56 b^{3}}-\frac {a^{3}}{504 b^{4}}}{\left (b x +a \right )^{9}}\)	\(44\)
parallelrisch	\(\frac {-84 b^{8} x^{3}-36 a \,b^{7} x^{2}-9 a^{2} b^{6} x -a^{3} b^{5}}{504 b^{9} \left (b x +a \right )^{9}}\)	\(48\)
default	\(\frac {a^{3}}{9 b^{4} \left (b x +a \right )^{9}}-\frac {3 a^{2}}{8 b^{4} \left (b x +a \right )^{8}}+\frac {3 a}{7 b^{4} \left (b x +a \right )^{7}}-\frac {1}{6 b^{4} \left (b x +a \right )^{6}}\)	\(57\)

[In]

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

[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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (56) = 112\).

Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.05 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=-\frac {84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \, {\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (60) = 120\).

Time = 0.37 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.17 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=\frac {- a^{3} - 9 a^{2} b x - 36 a b^{2} x^{2} - 84 b^{3} x^{3}}{504 a^{9} b^{4} + 4536 a^{8} b^{5} x + 18144 a^{7} b^{6} x^{2} + 42336 a^{6} b^{7} x^{3} + 63504 a^{5} b^{8} x^{4} + 63504 a^{4} b^{9} x^{5} + 42336 a^{3} b^{10} x^{6} + 18144 a^{2} b^{11} x^{7} + 4536 a b^{12} x^{8} + 504 b^{13} x^{9}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (56) = 112\).

Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.05 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=-\frac {84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \, {\left (b^{13} x^{9} + 9 \, a b^{12} x^{8} + 36 \, a^{2} b^{11} x^{7} + 84 \, a^{3} b^{10} x^{6} + 126 \, a^{4} b^{9} x^{5} + 126 \, a^{5} b^{8} x^{4} + 84 \, a^{6} b^{7} x^{3} + 36 \, a^{7} b^{6} x^{2} + 9 \, a^{8} b^{5} x + a^{9} b^{4}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.62 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=-\frac {84 \, b^{3} x^{3} + 36 \, a b^{2} x^{2} + 9 \, a^{2} b x + a^{3}}{504 \, {\left (b x + a\right )}^{9} b^{4}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.75 \[ \int \frac {x^3}{(a+b x)^{10}} \, dx=\frac {\frac {3\,a}{7\,{\left (a+b\,x\right )}^7}-\frac {1}{6\,{\left (a+b\,x\right )}^6}-\frac {3\,a^2}{8\,{\left (a+b\,x\right )}^8}+\frac {a^3}{9\,{\left (a+b\,x\right )}^9}}{b^4} \]

[In]

int(x^3/(a + b*x)^10,x)

[Out]

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