\(\int \frac {x^4}{(a+b x)^7} \, dx\) [213]

Optimal result

Integrand size = 11, antiderivative size = 35 \[ \int \frac {x^4}{(a+b x)^7} \, dx=\frac {x^5}{6 a (a+b x)^6}+\frac {x^5}{30 a^2 (a+b x)^5} \]

[Out]

1/6*x^5/a/(b*x+a)^6+1/30*x^5/a^2/(b*x+a)^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {x^4}{(a+b x)^7} \, dx=\frac {x^5}{30 a^2 (a+b x)^5}+\frac {x^5}{6 a (a+b x)^6} \]

[In]

Int[x^4/(a + b*x)^7,x]

[Out]

x^5/(6*a*(a + b*x)^6) + x^5/(30*a^2*(a + b*x)^5)

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 47

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*} \text {integral}& = \frac {x^5}{6 a (a+b x)^6}+\frac {\int \frac {x^4}{(a+b x)^6} \, dx}{6 a} \\ & = \frac {x^5}{6 a (a+b x)^6}+\frac {x^5}{30 a^2 (a+b x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {x^4}{(a+b x)^7} \, dx=-\frac {a^4+6 a^3 b x+15 a^2 b^2 x^2+20 a b^3 x^3+15 b^4 x^4}{30 b^5 (a+b x)^6} \]

[In]

Integrate[x^4/(a + b*x)^7,x]

[Out]

-1/30*(a^4 + 6*a^3*b*x + 15*a^2*b^2*x^2 + 20*a*b^3*x^3 + 15*b^4*x^4)/(b^5*(a + b*x)^6)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49

method	result	size
gosper	\(-\frac {15 b^{4} x^{4}+20 a \,b^{3} x^{3}+15 a^{2} b^{2} x^{2}+6 a^{3} b x +a^{4}}{30 \left (b x +a \right )^{6} b^{5}}\)	\(52\)
norman	\(\frac {-\frac {x^{4}}{2 b}-\frac {2 a \,x^{3}}{3 b^{2}}-\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a^{3} x}{5 b^{4}}-\frac {a^{4}}{30 b^{5}}}{\left (b x +a \right )^{6}}\)	\(55\)
risch	\(\frac {-\frac {x^{4}}{2 b}-\frac {2 a \,x^{3}}{3 b^{2}}-\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a^{3} x}{5 b^{4}}-\frac {a^{4}}{30 b^{5}}}{\left (b x +a \right )^{6}}\)	\(55\)
parallelrisch	\(\frac {-15 b^{5} x^{4}-20 a \,b^{4} x^{3}-15 a^{2} b^{3} x^{2}-6 a^{3} b^{2} x -a^{4} b}{30 b^{6} \left (b x +a \right )^{6}}\)	\(57\)
default	\(\frac {4 a^{3}}{5 b^{5} \left (b x +a \right )^{5}}-\frac {a^{4}}{6 b^{5} \left (b x +a \right )^{6}}-\frac {3 a^{2}}{2 b^{5} \left (b x +a \right )^{4}}+\frac {4 a}{3 b^{5} \left (b x +a \right )^{3}}-\frac {1}{2 b^{5} \left (b x +a \right )^{2}}\)	\(72\)

[In]

int(x^4/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

-1/30*(15*b^4*x^4+20*a*b^3*x^3+15*a^2*b^2*x^2+6*a^3*b*x+a^4)/(b*x+a)^6/b^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (31) = 62\).

Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.11 \[ \int \frac {x^4}{(a+b x)^7} \, dx=-\frac {15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \, {\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \]

[In]

integrate(x^4/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/(b^11*x^6 + 6*a*b^10*x^5 + 15*a^2*b^9*x^4
+ 20*a^3*b^8*x^3 + 15*a^4*b^7*x^2 + 6*a^5*b^6*x + a^6*b^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (27) = 54\).

Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.31 \[ \int \frac {x^4}{(a+b x)^7} \, dx=\frac {- a^{4} - 6 a^{3} b x - 15 a^{2} b^{2} x^{2} - 20 a b^{3} x^{3} - 15 b^{4} x^{4}}{30 a^{6} b^{5} + 180 a^{5} b^{6} x + 450 a^{4} b^{7} x^{2} + 600 a^{3} b^{8} x^{3} + 450 a^{2} b^{9} x^{4} + 180 a b^{10} x^{5} + 30 b^{11} x^{6}} \]

[In]

integrate(x**4/(b*x+a)**7,x)

[Out]

(-a**4 - 6*a**3*b*x - 15*a**2*b**2*x**2 - 20*a*b**3*x**3 - 15*b**4*x**4)/(30*a**6*b**5 + 180*a**5*b**6*x + 450
*a**4*b**7*x**2 + 600*a**3*b**8*x**3 + 450*a**2*b**9*x**4 + 180*a*b**10*x**5 + 30*b**11*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (31) = 62\).

Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.11 \[ \int \frac {x^4}{(a+b x)^7} \, dx=-\frac {15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \, {\left (b^{11} x^{6} + 6 \, a b^{10} x^{5} + 15 \, a^{2} b^{9} x^{4} + 20 \, a^{3} b^{8} x^{3} + 15 \, a^{4} b^{7} x^{2} + 6 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} \]

[In]

integrate(x^4/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/(b^11*x^6 + 6*a*b^10*x^5 + 15*a^2*b^9*x^4
+ 20*a^3*b^8*x^3 + 15*a^4*b^7*x^2 + 6*a^5*b^6*x + a^6*b^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \frac {x^4}{(a+b x)^7} \, dx=-\frac {15 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{30 \, {\left (b x + a\right )}^{6} b^{5}} \]

[In]

integrate(x^4/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/30*(15*b^4*x^4 + 20*a*b^3*x^3 + 15*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/((b*x + a)^6*b^5)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {x^4}{(a+b x)^7} \, dx=\frac {x^5\,\left (6\,a+b\,x\right )}{30\,a^2\,{\left (a+b\,x\right )}^6} \]

[In]

int(x^4/(a + b*x)^7,x)

[Out]

(x^5*(6*a + b*x))/(30*a^2*(a + b*x)^6)