\(\int \frac {1}{(a+b \sqrt {x})^5 x^3} \, dx\) [2222]

Optimal result

Integrand size = 15, antiderivative size = 156 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=\frac {b^4}{2 a^5 \left (a+b \sqrt {x}\right )^4}+\frac {10 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^3}+\frac {15 b^4}{a^7 \left (a+b \sqrt {x}\right )^2}+\frac {70 b^4}{a^8 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^5 x^2}+\frac {10 b}{3 a^6 x^{3/2}}-\frac {15 b^2}{a^7 x}+\frac {70 b^3}{a^8 \sqrt {x}}-\frac {140 b^4 \log \left (a+b \sqrt {x}\right )}{a^9}+\frac {70 b^4 \log (x)}{a^9} \]

[Out]

-1/2/a^5/x^2+10/3*b/a^6/x^(3/2)-15*b^2/a^7/x+70*b^4*ln(x)/a^9-140*b^4*ln(a+b*x^(1/2))/a^9+70*b^3/a^8/x^(1/2)+1
/2*b^4/a^5/(a+b*x^(1/2))^4+10/3*b^4/a^6/(a+b*x^(1/2))^3+15*b^4/a^7/(a+b*x^(1/2))^2+70*b^4/a^8/(a+b*x^(1/2))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 156, 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, 46} \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=-\frac {140 b^4 \log \left (a+b \sqrt {x}\right )}{a^9}+\frac {70 b^4 \log (x)}{a^9}+\frac {70 b^4}{a^8 \left (a+b \sqrt {x}\right )}+\frac {70 b^3}{a^8 \sqrt {x}}+\frac {15 b^4}{a^7 \left (a+b \sqrt {x}\right )^2}-\frac {15 b^2}{a^7 x}+\frac {10 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^3}+\frac {10 b}{3 a^6 x^{3/2}}+\frac {b^4}{2 a^5 \left (a+b \sqrt {x}\right )^4}-\frac {1}{2 a^5 x^2} \]

[In]

Int[1/((a + b*Sqrt[x])^5*x^3),x]

[Out]

b^4/(2*a^5*(a + b*Sqrt[x])^4) + (10*b^4)/(3*a^6*(a + b*Sqrt[x])^3) + (15*b^4)/(a^7*(a + b*Sqrt[x])^2) + (70*b^
4)/(a^8*(a + b*Sqrt[x])) - 1/(2*a^5*x^2) + (10*b)/(3*a^6*x^(3/2)) - (15*b^2)/(a^7*x) + (70*b^3)/(a^8*Sqrt[x])
- (140*b^4*Log[a + b*Sqrt[x]])/a^9 + (70*b^4*Log[x])/a^9

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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 {1}{x^5 (a+b x)^5} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{a^5 x^5}-\frac {5 b}{a^6 x^4}+\frac {15 b^2}{a^7 x^3}-\frac {35 b^3}{a^8 x^2}+\frac {70 b^4}{a^9 x}-\frac {b^5}{a^5 (a+b x)^5}-\frac {5 b^5}{a^6 (a+b x)^4}-\frac {15 b^5}{a^7 (a+b x)^3}-\frac {35 b^5}{a^8 (a+b x)^2}-\frac {70 b^5}{a^9 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {b^4}{2 a^5 \left (a+b \sqrt {x}\right )^4}+\frac {10 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^3}+\frac {15 b^4}{a^7 \left (a+b \sqrt {x}\right )^2}+\frac {70 b^4}{a^8 \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^5 x^2}+\frac {10 b}{3 a^6 x^{3/2}}-\frac {15 b^2}{a^7 x}+\frac {70 b^3}{a^8 \sqrt {x}}-\frac {140 b^4 \log \left (a+b \sqrt {x}\right )}{a^9}+\frac {70 b^4 \log (x)}{a^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=\frac {\frac {a \left (-3 a^7+8 a^6 b \sqrt {x}-28 a^5 b^2 x+168 a^4 b^3 x^{3/2}+1750 a^3 b^4 x^2+3640 a^2 b^5 x^{5/2}+2940 a b^6 x^3+840 b^7 x^{7/2}\right )}{\left (a+b \sqrt {x}\right )^4 x^2}-840 b^4 \log \left (a+b \sqrt {x}\right )+420 b^4 \log (x)}{6 a^9} \]

[In]

Integrate[1/((a + b*Sqrt[x])^5*x^3),x]

[Out]

((a*(-3*a^7 + 8*a^6*b*Sqrt[x] - 28*a^5*b^2*x + 168*a^4*b^3*x^(3/2) + 1750*a^3*b^4*x^2 + 3640*a^2*b^5*x^(5/2) +
2940*a*b^6*x^3 + 840*b^7*x^(7/2)))/((a + b*Sqrt[x])^4*x^2) - 840*b^4*Log[a + b*Sqrt[x]] + 420*b^4*Log[x])/(6*
a^9)

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(-\frac {1}{2 a^{5} x^{2}}+\frac {10 b}{3 a^{6} x^{\frac {3}{2}}}-\frac {15 b^{2}}{a^{7} x}+\frac {70 b^{4} \ln \left (x \right )}{a^{9}}-\frac {140 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{9}}+\frac {70 b^{3}}{a^{8} \sqrt {x}}+\frac {b^{4}}{2 a^{5} \left (a +b \sqrt {x}\right )^{4}}+\frac {10 b^{4}}{3 a^{6} \left (a +b \sqrt {x}\right )^{3}}+\frac {15 b^{4}}{a^{7} \left (a +b \sqrt {x}\right )^{2}}+\frac {70 b^{4}}{a^{8} \left (a +b \sqrt {x}\right )}\)	\(135\)
default	\(-\frac {1}{2 a^{5} x^{2}}+\frac {10 b}{3 a^{6} x^{\frac {3}{2}}}-\frac {15 b^{2}}{a^{7} x}+\frac {70 b^{4} \ln \left (x \right )}{a^{9}}-\frac {140 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{9}}+\frac {70 b^{3}}{a^{8} \sqrt {x}}+\frac {b^{4}}{2 a^{5} \left (a +b \sqrt {x}\right )^{4}}+\frac {10 b^{4}}{3 a^{6} \left (a +b \sqrt {x}\right )^{3}}+\frac {15 b^{4}}{a^{7} \left (a +b \sqrt {x}\right )^{2}}+\frac {70 b^{4}}{a^{8} \left (a +b \sqrt {x}\right )}\)	\(135\)

[In]

int(1/x^3/(a+b*x^(1/2))^5,x,method=_RETURNVERBOSE)

[Out]

-1/2/a^5/x^2+10/3*b/a^6/x^(3/2)-15*b^2/a^7/x+70*b^4*ln(x)/a^9-140*b^4*ln(a+b*x^(1/2))/a^9+70*b^3/a^8/x^(1/2)+1
/2*b^4/a^5/(a+b*x^(1/2))^4+10/3*b^4/a^6/(a+b*x^(1/2))^3+15*b^4/a^7/(a+b*x^(1/2))^2+70*b^4/a^8/(a+b*x^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (134) = 268\).

Time = 0.27 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=-\frac {420 \, a^{2} b^{10} x^{5} - 1470 \, a^{4} b^{8} x^{4} + 1820 \, a^{6} b^{6} x^{3} - 875 \, a^{8} b^{4} x^{2} + 78 \, a^{10} b^{2} x + 3 \, a^{12} + 840 \, {\left (b^{12} x^{6} - 4 \, a^{2} b^{10} x^{5} + 6 \, a^{4} b^{8} x^{4} - 4 \, a^{6} b^{6} x^{3} + a^{8} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 840 \, {\left (b^{12} x^{6} - 4 \, a^{2} b^{10} x^{5} + 6 \, a^{4} b^{8} x^{4} - 4 \, a^{6} b^{6} x^{3} + a^{8} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (210 \, a b^{11} x^{5} - 770 \, a^{3} b^{9} x^{4} + 1022 \, a^{5} b^{7} x^{3} - 558 \, a^{7} b^{5} x^{2} + 85 \, a^{9} b^{3} x + 5 \, a^{11} b\right )} \sqrt {x}}{6 \, {\left (a^{9} b^{8} x^{6} - 4 \, a^{11} b^{6} x^{5} + 6 \, a^{13} b^{4} x^{4} - 4 \, a^{15} b^{2} x^{3} + a^{17} x^{2}\right )}} \]

[In]

integrate(1/x^3/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

-1/6*(420*a^2*b^10*x^5 - 1470*a^4*b^8*x^4 + 1820*a^6*b^6*x^3 - 875*a^8*b^4*x^2 + 78*a^10*b^2*x + 3*a^12 + 840*
(b^12*x^6 - 4*a^2*b^10*x^5 + 6*a^4*b^8*x^4 - 4*a^6*b^6*x^3 + a^8*b^4*x^2)*log(b*sqrt(x) + a) - 840*(b^12*x^6 -
4*a^2*b^10*x^5 + 6*a^4*b^8*x^4 - 4*a^6*b^6*x^3 + a^8*b^4*x^2)*log(sqrt(x)) - 4*(210*a*b^11*x^5 - 770*a^3*b^9*
x^4 + 1022*a^5*b^7*x^3 - 558*a^7*b^5*x^2 + 85*a^9*b^3*x + 5*a^11*b)*sqrt(x))/(a^9*b^8*x^6 - 4*a^11*b^6*x^5 + 6
*a^13*b^4*x^4 - 4*a^15*b^2*x^3 + a^17*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1391 vs. \(2 (151) = 302\).

Time = 3.09 (sec) , antiderivative size = 1391, normalized size of antiderivative = 8.92 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/x**3/(a+b*x**(1/2))**5,x)

[Out]

Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-1/(2*a**5*x**2), Eq(b, 0)), (-2/(9*b**5*x**(9/2)), Eq(a, 0)),
(zoo/x**2, Eq(a, -b*sqrt(x))), (-3*a**8*sqrt(x)/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2)
+ 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 8*a**7*b*x/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*
x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) - 28*a**6*b**2*x**(3/2)/(6*a**13*x**(5/2) + 24*a**12*b*x
**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 168*a**5*b**3*x**2/(6*a**13*x**(5/
2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 420*a**4*b**4*x**
(5/2)*log(x)/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x
**(9/2)) - 840*a**4*b**4*x**(5/2)*log(a/b + sqrt(x))/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7
/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 1750*a**4*b**4*x**(5/2)/(6*a**13*x**(5/2) + 24*a**12*b*x**3
+ 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 1680*a**3*b**5*x**3*log(x)/(6*a**13*x
**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) - 3360*a**3*b*
*5*x**3*log(a/b + sqrt(x))/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 +
6*a**9*b**4*x**(9/2)) + 3640*a**3*b**5*x**3/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24
*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 2520*a**2*b**6*x**(7/2)*log(x)/(6*a**13*x**(5/2) + 24*a**12*b*x**3
+ 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) - 5040*a**2*b**6*x**(7/2)*log(a/b + sqrt
(x))/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2))
+ 2940*a**2*b**6*x**(7/2)/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 +
6*a**9*b**4*x**(9/2)) + 1680*a*b**7*x**4*log(x)/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2)
+ 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) - 3360*a*b**7*x**4*log(a/b + sqrt(x))/(6*a**13*x**(5/2) + 24*a**1
2*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 840*a*b**7*x**4/(6*a**13*x**(
5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)) + 420*b**8*x**(9/
2)*log(x)/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24*a**10*b**3*x**4 + 6*a**9*b**4*x**(
9/2)) - 840*b**8*x**(9/2)*log(a/b + sqrt(x))/(6*a**13*x**(5/2) + 24*a**12*b*x**3 + 36*a**11*b**2*x**(7/2) + 24
*a**10*b**3*x**4 + 6*a**9*b**4*x**(9/2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=\frac {840 \, b^{7} x^{\frac {7}{2}} + 2940 \, a b^{6} x^{3} + 3640 \, a^{2} b^{5} x^{\frac {5}{2}} + 1750 \, a^{3} b^{4} x^{2} + 168 \, a^{4} b^{3} x^{\frac {3}{2}} - 28 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt {x} - 3 \, a^{7}}{6 \, {\left (a^{8} b^{4} x^{4} + 4 \, a^{9} b^{3} x^{\frac {7}{2}} + 6 \, a^{10} b^{2} x^{3} + 4 \, a^{11} b x^{\frac {5}{2}} + a^{12} x^{2}\right )}} - \frac {140 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{9}} + \frac {70 \, b^{4} \log \left (x\right )}{a^{9}} \]

[In]

integrate(1/x^3/(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

1/6*(840*b^7*x^(7/2) + 2940*a*b^6*x^3 + 3640*a^2*b^5*x^(5/2) + 1750*a^3*b^4*x^2 + 168*a^4*b^3*x^(3/2) - 28*a^5
*b^2*x + 8*a^6*b*sqrt(x) - 3*a^7)/(a^8*b^4*x^4 + 4*a^9*b^3*x^(7/2) + 6*a^10*b^2*x^3 + 4*a^11*b*x^(5/2) + a^12*
x^2) - 140*b^4*log(b*sqrt(x) + a)/a^9 + 70*b^4*log(x)/a^9

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=-\frac {140 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{9}} + \frac {70 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac {840 \, b^{7} x^{\frac {7}{2}} + 2940 \, a b^{6} x^{3} + 3640 \, a^{2} b^{5} x^{\frac {5}{2}} + 1750 \, a^{3} b^{4} x^{2} + 168 \, a^{4} b^{3} x^{\frac {3}{2}} - 28 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt {x} - 3 \, a^{7}}{6 \, {\left (b x + a \sqrt {x}\right )}^{4} a^{8}} \]

[In]

integrate(1/x^3/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

-140*b^4*log(abs(b*sqrt(x) + a))/a^9 + 70*b^4*log(abs(x))/a^9 + 1/6*(840*b^7*x^(7/2) + 2940*a*b^6*x^3 + 3640*a
^2*b^5*x^(5/2) + 1750*a^3*b^4*x^2 + 168*a^4*b^3*x^(3/2) - 28*a^5*b^2*x + 8*a^6*b*sqrt(x) - 3*a^7)/((b*x + a*sq
rt(x))^4*a^8)

Mupad [B] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^5 x^3} \, dx=\frac {\frac {4\,b\,\sqrt {x}}{3\,a^2}-\frac {1}{2\,a}-\frac {14\,b^2\,x}{3\,a^3}+\frac {875\,b^4\,x^2}{3\,a^5}+\frac {28\,b^3\,x^{3/2}}{a^4}+\frac {490\,b^6\,x^3}{a^7}+\frac {1820\,b^5\,x^{5/2}}{3\,a^6}+\frac {140\,b^7\,x^{7/2}}{a^8}}{a^4\,x^2+b^4\,x^4+4\,a^3\,b\,x^{5/2}+4\,a\,b^3\,x^{7/2}+6\,a^2\,b^2\,x^3}-\frac {280\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^9} \]

[In]

int(1/(x^3*(a + b*x^(1/2))^5),x)

[Out]

((4*b*x^(1/2))/(3*a^2) - 1/(2*a) - (14*b^2*x)/(3*a^3) + (875*b^4*x^2)/(3*a^5) + (28*b^3*x^(3/2))/a^4 + (490*b^
6*x^3)/a^7 + (1820*b^5*x^(5/2))/(3*a^6) + (140*b^7*x^(7/2))/a^8)/(a^4*x^2 + b^4*x^4 + 4*a^3*b*x^(5/2) + 4*a*b^
3*x^(7/2) + 6*a^2*b^2*x^3) - (280*b^4*atanh((2*b*x^(1/2))/a + 1))/a^9