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

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

[Out]

-1/2/a^3/x^2+2*b/a^4/x^(3/2)-6*b^2/a^5/x+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))

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Rubi [A]  time = 0.07, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.18, size = 104, normalized size = 0.94 \[ \frac {\frac {a \left (-a^5+2 a^4 b \sqrt {x}-5 a^3 b^2 x+20 a^2 b^3 x^{3/2}+90 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^2 \left (a+b \sqrt {x}\right )^2}-60 b^4 \log \left (a+b \sqrt {x}\right )+30 b^4 \log (x)}{2 a^7} \]

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)

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fricas [A]  time = 0.85, size = 182, normalized size = 1.64 \[ -\frac {30 \, a^{2} b^{6} x^{3} - 45 \, a^{4} b^{4} x^{2} + 10 \, a^{6} b^{2} x + a^{8} + 60 \, {\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 60 \, {\left (b^{8} x^{4} - 2 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (15 \, a b^{7} x^{3} - 25 \, a^{3} b^{5} x^{2} + 8 \, a^{5} b^{3} x + a^{7} b\right )} \sqrt {x}}{2 \, {\left (a^{7} b^{4} x^{4} - 2 \, a^{9} b^{2} x^{3} + a^{11} x^{2}\right )}} \]

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)

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giac [A]  time = 0.16, size = 101, normalized size = 0.91 \[ -\frac {30 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{7}} + \frac {15 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{2}} + 90 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac {3}{2}} - 5 \, a^{4} b^{2} x + 2 \, a^{5} b \sqrt {x} - a^{6}}{2 \, {\left (b \sqrt {x} + a\right )}^{2} a^{7} x^{2}} \]

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)

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maple [A]  time = 0.01, size = 100, normalized size = 0.90 \[ \frac {b^{4}}{\left (b \sqrt {x}+a \right )^{2} a^{5}}+\frac {10 b^{4}}{\left (b \sqrt {x}+a \right ) a^{6}}+\frac {15 b^{4} \ln \relax (x )}{a^{7}}-\frac {30 b^{4} \ln \left (b \sqrt {x}+a \right )}{a^{7}}+\frac {20 b^{3}}{a^{6} \sqrt {x}}-\frac {6 b^{2}}{a^{5} x}+\frac {2 b}{a^{4} x^{\frac {3}{2}}}-\frac {1}{2 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.88, size = 110, normalized size = 0.99 \[ \frac {60 \, b^{5} x^{\frac {5}{2}} + 90 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac {3}{2}} - 5 \, a^{3} b^{2} x + 2 \, a^{4} b \sqrt {x} - a^{5}}{2 \, {\left (a^{6} b^{2} x^{3} + 2 \, a^{7} b x^{\frac {5}{2}} + a^{8} x^{2}\right )}} - \frac {30 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{7}} + \frac {15 \, b^{4} \log \relax (x)}{a^{7}} \]

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

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mupad [B]  time = 0.11, size = 102, normalized size = 0.92 \[ \frac {\frac {b\,\sqrt {x}}{a^2}-\frac {1}{2\,a}-\frac {5\,b^2\,x}{2\,a^3}+\frac {45\,b^4\,x^2}{a^5}+\frac {10\,b^3\,x^{3/2}}{a^4}+\frac {30\,b^5\,x^{5/2}}{a^6}}{a^2\,x^2+b^2\,x^3+2\,a\,b\,x^{5/2}}-\frac {60\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^(1/2))^3),x)

[Out]

((b*x^(1/2))/a^2 - 1/(2*a) - (5*b^2*x)/(2*a^3) + (45*b^4*x^2)/a^5 + (10*b^3*x^(3/2))/a^4 + (30*b^5*x^(5/2))/a^
6)/(a^2*x^2 + b^2*x^3 + 2*a*b*x^(5/2)) - (60*b^4*atanh((2*b*x^(1/2))/a + 1))/a^7

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sympy [A]  time = 11.06, size = 612, normalized size = 5.51 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {1}{2 a^{3} x^{2}} & \text {for}\: b = 0 \\- \frac {a^{6} \sqrt {x}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {2 a^{5} b x}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {5 a^{4} b^{2} x^{\frac {3}{2}}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {20 a^{3} b^{3} x^{2}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {30 a^{2} b^{4} x^{\frac {5}{2}} \log {\relax (x )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {60 a^{2} b^{4} x^{\frac {5}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {90 a^{2} b^{4} x^{\frac {5}{2}}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {60 a b^{5} x^{3} \log {\relax (x )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {120 a b^{5} x^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {60 a b^{5} x^{3}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} + \frac {30 b^{6} x^{\frac {7}{2}} \log {\relax (x )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} - \frac {60 b^{6} x^{\frac {7}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{2 a^{9} x^{\frac {5}{2}} + 4 a^{8} b x^{3} + 2 a^{7} b^{2} x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]

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)), (-2/(7*b**3*x**(7/2)), Eq(a, 0)), (-1/(2*a**3*x**2), Eq(b, 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))

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