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

Optimal. Leaf size=30 \[ \frac{a}{6 b^2 (a+b x)^6}-\frac{1}{5 b^2 (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0134989, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a}{6 b^2 (a+b x)^6}-\frac{1}{5 b^2 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x)^7,x]

[Out]

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

Rule 43

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

Mathematica [A]  time = 0.0105081, size = 20, normalized size = 0.67 \[ -\frac{a+6 b x}{30 b^2 (a+b x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x)^7,x]

[Out]

-(a + 6*b*x)/(30*b^2*(a + b*x)^6)

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Maple [A]  time = 0.004, size = 27, normalized size = 0.9 \begin{align*}{\frac{a}{6\,{b}^{2} \left ( bx+a \right ) ^{6}}}-{\frac{1}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.08307, size = 103, normalized size = 3.43 \begin{align*} -\frac{6 \, b x + a}{30 \,{\left (b^{8} x^{6} + 6 \, a b^{7} x^{5} + 15 \, a^{2} b^{6} x^{4} + 20 \, a^{3} b^{5} x^{3} + 15 \, a^{4} b^{4} x^{2} + 6 \, a^{5} b^{3} x + a^{6} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.51044, size = 159, normalized size = 5.3 \begin{align*} -\frac{6 \, b x + a}{30 \,{\left (b^{8} x^{6} + 6 \, a b^{7} x^{5} + 15 \, a^{2} b^{6} x^{4} + 20 \, a^{3} b^{5} x^{3} + 15 \, a^{4} b^{4} x^{2} + 6 \, a^{5} b^{3} x + a^{6} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.77669, size = 80, normalized size = 2.67 \begin{align*} - \frac{a + 6 b x}{30 a^{6} b^{2} + 180 a^{5} b^{3} x + 450 a^{4} b^{4} x^{2} + 600 a^{3} b^{5} x^{3} + 450 a^{2} b^{6} x^{4} + 180 a b^{7} x^{5} + 30 b^{8} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20376, size = 24, normalized size = 0.8 \begin{align*} -\frac{6 \, b x + a}{30 \,{\left (b x + a\right )}^{6} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(6*b*x + a)/((b*x + a)^6*b^2)