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*x^3)/(7*a*(a + b*Sqrt[x])^7) + x^3/(21*a^2*(a + b*Sqrt[x])^6)

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Rubi [A]  time = 0.0141991, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 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 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 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^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 = 0.0074807, 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)

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Maple [B]  time = 0.006, size = 99, normalized size = 2.3 \begin{align*} -5\,{\frac{{a}^{2}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{4}}}+{\frac{10\,a}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-3}}-{\frac{5\,{a}^{4}}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-6}}+4\,{\frac{{a}^{3}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{5}}}+{\frac{2\,{a}^{5}}{7\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-7}}-{\frac{1}{{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.960847, size = 132, normalized size = 3.07 \begin{align*} -\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}} \end{align*}

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)

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Fricas [B]  time = 1.21617, size = 408, normalized size = 9.49 \begin{align*} -\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 )}} \end{align*}

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)

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Sympy [A]  time = 7.79924, size = 204, normalized size = 4.74 \begin{align*} \begin{cases} \frac{7 a x^{3}}{21 a^{9} + 147 a^{8} b \sqrt{x} + 441 a^{7} b^{2} x + 735 a^{6} b^{3} x^{\frac{3}{2}} + 735 a^{5} b^{4} x^{2} + 441 a^{4} b^{5} x^{\frac{5}{2}} + 147 a^{3} b^{6} x^{3} + 21 a^{2} b^{7} x^{\frac{7}{2}}} + \frac{b x^{\frac{7}{2}}}{21 a^{9} + 147 a^{8} b \sqrt{x} + 441 a^{7} b^{2} x + 735 a^{6} b^{3} x^{\frac{3}{2}} + 735 a^{5} b^{4} x^{2} + 441 a^{4} b^{5} x^{\frac{5}{2}} + 147 a^{3} b^{6} x^{3} + 21 a^{2} b^{7} x^{\frac{7}{2}}} & \text{for}\: a \neq 0 \\- \frac{1}{b^{8} x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((7*a*x**3/(21*a**9 + 147*a**8*b*sqrt(x) + 441*a**7*b**2*x + 735*a**6*b**3*x**(3/2) + 735*a**5*b**4*x
**2 + 441*a**4*b**5*x**(5/2) + 147*a**3*b**6*x**3 + 21*a**2*b**7*x**(7/2)) + b*x**(7/2)/(21*a**9 + 147*a**8*b*
sqrt(x) + 441*a**7*b**2*x + 735*a**6*b**3*x**(3/2) + 735*a**5*b**4*x**2 + 441*a**4*b**5*x**(5/2) + 147*a**3*b*
*6*x**3 + 21*a**2*b**7*x**(7/2)), Ne(a, 0)), (-1/(b**8*x), True))

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Giac [A]  time = 1.10126, size = 86, normalized size = 2. \begin{align*} -\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}} \end{align*}

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)