\(\int \frac {x^2}{(a+b \sqrt {x})^5} \, dx\) [2217]

Optimal result

Integrand size = 15, antiderivative size = 107 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {a^5}{2 b^6 \left (a+b \sqrt {x}\right )^4}-\frac {10 a^4}{3 b^6 \left (a+b \sqrt {x}\right )^3}+\frac {10 a^3}{b^6 \left (a+b \sqrt {x}\right )^2}-\frac {20 a^2}{b^6 \left (a+b \sqrt {x}\right )}+\frac {2 \sqrt {x}}{b^5}-\frac {10 a \log \left (a+b \sqrt {x}\right )}{b^6} \]

[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))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {a^5}{2 b^6 \left (a+b \sqrt {x}\right )^4}-\frac {10 a^4}{3 b^6 \left (a+b \sqrt {x}\right )^3}+\frac {10 a^3}{b^6 \left (a+b \sqrt {x}\right )^2}-\frac {20 a^2}{b^6 \left (a+b \sqrt {x}\right )}-\frac {10 a \log \left (a+b \sqrt {x}\right )}{b^6}+\frac {2 \sqrt {x}}{b^5} \]

[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 45

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])

Rule 272

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*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^5}{(a+b x)^5} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {-77 a^5-248 a^4 b \sqrt {x}-252 a^3 b^2 x-48 a^2 b^3 x^{3/2}+48 a b^4 x^2+12 b^5 x^{5/2}}{6 b^6 \left (a+b \sqrt {x}\right )^4}-\frac {10 a \log \left (a+b \sqrt {x}\right )}{b^6} \]

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

Maple [A] (verified)

Time = 3.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(-\frac {10 a \ln \left (a +b \sqrt {x}\right )}{b^{6}}+\frac {2 \sqrt {x}}{b^{5}}+\frac {a^{5}}{2 b^{6} \left (a +b \sqrt {x}\right )^{4}}-\frac {10 a^{4}}{3 b^{6} \left (a +b \sqrt {x}\right )^{3}}+\frac {10 a^{3}}{b^{6} \left (a +b \sqrt {x}\right )^{2}}-\frac {20 a^{2}}{b^{6} \left (a +b \sqrt {x}\right )}\)	\(92\)
default	\(-\frac {10 a \ln \left (a +b \sqrt {x}\right )}{b^{6}}+\frac {2 \sqrt {x}}{b^{5}}+\frac {a^{5}}{2 b^{6} \left (a +b \sqrt {x}\right )^{4}}-\frac {10 a^{4}}{3 b^{6} \left (a +b \sqrt {x}\right )^{3}}+\frac {10 a^{3}}{b^{6} \left (a +b \sqrt {x}\right )^{2}}-\frac {20 a^{2}}{b^{6} \left (a +b \sqrt {x}\right )}\)	\(92\)

[In]

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

[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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (91) = 182\).

Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.79 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {180 \, a^{3} b^{6} x^{3} - 357 \, a^{5} b^{4} x^{2} + 278 \, a^{7} b^{2} x - 77 \, a^{9} - 60 \, {\left (a b^{8} x^{4} - 4 \, a^{3} b^{6} x^{3} + 6 \, a^{5} b^{4} x^{2} - 4 \, a^{7} b^{2} x + a^{9}\right )} \log \left (b \sqrt {x} + a\right ) + 4 \, {\left (3 \, b^{9} x^{4} - 42 \, a^{2} b^{7} x^{3} + 73 \, a^{4} b^{5} x^{2} - 55 \, a^{6} b^{3} x + 15 \, a^{8} b\right )} \sqrt {x}}{6 \, {\left (b^{14} x^{4} - 4 \, a^{2} b^{12} x^{3} + 6 \, a^{4} b^{10} x^{2} - 4 \, a^{6} b^{8} x + a^{8} b^{6}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (100) = 200\).

Time = 0.63 (sec) , antiderivative size = 704, normalized size of antiderivative = 6.58 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x^{3}}{3 a^{5}} & \text {for}\: b = 0 \\\tilde {\infty } x^{3} & \text {for}\: a = - b \sqrt {x} \\- \frac {60 a^{5} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {125 a^{5}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {240 a^{4} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {440 a^{4} b \sqrt {x}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {360 a^{3} b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {540 a^{3} b^{2} x}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {240 a^{2} b^{3} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {240 a^{2} b^{3} x^{\frac {3}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} - \frac {60 a b^{4} x^{2} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} + \frac {12 b^{5} x^{\frac {5}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt {x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac {3}{2}} + 6 b^{10} x^{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(b, 0)), (x**3/(3*a**5), Eq(b, 0)), (zoo*x**3, Eq(a, -b*sqrt(x))), (-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) - 125*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) - 440*a**4*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) - 540*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) - 240*a**2*b**3*x**(3/2)/(6*a**4*b**6 + 24*a**3*b**7*sqr
t(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) + 12*b**5*x**(5/2)/(6*a**4*b**6 + 2
4*a**3*b**7*sqrt(x) + 36*a**2*b**8*x + 24*a*b**9*x**(3/2) + 6*b**10*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {10 \, a \log \left (b \sqrt {x} + a\right )}{b^{6}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}}{b^{6}} - \frac {20 \, a^{2}}{{\left (b \sqrt {x} + a\right )} b^{6}} + \frac {10 \, a^{3}}{{\left (b \sqrt {x} + a\right )}^{2} b^{6}} - \frac {10 \, a^{4}}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{6}} + \frac {a^{5}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{6}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=-\frac {10 \, a \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{6}} + \frac {2 \, \sqrt {x}}{b^{5}} - \frac {120 \, a^{2} b^{3} x^{\frac {3}{2}} + 300 \, a^{3} b^{2} x + 260 \, a^{4} b \sqrt {x} + 77 \, a^{5}}{6 \, {\left (b \sqrt {x} + a\right )}^{4} b^{6}} \]

[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)

Mupad [B] (verification not implemented)

Time = 5.74 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99 \[ \int \frac {x^2}{\left (a+b \sqrt {x}\right )^5} \, dx=\frac {2\,\sqrt {x}}{b^5}-\frac {\frac {77\,a^5}{6\,b}+\frac {130\,a^4\,\sqrt {x}}{3}+20\,a^2\,b^2\,x^{3/2}+50\,a^3\,b\,x}{a^4\,b^5+b^9\,x^2+6\,a^2\,b^7\,x+4\,a\,b^8\,x^{3/2}+4\,a^3\,b^6\,\sqrt {x}}-\frac {10\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b^6} \]

[In]

int(x^2/(a + b*x^(1/2))^5,x)

[Out]

(2*x^(1/2))/b^5 - ((77*a^5)/(6*b) + (130*a^4*x^(1/2))/3 + 20*a^2*b^2*x^(3/2) + 50*a^3*b*x)/(a^4*b^5 + b^9*x^2
+ 6*a^2*b^7*x + 4*a*b^8*x^(3/2) + 4*a^3*b^6*x^(1/2)) - (10*a*log(a + b*x^(1/2)))/b^6