Integrand size = 15, antiderivative size = 103 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=-\frac {1}{3 a x^3}+\frac {2 b}{5 a^2 x^{5/2}}-\frac {b^2}{2 a^3 x^2}+\frac {2 b^3}{3 a^4 x^{3/2}}-\frac {b^4}{a^5 x}+\frac {2 b^5}{a^6 \sqrt {x}}-\frac {2 b^6 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7} \] Output:
-1/3/a/x^3+2/5*b/a^2/x^(5/2)-1/2*b^2/a^3/x^2+2/3*b^3/a^4/x^(3/2)-b^4/a^5/x +2*b^5/a^6/x^(1/2)-2*b^6*ln(a+b*x^(1/2))/a^7+b^6*ln(x)/a^7
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=\frac {\frac {a \left (-10 a^5+12 a^4 b \sqrt {x}-15 a^3 b^2 x+20 a^2 b^3 x^{3/2}-30 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^3}-60 b^6 \log \left (a+b \sqrt {x}\right )+30 b^6 \log (x)}{30 a^7} \] Input:
Integrate[1/((a + b*Sqrt[x])*x^4),x]
Output:
((a*(-10*a^5 + 12*a^4*b*Sqrt[x] - 15*a^3*b^2*x + 20*a^2*b^3*x^(3/2) - 30*a *b^4*x^2 + 60*b^5*x^(5/2)))/x^3 - 60*b^6*Log[a + b*Sqrt[x]] + 30*b^6*Log[x ])/(30*a^7)
Time = 0.39 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^4 \left (a+b \sqrt {x}\right )} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right ) x^{7/2}}d\sqrt {x}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle 2 \int \left (-\frac {b^7}{a^7 \left (a+b \sqrt {x}\right )}+\frac {b^6}{a^7 \sqrt {x}}-\frac {b^5}{a^6 x}+\frac {b^4}{a^5 x^{3/2}}-\frac {b^3}{a^4 x^2}+\frac {b^2}{a^3 x^{5/2}}-\frac {b}{a^2 x^3}+\frac {1}{a x^{7/2}}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {b^6 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {b^6 \log \left (\sqrt {x}\right )}{a^7}+\frac {b^5}{a^6 \sqrt {x}}-\frac {b^4}{2 a^5 x}+\frac {b^3}{3 a^4 x^{3/2}}-\frac {b^2}{4 a^3 x^2}+\frac {b}{5 a^2 x^{5/2}}-\frac {1}{6 a x^3}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])*x^4),x]
Output:
2*(-1/6*1/(a*x^3) + b/(5*a^2*x^(5/2)) - b^2/(4*a^3*x^2) + b^3/(3*a^4*x^(3/ 2)) - b^4/(2*a^5*x) + b^5/(a^6*Sqrt[x]) - (b^6*Log[a + b*Sqrt[x]])/a^7 + ( b^6*Log[Sqrt[x]])/a^7)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.47 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {1}{3 a \,x^{3}}+\frac {2 b}{5 a^{2} x^{\frac {5}{2}}}-\frac {b^{2}}{2 a^{3} x^{2}}+\frac {2 b^{3}}{3 a^{4} x^{\frac {3}{2}}}-\frac {b^{4}}{a^{5} x}+\frac {2 b^{5}}{a^{6} \sqrt {x}}-\frac {2 b^{6} \ln \left (a +b \sqrt {x}\right )}{a^{7}}+\frac {b^{6} \ln \left (x \right )}{a^{7}}\) | \(88\) |
default | \(-\frac {1}{3 a \,x^{3}}+\frac {2 b}{5 a^{2} x^{\frac {5}{2}}}-\frac {b^{2}}{2 a^{3} x^{2}}+\frac {2 b^{3}}{3 a^{4} x^{\frac {3}{2}}}-\frac {b^{4}}{a^{5} x}+\frac {2 b^{5}}{a^{6} \sqrt {x}}-\frac {2 b^{6} \ln \left (a +b \sqrt {x}\right )}{a^{7}}+\frac {b^{6} \ln \left (x \right )}{a^{7}}\) | \(88\) |
Input:
int(1/(a+b*x^(1/2))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3/a/x^3+2/5*b/a^2/x^(5/2)-1/2*b^2/a^3/x^2+2/3*b^3/a^4/x^(3/2)-b^4/a^5/x +2*b^5/a^6/x^(1/2)-2*b^6*ln(a+b*x^(1/2))/a^7+b^6*ln(x)/a^7
Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=-\frac {60 \, b^{6} x^{3} \log \left (b \sqrt {x} + a\right ) - 60 \, b^{6} x^{3} \log \left (\sqrt {x}\right ) + 30 \, a^{2} b^{4} x^{2} + 15 \, a^{4} b^{2} x + 10 \, a^{6} - 4 \, {\left (15 \, a b^{5} x^{2} + 5 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt {x}}{30 \, a^{7} x^{3}} \] Input:
integrate(1/(a+b*x^(1/2))/x^4,x, algorithm="fricas")
Output:
-1/30*(60*b^6*x^3*log(b*sqrt(x) + a) - 60*b^6*x^3*log(sqrt(x)) + 30*a^2*b^ 4*x^2 + 15*a^4*b^2*x + 10*a^6 - 4*(15*a*b^5*x^2 + 5*a^3*b^3*x + 3*a^5*b)*s qrt(x))/(a^7*x^3)
Time = 1.50 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{7 b x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {1}{3 a x^{3}} & \text {for}\: b = 0 \\- \frac {1}{3 a x^{3}} + \frac {2 b}{5 a^{2} x^{\frac {5}{2}}} - \frac {b^{2}}{2 a^{3} x^{2}} + \frac {2 b^{3}}{3 a^{4} x^{\frac {3}{2}}} - \frac {b^{4}}{a^{5} x} + \frac {2 b^{5}}{a^{6} \sqrt {x}} + \frac {b^{6} \log {\left (x \right )}}{a^{7}} - \frac {2 b^{6} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(1/2))/x**4,x)
Output:
Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b*x**(7/2)), Eq(a, 0 )), (-1/(3*a*x**3), Eq(b, 0)), (-1/(3*a*x**3) + 2*b/(5*a**2*x**(5/2)) - b* *2/(2*a**3*x**2) + 2*b**3/(3*a**4*x**(3/2)) - b**4/(a**5*x) + 2*b**5/(a**6 *sqrt(x)) + b**6*log(x)/a**7 - 2*b**6*log(a/b + sqrt(x))/a**7, True))
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=-\frac {2 \, b^{6} \log \left (b \sqrt {x} + a\right )}{a^{7}} + \frac {b^{6} \log \left (x\right )}{a^{7}} + \frac {60 \, b^{5} x^{\frac {5}{2}} - 30 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac {3}{2}} - 15 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt {x} - 10 \, a^{5}}{30 \, a^{6} x^{3}} \] Input:
integrate(1/(a+b*x^(1/2))/x^4,x, algorithm="maxima")
Output:
-2*b^6*log(b*sqrt(x) + a)/a^7 + b^6*log(x)/a^7 + 1/30*(60*b^5*x^(5/2) - 30 *a*b^4*x^2 + 20*a^2*b^3*x^(3/2) - 15*a^3*b^2*x + 12*a^4*b*sqrt(x) - 10*a^5 )/(a^6*x^3)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=-\frac {2 \, b^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{7}} + \frac {b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{2}} - 30 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac {3}{2}} - 15 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt {x} - 10 \, a^{6}}{30 \, a^{7} x^{3}} \] Input:
integrate(1/(a+b*x^(1/2))/x^4,x, algorithm="giac")
Output:
-2*b^6*log(abs(b*sqrt(x) + a))/a^7 + b^6*log(abs(x))/a^7 + 1/30*(60*a*b^5* x^(5/2) - 30*a^2*b^4*x^2 + 20*a^3*b^3*x^(3/2) - 15*a^4*b^2*x + 12*a^5*b*sq rt(x) - 10*a^6)/(a^7*x^3)
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=-\frac {\frac {1}{3\,a}-\frac {2\,b\,\sqrt {x}}{5\,a^2}+\frac {b^2\,x}{2\,a^3}+\frac {b^4\,x^2}{a^5}-\frac {2\,b^3\,x^{3/2}}{3\,a^4}-\frac {2\,b^5\,x^{5/2}}{a^6}}{x^3}-\frac {4\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^7} \] Input:
int(1/(x^4*(a + b*x^(1/2))),x)
Output:
- (1/(3*a) - (2*b*x^(1/2))/(5*a^2) + (b^2*x)/(2*a^3) + (b^4*x^2)/a^5 - (2* b^3*x^(3/2))/(3*a^4) - (2*b^5*x^(5/2))/a^6)/x^3 - (4*b^6*atanh((2*b*x^(1/2 ))/a + 1))/a^7
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b \sqrt {x}\right ) x^4} \, dx=\frac {12 \sqrt {x}\, a^{5} b +20 \sqrt {x}\, a^{3} b^{3} x +60 \sqrt {x}\, a \,b^{5} x^{2}-60 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{6} x^{3}+60 \,\mathrm {log}\left (\sqrt {x}\right ) b^{6} x^{3}-10 a^{6}-15 a^{4} b^{2} x -30 a^{2} b^{4} x^{2}}{30 a^{7} x^{3}} \] Input:
int(1/(a+b*x^(1/2))/x^4,x)
Output:
(12*sqrt(x)*a**5*b + 20*sqrt(x)*a**3*b**3*x + 60*sqrt(x)*a*b**5*x**2 - 60* log(sqrt(x)*b + a)*b**6*x**3 + 60*log(sqrt(x))*b**6*x**3 - 10*a**6 - 15*a* *4*b**2*x - 30*a**2*b**4*x**2)/(30*a**7*x**3)