\(\int \frac {x^3}{(a+b x)^7} \, dx\) [214]

Optimal result

Integrand size = 11, antiderivative size = 52 \[ \int \frac {x^3}{(a+b x)^7} \, dx=\frac {x^4}{6 a (a+b x)^6}+\frac {x^4}{15 a^2 (a+b x)^5}+\frac {x^4}{60 a^3 (a+b x)^4} \]

[Out]

1/6*x^4/a/(b*x+a)^6+1/15*x^4/a^2/(b*x+a)^5+1/60*x^4/a^3/(b*x+a)^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^3}{(a+b x)^7} \, dx=\frac {a^3}{6 b^4 (a+b x)^6}-\frac {3 a^2}{5 b^4 (a+b x)^5}+\frac {3 a}{4 b^4 (a+b x)^4}-\frac {1}{3 b^4 (a+b x)^3} \]

[In]

Int[x^3/(a + b*x)^7,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{(a+b x)^7} \, dx=-\frac {a^3+6 a^2 b x+15 a b^2 x^2+20 b^3 x^3}{60 b^4 (a+b x)^6} \]

[In]

Integrate[x^3/(a + b*x)^7,x]

[Out]

-1/60*(a^3 + 6*a^2*b*x + 15*a*b^2*x^2 + 20*b^3*x^3)/(b^4*(a + b*x)^6)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.79

method	result	size
gosper	\(-\frac {20 b^{3} x^{3}+15 a \,b^{2} x^{2}+6 a^{2} b x +a^{3}}{60 \left (b x +a \right )^{6} b^{4}}\)	\(41\)
norman	\(\frac {-\frac {x^{3}}{3 b}-\frac {a \,x^{2}}{4 b^{2}}-\frac {a^{2} x}{10 b^{3}}-\frac {a^{3}}{60 b^{4}}}{\left (b x +a \right )^{6}}\)	\(44\)
risch	\(\frac {-\frac {x^{3}}{3 b}-\frac {a \,x^{2}}{4 b^{2}}-\frac {a^{2} x}{10 b^{3}}-\frac {a^{3}}{60 b^{4}}}{\left (b x +a \right )^{6}}\)	\(44\)
parallelrisch	\(\frac {-20 b^{5} x^{3}-15 a \,b^{4} x^{2}-6 a^{2} b^{3} x -a^{3} b^{2}}{60 b^{6} \left (b x +a \right )^{6}}\)	\(48\)
default	\(-\frac {3 a^{2}}{5 b^{4} \left (b x +a \right )^{5}}+\frac {a^{3}}{6 b^{4} \left (b x +a \right )^{6}}+\frac {3 a}{4 b^{4} \left (b x +a \right )^{4}}-\frac {1}{3 b^{4} \left (b x +a \right )^{3}}\)	\(57\)

[In]

int(x^3/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

-1/60*(20*b^3*x^3+15*a*b^2*x^2+6*a^2*b*x+a^3)/(b*x+a)^6/b^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (46) = 92\).

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{(a+b x)^7} \, dx=-\frac {20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]

[In]

integrate(x^3/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3
+ 15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (42) = 84\).

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.00 \[ \int \frac {x^3}{(a+b x)^7} \, dx=\frac {- a^{3} - 6 a^{2} b x - 15 a b^{2} x^{2} - 20 b^{3} x^{3}}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \]

[In]

integrate(x**3/(b*x+a)**7,x)

[Out]

(-a**3 - 6*a**2*b*x - 15*a*b**2*x**2 - 20*b**3*x**3)/(60*a**6*b**4 + 360*a**5*b**5*x + 900*a**4*b**6*x**2 + 12
00*a**3*b**7*x**3 + 900*a**2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (46) = 92\).

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.88 \[ \int \frac {x^3}{(a+b x)^7} \, dx=-\frac {20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \]

[In]

integrate(x^3/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3
+ 15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^4)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{(a+b x)^7} \, dx=-\frac {20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \, {\left (b x + a\right )}^{6} b^{4}} \]

[In]

integrate(x^3/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)/((b*x + a)^6*b^4)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{(a+b x)^7} \, dx=\frac {\frac {3\,a}{4\,{\left (a+b\,x\right )}^4}-\frac {1}{3\,{\left (a+b\,x\right )}^3}-\frac {3\,a^2}{5\,{\left (a+b\,x\right )}^5}+\frac {a^3}{6\,{\left (a+b\,x\right )}^6}}{b^4} \]

[In]

int(x^3/(a + b*x)^7,x)

[Out]

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