3.2222 \(\int \frac{1}{(a+b \sqrt{x})^5 x^3} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.10857, antiderivative size = 156, 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, 44} \[ \frac{70 b^4}{a^8 \left (a+b \sqrt{x}\right )}+\frac{15 b^4}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{10 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^3}+\frac{b^4}{2 a^5 \left (a+b \sqrt{x}\right )^4}+\frac{70 b^3}{a^8 \sqrt{x}}-\frac{15 b^2}{a^7 x}-\frac{140 b^4 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{70 b^4 \log (x)}{a^9}+\frac{10 b}{3 a^6 x^{3/2}}-\frac{1}{2 a^5 x^2} \]

Antiderivative was successfully verified.

[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 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 44

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] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^5 x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.147853, size = 128, normalized size = 0.82 \[ \frac{\frac{a \left (168 a^4 b^3 x^{3/2}+1750 a^3 b^4 x^2+3640 a^2 b^5 x^{5/2}-28 a^5 b^2 x+8 a^6 b \sqrt{x}-3 a^7+2940 a b^6 x^3+840 b^7 x^{7/2}\right )}{x^2 \left (a+b \sqrt{x}\right )^4}-840 b^4 \log \left (a+b \sqrt{x}\right )+420 b^4 \log (x)}{6 a^9} \]

Antiderivative was successfully verified.

[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]  time = 0.012, size = 135, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{5}{x}^{2}}}+{\frac{10\,b}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}-15\,{\frac{{b}^{2}}{{a}^{7}x}}+70\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{9}}}-140\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{9}}}+70\,{\frac{{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}}+15\,{\frac{{b}^{4}}{{a}^{7} \left ( a+b\sqrt{x} \right ) ^{2}}}+70\,{\frac{{b}^{4}}{{a}^{8} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

________________________________________________________________________________________

Maxima [A]  time = 0.983064, size = 208, normalized size = 1.33 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [B]  time = 1.39808, size = 653, normalized size = 4.19 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.6177, size = 161, normalized size = 1.03 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)