3.2213 \(\int \frac{1}{(a+b \sqrt{x})^3 x^3} \, dx\)

Optimal. Leaf size=111 \[ \frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}+\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{20 b^3}{a^6 \sqrt{x}}-\frac{6 b^2}{a^5 x}-\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{2 b}{a^4 x^{3/2}}-\frac{1}{2 a^3 x^2} \]

[Out]

b^4/(a^5*(a + b*Sqrt[x])^2) + (10*b^4)/(a^6*(a + b*Sqrt[x])) - 1/(2*a^3*x^2) + (2*b)/(a^4*x^(3/2)) - (6*b^2)/(
a^5*x) + (20*b^3)/(a^6*Sqrt[x]) - (30*b^4*Log[a + b*Sqrt[x]])/a^7 + (15*b^4*Log[x])/a^7

________________________________________________________________________________________

Rubi [A]  time = 0.0714815, antiderivative size = 111, 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} \[ \frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}+\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{20 b^3}{a^6 \sqrt{x}}-\frac{6 b^2}{a^5 x}-\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}+\frac{2 b}{a^4 x^{3/2}}-\frac{1}{2 a^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^3*x^3),x]

[Out]

b^4/(a^5*(a + b*Sqrt[x])^2) + (10*b^4)/(a^6*(a + b*Sqrt[x])) - 1/(2*a^3*x^2) + (2*b)/(a^4*x^(3/2)) - (6*b^2)/(
a^5*x) + (20*b^3)/(a^6*Sqrt[x]) - (30*b^4*Log[a + b*Sqrt[x]])/a^7 + (15*b^4*Log[x])/a^7

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 )^3 x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^5}-\frac{3 b}{a^4 x^4}+\frac{6 b^2}{a^5 x^3}-\frac{10 b^3}{a^6 x^2}+\frac{15 b^4}{a^7 x}-\frac{b^5}{a^5 (a+b x)^3}-\frac{5 b^5}{a^6 (a+b x)^2}-\frac{15 b^5}{a^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b^4}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{10 b^4}{a^6 \left (a+b \sqrt{x}\right )}-\frac{1}{2 a^3 x^2}+\frac{2 b}{a^4 x^{3/2}}-\frac{6 b^2}{a^5 x}+\frac{20 b^3}{a^6 \sqrt{x}}-\frac{30 b^4 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^4 \log (x)}{a^7}\\ \end{align*}

Mathematica [A]  time = 0.119479, size = 104, normalized size = 0.94 \[ \frac{\frac{a \left (20 a^2 b^3 x^{3/2}-5 a^3 b^2 x+2 a^4 b \sqrt{x}-a^5+90 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^2 \left (a+b \sqrt{x}\right )^2}-60 b^4 \log \left (a+b \sqrt{x}\right )+30 b^4 \log (x)}{2 a^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(-a^5 + 2*a^4*b*Sqrt[x] - 5*a^3*b^2*x + 20*a^2*b^3*x^(3/2) + 90*a*b^4*x^2 + 60*b^5*x^(5/2)))/((a + b*Sqrt[
x])^2*x^2) - 60*b^4*Log[a + b*Sqrt[x]] + 30*b^4*Log[x])/(2*a^7)

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 100, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}{a}^{3}}}+2\,{\frac{b}{{a}^{4}{x}^{3/2}}}-6\,{\frac{{b}^{2}}{x{a}^{5}}}+15\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{7}}}-30\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+20\,{\frac{{b}^{3}}{{a}^{6}\sqrt{x}}}+{\frac{{b}^{4}}{{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-2}}+10\,{\frac{{b}^{4}}{{a}^{6} \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))^3,x)

[Out]

-1/2/x^2/a^3+2*b/a^4/x^(3/2)-6*b^2/x/a^5+15*b^4*ln(x)/a^7-30*b^4*ln(a+b*x^(1/2))/a^7+20*b^3/a^6/x^(1/2)+b^4/a^
5/(a+b*x^(1/2))^2+10*b^4/a^6/(a+b*x^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.01847, size = 149, normalized size = 1.34 \begin{align*} \frac{60 \, b^{5} x^{\frac{5}{2}} + 90 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} - 5 \, a^{3} b^{2} x + 2 \, a^{4} b \sqrt{x} - a^{5}}{2 \,{\left (a^{6} b^{2} x^{3} + 2 \, a^{7} b x^{\frac{5}{2}} + a^{8} x^{2}\right )}} - \frac{30 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{15 \, b^{4} \log \left (x\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(60*b^5*x^(5/2) + 90*a*b^4*x^2 + 20*a^2*b^3*x^(3/2) - 5*a^3*b^2*x + 2*a^4*b*sqrt(x) - a^5)/(a^6*b^2*x^3 +
2*a^7*b*x^(5/2) + a^8*x^2) - 30*b^4*log(b*sqrt(x) + a)/a^7 + 15*b^4*log(x)/a^7

________________________________________________________________________________________

Fricas [A]  time = 1.37366, size = 392, normalized size = 3.53 \begin{align*} -\frac{30 \, a^{2} b^{6} x^{3} - 45 \, a^{4} b^{4} x^{2} + 10 \, a^{6} b^{2} x + a^{8} + 60 \,{\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (b \sqrt{x} + a\right ) - 60 \,{\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (\sqrt{x}\right ) - 4 \,{\left (15 \, a b^{7} x^{3} - 25 \, a^{3} b^{5} x^{2} + 8 \, a^{5} b^{3} x + a^{7} b\right )} \sqrt{x}}{2 \,{\left (a^{7} b^{4} x^{4} - 2 \, a^{9} b^{2} x^{3} + a^{11} 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))^3,x, algorithm="fricas")

[Out]

-1/2*(30*a^2*b^6*x^3 - 45*a^4*b^4*x^2 + 10*a^6*b^2*x + a^8 + 60*(b^8*x^4 - 2*a^2*b^6*x^3 + a^4*b^4*x^2)*log(b*
sqrt(x) + a) - 60*(b^8*x^4 - 2*a^2*b^6*x^3 + a^4*b^4*x^2)*log(sqrt(x)) - 4*(15*a*b^7*x^3 - 25*a^3*b^5*x^2 + 8*
a^5*b^3*x + a^7*b)*sqrt(x))/(a^7*b^4*x^4 - 2*a^9*b^2*x^3 + a^11*x^2)

________________________________________________________________________________________

Sympy [A]  time = 8.80979, size = 612, normalized size = 5.51 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 a^{3} x^{2}} & \text{for}\: b = 0 \\- \frac{2}{7 b^{3} x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{a^{6} \sqrt{x}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{2 a^{5} b x}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{5 a^{4} b^{2} x^{\frac{3}{2}}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{20 a^{3} b^{3} x^{2}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{30 a^{2} b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{60 a^{2} b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{90 a^{2} b^{4} x^{\frac{5}{2}}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{60 a b^{5} x^{3} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{120 a b^{5} x^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{60 a b^{5} x^{3}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} + \frac{30 b^{6} x^{\frac{7}{2}} \log{\left (x \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} - \frac{60 b^{6} x^{\frac{7}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{2 a^{9} x^{\frac{5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-1/(2*a**3*x**2), Eq(b, 0)), (-2/(7*b**3*x**(7/2)), Eq(a, 0)),
(-a**6*sqrt(x)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 2*a**5*b*x/(2*a**9*x**(5/2) + 4*a**
8*b*x**3 + 2*a**7*b**2*x**(7/2)) - 5*a**4*b**2*x**(3/2)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2
)) + 20*a**3*b**3*x**2/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 30*a**2*b**4*x**(5/2)*log(x)
/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) - 60*a**2*b**4*x**(5/2)*log(a/b + sqrt(x))/(2*a**9*x
**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 90*a**2*b**4*x**(5/2)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a
**7*b**2*x**(7/2)) + 60*a*b**5*x**3*log(x)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) - 120*a*b*
*5*x**3*log(a/b + sqrt(x))/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 60*a*b**5*x**3/(2*a**9*x
**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(7/2)) + 30*b**6*x**(7/2)*log(x)/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2
*a**7*b**2*x**(7/2)) - 60*b**6*x**(7/2)*log(a/b + sqrt(x))/(2*a**9*x**(5/2) + 4*a**8*b*x**3 + 2*a**7*b**2*x**(
7/2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.11759, size = 136, normalized size = 1.23 \begin{align*} -\frac{30 \, b^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} + 90 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac{3}{2}} - 5 \, a^{4} b^{2} x + 2 \, a^{5} b \sqrt{x} - a^{6}}{2 \,{\left (b \sqrt{x} + a\right )}^{2} a^{7} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30*b^4*log(abs(b*sqrt(x) + a))/a^7 + 15*b^4*log(abs(x))/a^7 + 1/2*(60*a*b^5*x^(5/2) + 90*a^2*b^4*x^2 + 20*a^3
*b^3*x^(3/2) - 5*a^4*b^2*x + 2*a^5*b*sqrt(x) - a^6)/((b*sqrt(x) + a)^2*a^7*x^2)