3.2226 \(\int \frac {x^2}{(a+b \sqrt {x})^8} \, dx\)

Optimal. Leaf size=43 \[ \frac {x^3}{21 a^2 \left (a+b \sqrt {x}\right )^6}+\frac {2 x^3}{7 a \left (a+b \sqrt {x}\right )^7} \]

[Out]

2/7*x^3/a/(a+b*x^(1/2))^7+1/21*x^3/a^2/(a+b*x^(1/2))^6

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ \frac {x^3}{21 a^2 \left (a+b \sqrt {x}\right )^6}+\frac {2 x^3}{7 a \left (a+b \sqrt {x}\right )^7} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Sqrt[x])^8,x]

[Out]

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

Rule 45

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])

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]]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b \sqrt {x}\right )^8} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^8} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^3}{7 a \left (a+b \sqrt {x}\right )^7}+\frac {2 \operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^7} \, dx,x,\sqrt {x}\right )}{7 a}\\ &=\frac {2 x^3}{7 a \left (a+b \sqrt {x}\right )^7}+\frac {x^3}{21 a^2 \left (a+b \sqrt {x}\right )^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 32, normalized size = 0.74 \[ \frac {x^3 \left (7 a+b \sqrt {x}\right )}{21 a^2 \left (a+b \sqrt {x}\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Sqrt[x])^8,x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.08, size = 188, normalized size = 4.37 \[ -\frac {21 \, b^{12} x^{6} + 231 \, a^{2} b^{10} x^{5} + 105 \, a^{4} b^{8} x^{4} + 42 \, a^{6} b^{6} x^{3} - 21 \, a^{8} b^{4} x^{2} + 7 \, a^{10} b^{2} x - a^{12} - 16 \, {\left (7 \, a b^{11} x^{5} + 14 \, a^{3} b^{9} x^{4} + 3 \, a^{5} b^{7} x^{3}\right )} \sqrt {x}}{21 \, {\left (b^{20} x^{7} - 7 \, a^{2} b^{18} x^{6} + 21 \, a^{4} b^{16} x^{5} - 35 \, a^{6} b^{14} x^{4} + 35 \, a^{8} b^{12} x^{3} - 21 \, a^{10} b^{10} x^{2} + 7 \, a^{12} b^{8} x - a^{14} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(21*b^12*x^6 + 231*a^2*b^10*x^5 + 105*a^4*b^8*x^4 + 42*a^6*b^6*x^3 - 21*a^8*b^4*x^2 + 7*a^10*b^2*x - a^1
2 - 16*(7*a*b^11*x^5 + 14*a^3*b^9*x^4 + 3*a^5*b^7*x^3)*sqrt(x))/(b^20*x^7 - 7*a^2*b^18*x^6 + 21*a^4*b^16*x^5 -
35*a^6*b^14*x^4 + 35*a^8*b^12*x^3 - 21*a^10*b^10*x^2 + 7*a^12*b^8*x - a^14*b^6)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 64, normalized size = 1.49 \[ -\frac {21 \, b^{5} x^{\frac {5}{2}} + 35 \, a b^{4} x^{2} + 35 \, a^{2} b^{3} x^{\frac {3}{2}} + 21 \, a^{3} b^{2} x + 7 \, a^{4} b \sqrt {x} + a^{5}}{21 \, {\left (b \sqrt {x} + a\right )}^{7} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(21*b^5*x^(5/2) + 35*a*b^4*x^2 + 35*a^2*b^3*x^(3/2) + 21*a^3*b^2*x + 7*a^4*b*sqrt(x) + a^5)/((b*sqrt(x)
+ a)^7*b^6)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 99, normalized size = 2.30 \[ \frac {2 a^{5}}{7 \left (b \sqrt {x}+a \right )^{7} b^{6}}-\frac {5 a^{4}}{3 \left (b \sqrt {x}+a \right )^{6} b^{6}}+\frac {4 a^{3}}{\left (b \sqrt {x}+a \right )^{5} b^{6}}-\frac {5 a^{2}}{\left (b \sqrt {x}+a \right )^{4} b^{6}}+\frac {10 a}{3 \left (b \sqrt {x}+a \right )^{3} b^{6}}-\frac {1}{\left (b \sqrt {x}+a \right )^{2} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/3*a/b^6/(b*x^(1/2)+a)^3+4*a^3/b^6/(b*x^(1/2)+a)^5+2/7*a^5/b^6/(b*x^(1/2)+a)^7-5*a^2/b^6/(b*x^(1/2)+a)^4-5/3
*a^4/b^6/(b*x^(1/2)+a)^6-1/b^6/(b*x^(1/2)+a)^2

________________________________________________________________________________________

maxima [B]  time = 0.90, size = 98, normalized size = 2.28 \[ -\frac {1}{{\left (b \sqrt {x} + a\right )}^{2} b^{6}} + \frac {10 \, a}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{6}} - \frac {5 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{4} b^{6}} + \frac {4 \, a^{3}}{{\left (b \sqrt {x} + a\right )}^{5} b^{6}} - \frac {5 \, a^{4}}{3 \, {\left (b \sqrt {x} + a\right )}^{6} b^{6}} + \frac {2 \, a^{5}}{7 \, {\left (b \sqrt {x} + a\right )}^{7} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*sqrt(x) + a)^2*b^6) + 10/3*a/((b*sqrt(x) + a)^3*b^6) - 5*a^2/((b*sqrt(x) + a)^4*b^6) + 4*a^3/((b*sqrt(x
) + a)^5*b^6) - 5/3*a^4/((b*sqrt(x) + a)^6*b^6) + 2/7*a^5/((b*sqrt(x) + a)^7*b^6)

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 130, normalized size = 3.02 \[ -\frac {\frac {a^5}{21\,b^6}+\frac {x^{5/2}}{b}+\frac {5\,a\,x^2}{3\,b^2}+\frac {a^3\,x}{b^4}+\frac {5\,a^2\,x^{3/2}}{3\,b^3}+\frac {a^4\,\sqrt {x}}{3\,b^5}}{a^7+b^7\,x^{7/2}+21\,a^5\,b^2\,x+7\,a\,b^6\,x^3+7\,a^6\,b\,\sqrt {x}+35\,a^3\,b^4\,x^2+35\,a^4\,b^3\,x^{3/2}+21\,a^2\,b^5\,x^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^(1/2))^8,x)

[Out]

-(a^5/(21*b^6) + x^(5/2)/b + (5*a*x^2)/(3*b^2) + (a^3*x)/b^4 + (5*a^2*x^(3/2))/(3*b^3) + (a^4*x^(1/2))/(3*b^5)
)/(a^7 + b^7*x^(7/2) + 21*a^5*b^2*x + 7*a*b^6*x^3 + 7*a^6*b*x^(1/2) + 35*a^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) + 21
*a^2*b^5*x^(5/2))

________________________________________________________________________________________

sympy [A]  time = 6.49, size = 619, normalized size = 14.40 \[ \begin {cases} - \frac {a^{5}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {7 a^{4} b \sqrt {x}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {21 a^{3} b^{2} x}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {35 a^{2} b^{3} x^{\frac {3}{2}}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {35 a b^{4} x^{2}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} - \frac {21 b^{5} x^{\frac {5}{2}}}{21 a^{7} b^{6} + 147 a^{6} b^{7} \sqrt {x} + 441 a^{5} b^{8} x + 735 a^{4} b^{9} x^{\frac {3}{2}} + 735 a^{3} b^{10} x^{2} + 441 a^{2} b^{11} x^{\frac {5}{2}} + 147 a b^{12} x^{3} + 21 b^{13} x^{\frac {7}{2}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{8}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**5/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b
**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 7*a**4*b*sqrt(x)/(21*a**7*b**6 +
147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5
/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 21*a**3*b**2*x/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*
b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*
x**(7/2)) - 35*a**2*b**3*x**(3/2)/(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(
3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 35*a*b**4*x**2/
(21*a**7*b**6 + 147*a**6*b**7*sqrt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a
**2*b**11*x**(5/2) + 147*a*b**12*x**3 + 21*b**13*x**(7/2)) - 21*b**5*x**(5/2)/(21*a**7*b**6 + 147*a**6*b**7*sq
rt(x) + 441*a**5*b**8*x + 735*a**4*b**9*x**(3/2) + 735*a**3*b**10*x**2 + 441*a**2*b**11*x**(5/2) + 147*a*b**12
*x**3 + 21*b**13*x**(7/2)), Ne(b, 0)), (x**3/(3*a**8), True))

________________________________________________________________________________________