3.2217 \(\int \frac{x^2}{(a+b \sqrt{x})^5} \, dx\)

Optimal. Leaf size=107 \[ \frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 \sqrt{x}}{b^5} \]

[Out]

a^5/(2*b^6*(a + b*Sqrt[x])^4) - (10*a^4)/(3*b^6*(a + b*Sqrt[x])^3) + (10*a^3)/(b^6*(a + b*Sqrt[x])^2) - (20*a^
2)/(b^6*(a + b*Sqrt[x])) + (2*Sqrt[x])/b^5 - (10*a*Log[a + b*Sqrt[x]])/b^6

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Rubi [A]  time = 0.06793, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 \sqrt{x}}{b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Sqrt[x])^5,x]

[Out]

a^5/(2*b^6*(a + b*Sqrt[x])^4) - (10*a^4)/(3*b^6*(a + b*Sqrt[x])^3) + (10*a^3)/(b^6*(a + b*Sqrt[x])^2) - (20*a^
2)/(b^6*(a + b*Sqrt[x])) + (2*Sqrt[x])/b^5 - (10*a*Log[a + b*Sqrt[x]])/b^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 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^2}{\left (a+b \sqrt{x}\right )^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{b^5}-\frac{a^5}{b^5 (a+b x)^5}+\frac{5 a^4}{b^5 (a+b x)^4}-\frac{10 a^3}{b^5 (a+b x)^3}+\frac{10 a^2}{b^5 (a+b x)^2}-\frac{5 a}{b^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}+\frac{2 \sqrt{x}}{b^5}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.069917, size = 100, normalized size = 0.93 \[ -\frac{48 a^2 b^3 x^{3/2}+252 a^3 b^2 x+248 a^4 b \sqrt{x}+77 a^5-48 a b^4 x^2+60 a \left (a+b \sqrt{x}\right )^4 \log \left (a+b \sqrt{x}\right )-12 b^5 x^{5/2}}{6 b^6 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Sqrt[x])^5,x]

[Out]

-(77*a^5 + 248*a^4*b*Sqrt[x] + 252*a^3*b^2*x + 48*a^2*b^3*x^(3/2) - 48*a*b^4*x^2 - 12*b^5*x^(5/2) + 60*a*(a +
b*Sqrt[x])^4*Log[a + b*Sqrt[x]])/(6*b^6*(a + b*Sqrt[x])^4)

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Maple [A]  time = 0.008, size = 92, normalized size = 0.9 \begin{align*} -10\,{\frac{a\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+2\,{\frac{\sqrt{x}}{{b}^{5}}}+{\frac{{a}^{5}}{2\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-4}}-{\frac{10\,{a}^{4}}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-3}}+10\,{\frac{{a}^{3}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{2}}}-20\,{\frac{{a}^{2}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*x^(1/2))^5,x)

[Out]

-10*a*ln(a+b*x^(1/2))/b^6+2*x^(1/2)/b^5+1/2*a^5/b^6/(a+b*x^(1/2))^4-10/3*a^4/b^6/(a+b*x^(1/2))^3+10*a^3/b^6/(a
+b*x^(1/2))^2-20*a^2/b^6/(a+b*x^(1/2))

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Maxima [A]  time = 0.988579, size = 128, normalized size = 1.2 \begin{align*} -\frac{10 \, a \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{6}} - \frac{20 \, a^{2}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{10 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} - \frac{10 \, a^{4}}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{6}} + \frac{a^{5}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

-10*a*log(b*sqrt(x) + a)/b^6 + 2*(b*sqrt(x) + a)/b^6 - 20*a^2/((b*sqrt(x) + a)*b^6) + 10*a^3/((b*sqrt(x) + a)^
2*b^6) - 10/3*a^4/((b*sqrt(x) + a)^3*b^6) + 1/2*a^5/((b*sqrt(x) + a)^4*b^6)

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Fricas [B]  time = 1.31901, size = 414, normalized size = 3.87 \begin{align*} \frac{180 \, a^{3} b^{6} x^{3} - 357 \, a^{5} b^{4} x^{2} + 278 \, a^{7} b^{2} x - 77 \, a^{9} - 60 \,{\left (a b^{8} x^{4} - 4 \, a^{3} b^{6} x^{3} + 6 \, a^{5} b^{4} x^{2} - 4 \, a^{7} b^{2} x + a^{9}\right )} \log \left (b \sqrt{x} + a\right ) + 4 \,{\left (3 \, b^{9} x^{4} - 42 \, a^{2} b^{7} x^{3} + 73 \, a^{4} b^{5} x^{2} - 55 \, a^{6} b^{3} x + 15 \, a^{8} b\right )} \sqrt{x}}{6 \,{\left (b^{14} x^{4} - 4 \, a^{2} b^{12} x^{3} + 6 \, a^{4} b^{10} x^{2} - 4 \, a^{6} b^{8} x + a^{8} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

1/6*(180*a^3*b^6*x^3 - 357*a^5*b^4*x^2 + 278*a^7*b^2*x - 77*a^9 - 60*(a*b^8*x^4 - 4*a^3*b^6*x^3 + 6*a^5*b^4*x^
2 - 4*a^7*b^2*x + a^9)*log(b*sqrt(x) + a) + 4*(3*b^9*x^4 - 42*a^2*b^7*x^3 + 73*a^4*b^5*x^2 - 55*a^6*b^3*x + 15
*a^8*b)*sqrt(x))/(b^14*x^4 - 4*a^2*b^12*x^3 + 6*a^4*b^10*x^2 - 4*a^6*b^8*x + a^8*b^6)

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Sympy [A]  time = 2.77458, size = 685, normalized size = 6.4 \begin{align*} \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{15 a^{5}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{360 a^{3} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{120 a^{3} b^{2} x}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{2} b^{3} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{200 a^{2} b^{3} x^{\frac{3}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{60 a b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{110 a b^{4} x^{2}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{12 b^{5} x^{\frac{5}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{5}} & \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))**5,x)

[Out]

Piecewise((-60*a**5*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/
2) + 6*b**10*x**2) - 15*a**5/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**
10*x**2) - 240*a**4*b*sqrt(x)*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b
**9*x**(3/2) + 6*b**10*x**2) - 360*a**3*b**2*x*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**
2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) + 120*a**3*b**2*x/(6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*
b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 240*a**2*b**3*x**(3/2)*log(a/b + sqrt(x))/(6*a**4*b**6 + 24*a**3
*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) + 200*a**2*b**3*x**(3/2)/(6*a**4*b**6 + 24
*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) - 60*a*b**4*x**2*log(a/b + sqrt(x))/(
6*a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) + 110*a*b**4*x**2/(6*
a**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2) + 12*b**5*x**(5/2)/(6*a
**4*b**6 + 24*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2), Ne(b, 0)), (x**3/(3*a**
5), True))

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Giac [A]  time = 1.10234, size = 99, normalized size = 0.93 \begin{align*} -\frac{10 \, a \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{2 \, \sqrt{x}}{b^{5}} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 300 \, a^{3} b^{2} x + 260 \, a^{4} b \sqrt{x} + 77 \, a^{5}}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

-10*a*log(abs(b*sqrt(x) + a))/b^6 + 2*sqrt(x)/b^5 - 1/6*(120*a^2*b^3*x^(3/2) + 300*a^3*b^2*x + 260*a^4*b*sqrt(
x) + 77*a^5)/((b*sqrt(x) + a)^4*b^6)