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

Optimal. Leaf size=17 \[ \frac{x^6}{6 a (a+b x)^6} \]

[Out]

x^6/(6*a*(a + b*x)^6)

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Rubi [A]  time = 0.0018992, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {37} \[ \frac{x^6}{6 a (a+b x)^6} \]

Antiderivative was successfully verified.

[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*} \int \frac{x^5}{(a+b x)^7} \, dx &=\frac{x^6}{6 a (a+b x)^6}\\ \end{align*}

Mathematica [B]  time = 0.0184956, size = 64, normalized size = 3.76 \[ -\frac{15 a^3 b^2 x^2+20 a^2 b^3 x^3+6 a^4 b x+a^5+15 a b^4 x^4+6 b^5 x^5}{6 b^6 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x)^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*b^6*(a + b*x)^6)

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Maple [B]  time = 0.007, size = 87, normalized size = 5.1 \begin{align*} -{\frac{{a}^{4}}{{b}^{6} \left ( bx+a \right ) ^{5}}}-{\frac{1}{{b}^{6} \left ( bx+a \right ) }}+{\frac{5\,{a}^{3}}{2\,{b}^{6} \left ( bx+a \right ) ^{4}}}+{\frac{5\,a}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{5}}{6\,{b}^{6} \left ( bx+a \right ) ^{6}}}-{\frac{10\,{a}^{2}}{3\,{b}^{6} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.06284, size = 162, normalized size = 9.53 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.4968, size = 252, normalized size = 14.82 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 0.99007, size = 128, normalized size = 7.53 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.18808, size = 84, normalized size = 4.94 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)