3.2227 \(\int \frac{x}{(a+b \sqrt{x})^8} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 a^3}{7 b^4 \left (a+b \sqrt{x}\right )^7}-\frac{a^2}{b^4 \left (a+b \sqrt{x}\right )^6}+\frac{6 a}{5 b^4 \left (a+b \sqrt{x}\right )^5}-\frac{1}{2 b^4 \left (a+b \sqrt{x}\right )^4} \]

[Out]

(2*a^3)/(7*b^4*(a + b*Sqrt[x])^7) - a^2/(b^4*(a + b*Sqrt[x])^6) + (6*a)/(5*b^4*(a + b*Sqrt[x])^5) - 1/(2*b^4*(
a + b*Sqrt[x])^4)

________________________________________________________________________________________

Rubi [A]  time = 0.049026, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{2 a^3}{7 b^4 \left (a+b \sqrt{x}\right )^7}-\frac{a^2}{b^4 \left (a+b \sqrt{x}\right )^6}+\frac{6 a}{5 b^4 \left (a+b \sqrt{x}\right )^5}-\frac{1}{2 b^4 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^8,x]

[Out]

(2*a^3)/(7*b^4*(a + b*Sqrt[x])^7) - a^2/(b^4*(a + b*Sqrt[x])^6) + (6*a)/(5*b^4*(a + b*Sqrt[x])^5) - 1/(2*b^4*(
a + b*Sqrt[x])^4)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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}{\left (a+b \sqrt{x}\right )^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^3 (a+b x)^8}+\frac{3 a^2}{b^3 (a+b x)^7}-\frac{3 a}{b^3 (a+b x)^6}+\frac{1}{b^3 (a+b x)^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^3}{7 b^4 \left (a+b \sqrt{x}\right )^7}-\frac{a^2}{b^4 \left (a+b \sqrt{x}\right )^6}+\frac{6 a}{5 b^4 \left (a+b \sqrt{x}\right )^5}-\frac{1}{2 b^4 \left (a+b \sqrt{x}\right )^4}\\ \end{align*}

Mathematica [A]  time = 0.0337104, size = 50, normalized size = 0.64 \[ -\frac{7 a^2 b \sqrt{x}+a^3+21 a b^2 x+35 b^3 x^{3/2}}{70 b^4 \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^8,x]

[Out]

-(a^3 + 7*a^2*b*Sqrt[x] + 21*a*b^2*x + 35*b^3*x^(3/2))/(70*b^4*(a + b*Sqrt[x])^7)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 65, normalized size = 0.8 \begin{align*}{\frac{2\,{a}^{3}}{7\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{{a}^{2}}{{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{6\,a}{5\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-5}}-{\frac{1}{2\,{b}^{4}} \left ( a+b\sqrt{x} \right ) ^{-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*x^(1/2))^8,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.967899, size = 86, normalized size = 1.1 \begin{align*} -\frac{1}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{4}} + \frac{6 \, a}{5 \,{\left (b \sqrt{x} + a\right )}^{5} b^{4}} - \frac{a^{2}}{{\left (b \sqrt{x} + a\right )}^{6} b^{4}} + \frac{2 \, a^{3}}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2))^8,x, algorithm="maxima")

[Out]

-1/2/((b*sqrt(x) + a)^4*b^4) + 6/5*a/((b*sqrt(x) + a)^5*b^4) - a^2/((b*sqrt(x) + a)^6*b^4) + 2/7*a^3/((b*sqrt(
x) + a)^7*b^4)

________________________________________________________________________________________

Fricas [B]  time = 1.29705, size = 379, normalized size = 4.86 \begin{align*} -\frac{35 \, b^{10} x^{5} + 595 \, a^{2} b^{8} x^{4} + 630 \, a^{4} b^{6} x^{3} + 14 \, a^{6} b^{4} x^{2} + 7 \, a^{8} b^{2} x - a^{10} - 32 \,{\left (7 \, a b^{9} x^{4} + 26 \, a^{3} b^{7} x^{3} + 7 \, a^{5} b^{5} x^{2}\right )} \sqrt{x}}{70 \,{\left (b^{18} x^{7} - 7 \, a^{2} b^{16} x^{6} + 21 \, a^{4} b^{14} x^{5} - 35 \, a^{6} b^{12} x^{4} + 35 \, a^{8} b^{10} x^{3} - 21 \, a^{10} b^{8} x^{2} + 7 \, a^{12} b^{6} x - a^{14} b^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2))^8,x, algorithm="fricas")

[Out]

-1/70*(35*b^10*x^5 + 595*a^2*b^8*x^4 + 630*a^4*b^6*x^3 + 14*a^6*b^4*x^2 + 7*a^8*b^2*x - a^10 - 32*(7*a*b^9*x^4
+ 26*a^3*b^7*x^3 + 7*a^5*b^5*x^2)*sqrt(x))/(b^18*x^7 - 7*a^2*b^16*x^6 + 21*a^4*b^14*x^5 - 35*a^6*b^12*x^4 + 3
5*a^8*b^10*x^3 - 21*a^10*b^8*x^2 + 7*a^12*b^6*x - a^14*b^4)

________________________________________________________________________________________

Sympy [A]  time = 6.21893, size = 416, normalized size = 5.33 \begin{align*} \begin{cases} \frac{35 a^{3} x^{2}}{70 a^{11} + 490 a^{10} b \sqrt{x} + 1470 a^{9} b^{2} x + 2450 a^{8} b^{3} x^{\frac{3}{2}} + 2450 a^{7} b^{4} x^{2} + 1470 a^{6} b^{5} x^{\frac{5}{2}} + 490 a^{5} b^{6} x^{3} + 70 a^{4} b^{7} x^{\frac{7}{2}}} + \frac{21 a^{2} b x^{\frac{5}{2}}}{70 a^{11} + 490 a^{10} b \sqrt{x} + 1470 a^{9} b^{2} x + 2450 a^{8} b^{3} x^{\frac{3}{2}} + 2450 a^{7} b^{4} x^{2} + 1470 a^{6} b^{5} x^{\frac{5}{2}} + 490 a^{5} b^{6} x^{3} + 70 a^{4} b^{7} x^{\frac{7}{2}}} + \frac{7 a b^{2} x^{3}}{70 a^{11} + 490 a^{10} b \sqrt{x} + 1470 a^{9} b^{2} x + 2450 a^{8} b^{3} x^{\frac{3}{2}} + 2450 a^{7} b^{4} x^{2} + 1470 a^{6} b^{5} x^{\frac{5}{2}} + 490 a^{5} b^{6} x^{3} + 70 a^{4} b^{7} x^{\frac{7}{2}}} + \frac{b^{3} x^{\frac{7}{2}}}{70 a^{11} + 490 a^{10} b \sqrt{x} + 1470 a^{9} b^{2} x + 2450 a^{8} b^{3} x^{\frac{3}{2}} + 2450 a^{7} b^{4} x^{2} + 1470 a^{6} b^{5} x^{\frac{5}{2}} + 490 a^{5} b^{6} x^{3} + 70 a^{4} b^{7} x^{\frac{7}{2}}} & \text{for}\: a \neq 0 \\- \frac{1}{2 b^{8} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((35*a**3*x**2/(70*a**11 + 490*a**10*b*sqrt(x) + 1470*a**9*b**2*x + 2450*a**8*b**3*x**(3/2) + 2450*a*
*7*b**4*x**2 + 1470*a**6*b**5*x**(5/2) + 490*a**5*b**6*x**3 + 70*a**4*b**7*x**(7/2)) + 21*a**2*b*x**(5/2)/(70*
a**11 + 490*a**10*b*sqrt(x) + 1470*a**9*b**2*x + 2450*a**8*b**3*x**(3/2) + 2450*a**7*b**4*x**2 + 1470*a**6*b**
5*x**(5/2) + 490*a**5*b**6*x**3 + 70*a**4*b**7*x**(7/2)) + 7*a*b**2*x**3/(70*a**11 + 490*a**10*b*sqrt(x) + 147
0*a**9*b**2*x + 2450*a**8*b**3*x**(3/2) + 2450*a**7*b**4*x**2 + 1470*a**6*b**5*x**(5/2) + 490*a**5*b**6*x**3 +
70*a**4*b**7*x**(7/2)) + b**3*x**(7/2)/(70*a**11 + 490*a**10*b*sqrt(x) + 1470*a**9*b**2*x + 2450*a**8*b**3*x*
*(3/2) + 2450*a**7*b**4*x**2 + 1470*a**6*b**5*x**(5/2) + 490*a**5*b**6*x**3 + 70*a**4*b**7*x**(7/2)), Ne(a, 0)
), (-1/(2*b**8*x**2), True))

________________________________________________________________________________________

Giac [A]  time = 1.24189, size = 57, normalized size = 0.73 \begin{align*} -\frac{35 \, b^{3} x^{\frac{3}{2}} + 21 \, a b^{2} x + 7 \, a^{2} b \sqrt{x} + a^{3}}{70 \,{\left (b \sqrt{x} + a\right )}^{7} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2))^8,x, algorithm="giac")

[Out]

-1/70*(35*b^3*x^(3/2) + 21*a*b^2*x + 7*a^2*b*sqrt(x) + a^3)/((b*sqrt(x) + a)^7*b^4)