\(\int \frac {x^5}{(a+b x)^7} \, dx\) [212]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {x^5}{(a+b x)^7} \, dx=\frac {x^6}{6 a (a+b x)^6} \]

[Out]

1/6*x^6/a/(b*x+a)^6

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \[ \int \frac {x^5}{(a+b x)^7} \, dx=\frac {x^6}{6 a (a+b x)^6} \]

[In]

Int[x^5/(a + b*x)^7,x]

[Out]

x^6/(6*a*(a + b*x)^6)

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]

Rubi steps \begin{align*} \text {integral}& = \frac {x^6}{6 a (a+b x)^6} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(17)=34\).

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.76 \[ \int \frac {x^5}{(a+b x)^7} \, dx=-\frac {a^5+6 a^4 b x+15 a^3 b^2 x^2+20 a^2 b^3 x^3+15 a b^4 x^4+6 b^5 x^5}{6 b^6 (a+b x)^6} \]

[In]

Integrate[x^5/(a + b*x)^7,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(15)=30\).

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.71

method	result	size
gosper	\(-\frac {6 b^{5} x^{5}+15 a \,b^{4} x^{4}+20 a^{2} b^{3} x^{3}+15 a^{3} b^{2} x^{2}+6 a^{4} b x +a^{5}}{6 \left (b x +a \right )^{6} b^{6}}\)	\(63\)
parallelrisch	\(\frac {-6 b^{5} x^{5}-15 a \,b^{4} x^{4}-20 a^{2} b^{3} x^{3}-15 a^{3} b^{2} x^{2}-6 a^{4} b x -a^{5}}{6 b^{6} \left (b x +a \right )^{6}}\)	\(65\)
norman	\(\frac {-\frac {x^{5}}{b}-\frac {5 a \,x^{4}}{2 b^{2}}-\frac {10 a^{2} x^{3}}{3 b^{3}}-\frac {5 a^{3} x^{2}}{2 b^{4}}-\frac {a^{4} x}{b^{5}}-\frac {a^{5}}{6 b^{6}}}{\left (b x +a \right )^{6}}\)	\(66\)
risch	\(\frac {-\frac {x^{5}}{b}-\frac {5 a \,x^{4}}{2 b^{2}}-\frac {10 a^{2} x^{3}}{3 b^{3}}-\frac {5 a^{3} x^{2}}{2 b^{4}}-\frac {a^{4} x}{b^{5}}-\frac {a^{5}}{6 b^{6}}}{\left (b x +a \right )^{6}}\)	\(66\)
default	\(-\frac {a^{4}}{b^{6} \left (b x +a \right )^{5}}+\frac {a^{5}}{6 b^{6} \left (b x +a \right )^{6}}+\frac {5 a^{3}}{2 b^{6} \left (b x +a \right )^{4}}-\frac {10 a^{2}}{3 b^{6} \left (b x +a \right )^{3}}+\frac {5 a}{2 b^{6} \left (b x +a \right )^{2}}-\frac {1}{\left (b x +a \right ) b^{6}}\)	\(87\)

[In]

int(x^5/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 7.06 \[ \int \frac {x^5}{(a+b x)^7} \, dx=-\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, {\left (b^{12} x^{6} + 6 \, a b^{11} x^{5} + 15 \, a^{2} b^{10} x^{4} + 20 \, a^{3} b^{9} x^{3} + 15 \, a^{4} b^{8} x^{2} + 6 \, a^{5} b^{7} x + a^{6} b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/(b^12*x^6 + 6*a*b^11*x^5 +
15*a^2*b^10*x^4 + 20*a^3*b^9*x^3 + 15*a^4*b^8*x^2 + 6*a^5*b^7*x + a^6*b^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (12) = 24\).

Time = 0.26 (sec) , antiderivative size = 128, normalized size of antiderivative = 7.53 \[ \int \frac {x^5}{(a+b x)^7} \, dx=\frac {- a^{5} - 6 a^{4} b x - 15 a^{3} b^{2} x^{2} - 20 a^{2} b^{3} x^{3} - 15 a b^{4} x^{4} - 6 b^{5} x^{5}}{6 a^{6} b^{6} + 36 a^{5} b^{7} x + 90 a^{4} b^{8} x^{2} + 120 a^{3} b^{9} x^{3} + 90 a^{2} b^{10} x^{4} + 36 a b^{11} x^{5} + 6 b^{12} x^{6}} \]

[In]

integrate(x**5/(b*x+a)**7,x)

[Out]

(-a**5 - 6*a**4*b*x - 15*a**3*b**2*x**2 - 20*a**2*b**3*x**3 - 15*a*b**4*x**4 - 6*b**5*x**5)/(6*a**6*b**6 + 36*
a**5*b**7*x + 90*a**4*b**8*x**2 + 120*a**3*b**9*x**3 + 90*a**2*b**10*x**4 + 36*a*b**11*x**5 + 6*b**12*x**6)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 7.06 \[ \int \frac {x^5}{(a+b x)^7} \, dx=-\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, {\left (b^{12} x^{6} + 6 \, a b^{11} x^{5} + 15 \, a^{2} b^{10} x^{4} + 20 \, a^{3} b^{9} x^{3} + 15 \, a^{4} b^{8} x^{2} + 6 \, a^{5} b^{7} x + a^{6} b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/(b^12*x^6 + 6*a*b^11*x^5 +
15*a^2*b^10*x^4 + 20*a^3*b^9*x^3 + 15*a^4*b^8*x^2 + 6*a^5*b^7*x + a^6*b^6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (15) = 30\).

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.65 \[ \int \frac {x^5}{(a+b x)^7} \, dx=-\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, {\left (b x + a\right )}^{6} b^{6}} \]

[In]

integrate(x^5/(b*x+a)^7,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.24 \[ \int \frac {x^5}{(a+b x)^7} \, dx=\frac {\frac {5\,a}{2\,{\left (a+b\,x\right )}^2}-\frac {1}{a+b\,x}-\frac {10\,a^2}{3\,{\left (a+b\,x\right )}^3}+\frac {5\,a^3}{2\,{\left (a+b\,x\right )}^4}-\frac {a^4}{{\left (a+b\,x\right )}^5}+\frac {a^5}{6\,{\left (a+b\,x\right )}^6}}{b^6} \]

[In]

int(x^5/(a + b*x)^7,x)

[Out]

((5*a)/(2*(a + b*x)^2) - 1/(a + b*x) - (10*a^2)/(3*(a + b*x)^3) + (5*a^3)/(2*(a + b*x)^4) - a^4/(a + b*x)^5 +
a^5/(6*(a + b*x)^6))/b^6