3.2221 \(\int \frac{1}{(a+b \sqrt{x})^5 x^2} \, dx\)

Optimal. Leaf size=126 \[ \frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4}-\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}+\frac{10 b}{a^6 \sqrt{x}}-\frac{1}{a^5 x} \]

[Out]

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

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Rubi [A]  time = 0.0853851, antiderivative size = 126, 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} \[ \frac{20 b^2}{a^6 \left (a+b \sqrt{x}\right )}+\frac{6 b^2}{a^5 \left (a+b \sqrt{x}\right )^2}+\frac{2 b^2}{a^4 \left (a+b \sqrt{x}\right )^3}+\frac{b^2}{2 a^3 \left (a+b \sqrt{x}\right )^4}-\frac{30 b^2 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{15 b^2 \log (x)}{a^7}+\frac{10 b}{a^6 \sqrt{x}}-\frac{1}{a^5 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^5*x^2),x]

[Out]

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

Mathematica [A]  time = 0.115404, size = 104, normalized size = 0.83 \[ \frac{\frac{a \left (260 a^2 b^3 x^{3/2}+125 a^3 b^2 x+12 a^4 b \sqrt{x}-2 a^5+210 a b^4 x^2+60 b^5 x^{5/2}\right )}{x \left (a+b \sqrt{x}\right )^4}-60 b^2 \log \left (a+b \sqrt{x}\right )+30 b^2 \log (x)}{2 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^5*x^2),x]

[Out]

((a*(-2*a^5 + 12*a^4*b*Sqrt[x] + 125*a^3*b^2*x + 260*a^2*b^3*x^(3/2) + 210*a*b^4*x^2 + 60*b^5*x^(5/2)))/((a +
b*Sqrt[x])^4*x) - 60*b^2*Log[a + b*Sqrt[x]] + 30*b^2*Log[x])/(2*a^7)

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Maple [A]  time = 0.012, size = 113, normalized size = 0.9 \begin{align*} -{\frac{1}{x{a}^{5}}}+15\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{7}}}-30\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+10\,{\frac{b}{{a}^{6}\sqrt{x}}}+{\frac{{b}^{2}}{2\,{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-4}}+2\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\sqrt{x} \right ) ^{3}}}+6\,{\frac{{b}^{2}}{{a}^{5} \left ( a+b\sqrt{x} \right ) ^{2}}}+20\,{\frac{{b}^{2}}{{a}^{6} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01853, size = 176, normalized size = 1.4 \begin{align*} \frac{60 \, b^{5} x^{\frac{5}{2}} + 210 \, a b^{4} x^{2} + 260 \, a^{2} b^{3} x^{\frac{3}{2}} + 125 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt{x} - 2 \, a^{5}}{2 \,{\left (a^{6} b^{4} x^{3} + 4 \, a^{7} b^{3} x^{\frac{5}{2}} + 6 \, a^{8} b^{2} x^{2} + 4 \, a^{9} b x^{\frac{3}{2}} + a^{10} x\right )}} - \frac{30 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{15 \, b^{2} \log \left (x\right )}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(60*b^5*x^(5/2) + 210*a*b^4*x^2 + 260*a^2*b^3*x^(3/2) + 125*a^3*b^2*x + 12*a^4*b*sqrt(x) - 2*a^5)/(a^6*b^4
*x^3 + 4*a^7*b^3*x^(5/2) + 6*a^8*b^2*x^2 + 4*a^9*b*x^(3/2) + a^10*x) - 30*b^2*log(b*sqrt(x) + a)/a^7 + 15*b^2*
log(x)/a^7

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Fricas [B]  time = 1.36896, size = 575, normalized size = 4.56 \begin{align*} -\frac{30 \, a^{2} b^{8} x^{4} - 105 \, a^{4} b^{6} x^{3} + 130 \, a^{6} b^{4} x^{2} - 65 \, a^{8} b^{2} x + 2 \, a^{10} + 60 \,{\left (b^{10} x^{5} - 4 \, a^{2} b^{8} x^{4} + 6 \, a^{4} b^{6} x^{3} - 4 \, a^{6} b^{4} x^{2} + a^{8} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) - 60 \,{\left (b^{10} x^{5} - 4 \, a^{2} b^{8} x^{4} + 6 \, a^{4} b^{6} x^{3} - 4 \, a^{6} b^{4} x^{2} + a^{8} b^{2} x\right )} \log \left (\sqrt{x}\right ) - 4 \,{\left (15 \, a b^{9} x^{4} - 55 \, a^{3} b^{7} x^{3} + 73 \, a^{5} b^{5} x^{2} - 40 \, a^{7} b^{3} x + 5 \, a^{9} b\right )} \sqrt{x}}{2 \,{\left (a^{7} b^{8} x^{5} - 4 \, a^{9} b^{6} x^{4} + 6 \, a^{11} b^{4} x^{3} - 4 \, a^{13} b^{2} x^{2} + a^{15} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(30*a^2*b^8*x^4 - 105*a^4*b^6*x^3 + 130*a^6*b^4*x^2 - 65*a^8*b^2*x + 2*a^10 + 60*(b^10*x^5 - 4*a^2*b^8*x^
4 + 6*a^4*b^6*x^3 - 4*a^6*b^4*x^2 + a^8*b^2*x)*log(b*sqrt(x) + a) - 60*(b^10*x^5 - 4*a^2*b^8*x^4 + 6*a^4*b^6*x
^3 - 4*a^6*b^4*x^2 + a^8*b^2*x)*log(sqrt(x)) - 4*(15*a*b^9*x^4 - 55*a^3*b^7*x^3 + 73*a^5*b^5*x^2 - 40*a^7*b^3*
x + 5*a^9*b)*sqrt(x))/(a^7*b^8*x^5 - 4*a^9*b^6*x^4 + 6*a^11*b^4*x^3 - 4*a^13*b^2*x^2 + a^15*x)

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Sympy [A]  time = 12.7859, size = 1229, normalized size = 9.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b**5*x**(7/2)), Eq(a, 0)), (-1/(a**5*x), Eq(b, 0)), (-2*
a**6*sqrt(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7
/2)) + 12*a**5*b*x/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4
*x**(7/2)) + 30*a**4*b**2*x**(3/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*
b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 60*a**4*b**2*x**(3/2)*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**
2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 120*a**3*b**3*x**2*log(x)/(2*a**11*x**(
3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 240*a**3*b**3*x**2*
log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4
*x**(7/2)) - 240*a**3*b**3*x**2/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3
+ 2*a**7*b**4*x**(7/2)) + 180*a**2*b**4*x**(5/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(
5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 360*a**2*b**4*x**(5/2)*log(a/b + sqrt(x))/(2*a**11*x**(3/2)
+ 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 540*a**2*b**4*x**(5/2)/(
2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 120*a*b
**5*x**3*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x*
*(7/2)) - 240*a*b**5*x**3*log(a/b + sqrt(x))/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a*
*8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 440*a*b**5*x**3/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5
/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) + 30*b**6*x**(7/2)*log(x)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 +
12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 60*b**6*x**(7/2)*log(a/b + sqrt(x))/(2*a**1
1*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)) - 125*b**6*x**(
7/2)/(2*a**11*x**(3/2) + 8*a**10*b*x**2 + 12*a**9*b**2*x**(5/2) + 8*a**8*b**3*x**3 + 2*a**7*b**4*x**(7/2)), Tr
ue))

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Giac [A]  time = 1.11892, size = 136, normalized size = 1.08 \begin{align*} -\frac{30 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{15 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} + 210 \, a^{2} b^{4} x^{2} + 260 \, a^{3} b^{3} x^{\frac{3}{2}} + 125 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt{x} - 2 \, a^{6}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} a^{7} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30*b^2*log(abs(b*sqrt(x) + a))/a^7 + 15*b^2*log(abs(x))/a^7 + 1/2*(60*a*b^5*x^(5/2) + 210*a^2*b^4*x^2 + 260*a
^3*b^3*x^(3/2) + 125*a^4*b^2*x + 12*a^5*b*sqrt(x) - 2*a^6)/((b*sqrt(x) + a)^4*a^7*x)