Integrand size = 15, antiderivative size = 217 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=\frac {2 b^4}{7 a^5 \left (a+b \sqrt {x}\right )^7}+\frac {5 b^4}{3 a^6 \left (a+b \sqrt {x}\right )^6}+\frac {6 b^4}{a^7 \left (a+b \sqrt {x}\right )^5}+\frac {35 b^4}{2 a^8 \left (a+b \sqrt {x}\right )^4}+\frac {140 b^4}{3 a^9 \left (a+b \sqrt {x}\right )^3}+\frac {126 b^4}{a^{10} \left (a+b \sqrt {x}\right )^2}+\frac {420 b^4}{a^{11} \left (a+b \sqrt {x}\right )}-\frac {1}{2 a^8 x^2}+\frac {16 b}{3 a^9 x^{3/2}}-\frac {36 b^2}{a^{10} x}+\frac {240 b^3}{a^{11} \sqrt {x}}-\frac {660 b^4 \log \left (a+b \sqrt {x}\right )}{a^{12}}+\frac {330 b^4 \log (x)}{a^{12}} \] Output:
2/7*b^4/a^5/(a+b*x^(1/2))^7+5/3*b^4/a^6/(a+b*x^(1/2))^6+6*b^4/a^7/(a+b*x^( 1/2))^5+35/2*b^4/a^8/(a+b*x^(1/2))^4+140/3*b^4/a^9/(a+b*x^(1/2))^3+126*b^4 /a^10/(a+b*x^(1/2))^2+420*b^4/a^11/(a+b*x^(1/2))-1/2/a^8/x^2+16/3*b/a^9/x^ (3/2)-36*b^2/a^10/x+240*b^3/a^11/x^(1/2)-660*b^4*ln(a+b*x^(1/2))/a^12+330* b^4*ln(x)/a^12
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=\frac {\frac {a \left (-21 a^{10}+77 a^9 b \sqrt {x}-385 a^8 b^2 x+3465 a^7 b^3 x^{3/2}+71874 a^6 b^4 x^2+309078 a^5 b^5 x^{5/2}+636174 a^4 b^6 x^3+736890 a^3 b^7 x^{7/2}+494340 a^2 b^8 x^4+180180 a b^9 x^{9/2}+27720 b^{10} x^5\right )}{\left (a+b \sqrt {x}\right )^7 x^2}-27720 b^4 \log \left (a+b \sqrt {x}\right )+13860 b^4 \log (x)}{42 a^{12}} \] Input:
Integrate[1/((a + b*Sqrt[x])^8*x^3),x]
Output:
((a*(-21*a^10 + 77*a^9*b*Sqrt[x] - 385*a^8*b^2*x + 3465*a^7*b^3*x^(3/2) + 71874*a^6*b^4*x^2 + 309078*a^5*b^5*x^(5/2) + 636174*a^4*b^6*x^3 + 736890*a ^3*b^7*x^(7/2) + 494340*a^2*b^8*x^4 + 180180*a*b^9*x^(9/2) + 27720*b^10*x^ 5))/((a + b*Sqrt[x])^7*x^2) - 27720*b^4*Log[a + b*Sqrt[x]] + 13860*b^4*Log [x])/(42*a^12)
Time = 0.65 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b \sqrt {x}\right )^8} \, dx\) |
| \(\Big \downarrow \) 798 |
| \(\displaystyle 2 \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^{5/2}}d\sqrt {x}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 2 \int \left (-\frac {330 b^5}{a^{12} \left (a+b \sqrt {x}\right )}-\frac {210 b^5}{a^{11} \left (a+b \sqrt {x}\right )^2}-\frac {126 b^5}{a^{10} \left (a+b \sqrt {x}\right )^3}-\frac {70 b^5}{a^9 \left (a+b \sqrt {x}\right )^4}-\frac {35 b^5}{a^8 \left (a+b \sqrt {x}\right )^5}-\frac {15 b^5}{a^7 \left (a+b \sqrt {x}\right )^6}-\frac {5 b^5}{a^6 \left (a+b \sqrt {x}\right )^7}-\frac {b^5}{a^5 \left (a+b \sqrt {x}\right )^8}+\frac {330 b^4}{a^{12} \sqrt {x}}-\frac {120 b^3}{a^{11} x}+\frac {36 b^2}{a^{10} x^{3/2}}-\frac {8 b}{a^9 x^2}+\frac {1}{a^8 x^{5/2}}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {330 b^4 \log \left (a+b \sqrt {x}\right )}{a^{12}}+\frac {330 b^4 \log \left (\sqrt {x}\right )}{a^{12}}+\frac {210 b^4}{a^{11} \left (a+b \sqrt {x}\right )}+\frac {120 b^3}{a^{11} \sqrt {x}}+\frac {63 b^4}{a^{10} \left (a+b \sqrt {x}\right )^2}-\frac {18 b^2}{a^{10} x}+\frac {70 b^4}{3 a^9 \left (a+b \sqrt {x}\right )^3}+\frac {8 b}{3 a^9 x^{3/2}}+\frac {35 b^4}{4 a^8 \left (a+b \sqrt {x}\right )^4}-\frac {1}{4 a^8 x^2}+\frac {3 b^4}{a^7 \left (a+b \sqrt {x}\right )^5}+\frac {5 b^4}{6 a^6 \left (a+b \sqrt {x}\right )^6}+\frac {b^4}{7 a^5 \left (a+b \sqrt {x}\right )^7}\right )\) |
Input:
Int[1/((a + b*Sqrt[x])^8*x^3),x]
Output:
2*(b^4/(7*a^5*(a + b*Sqrt[x])^7) + (5*b^4)/(6*a^6*(a + b*Sqrt[x])^6) + (3* b^4)/(a^7*(a + b*Sqrt[x])^5) + (35*b^4)/(4*a^8*(a + b*Sqrt[x])^4) + (70*b^ 4)/(3*a^9*(a + b*Sqrt[x])^3) + (63*b^4)/(a^10*(a + b*Sqrt[x])^2) + (210*b^ 4)/(a^11*(a + b*Sqrt[x])) - 1/(4*a^8*x^2) + (8*b)/(3*a^9*x^(3/2)) - (18*b^ 2)/(a^10*x) + (120*b^3)/(a^11*Sqrt[x]) - (330*b^4*Log[a + b*Sqrt[x]])/a^12 + (330*b^4*Log[Sqrt[x]])/a^12)
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 = 0.48 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {2 b^{4}}{7 a^{5} \left (a +b \sqrt {x}\right )^{7}}+\frac {5 b^{4}}{3 a^{6} \left (a +b \sqrt {x}\right )^{6}}+\frac {6 b^{4}}{a^{7} \left (a +b \sqrt {x}\right )^{5}}+\frac {35 b^{4}}{2 a^{8} \left (a +b \sqrt {x}\right )^{4}}+\frac {140 b^{4}}{3 a^{9} \left (a +b \sqrt {x}\right )^{3}}+\frac {126 b^{4}}{a^{10} \left (a +b \sqrt {x}\right )^{2}}+\frac {420 b^{4}}{a^{11} \left (a +b \sqrt {x}\right )}-\frac {1}{2 a^{8} x^{2}}+\frac {16 b}{3 a^{9} x^{\frac {3}{2}}}-\frac {36 b^{2}}{a^{10} x}+\frac {240 b^{3}}{a^{11} \sqrt {x}}-\frac {660 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{12}}+\frac {330 b^{4} \ln \left (x \right )}{a^{12}}\) | \(186\) |
| default | \(\frac {2 b^{4}}{7 a^{5} \left (a +b \sqrt {x}\right )^{7}}+\frac {5 b^{4}}{3 a^{6} \left (a +b \sqrt {x}\right )^{6}}+\frac {6 b^{4}}{a^{7} \left (a +b \sqrt {x}\right )^{5}}+\frac {35 b^{4}}{2 a^{8} \left (a +b \sqrt {x}\right )^{4}}+\frac {140 b^{4}}{3 a^{9} \left (a +b \sqrt {x}\right )^{3}}+\frac {126 b^{4}}{a^{10} \left (a +b \sqrt {x}\right )^{2}}+\frac {420 b^{4}}{a^{11} \left (a +b \sqrt {x}\right )}-\frac {1}{2 a^{8} x^{2}}+\frac {16 b}{3 a^{9} x^{\frac {3}{2}}}-\frac {36 b^{2}}{a^{10} x}+\frac {240 b^{3}}{a^{11} \sqrt {x}}-\frac {660 b^{4} \ln \left (a +b \sqrt {x}\right )}{a^{12}}+\frac {330 b^{4} \ln \left (x \right )}{a^{12}}\) | \(186\) |
Input:
int(1/(a+b*x^(1/2))^8/x^3,x,method=_RETURNVERBOSE)
Output:
2/7*b^4/a^5/(a+b*x^(1/2))^7+5/3*b^4/a^6/(a+b*x^(1/2))^6+6*b^4/a^7/(a+b*x^( 1/2))^5+35/2*b^4/a^8/(a+b*x^(1/2))^4+140/3*b^4/a^9/(a+b*x^(1/2))^3+126*b^4 /a^10/(a+b*x^(1/2))^2+420*b^4/a^11/(a+b*x^(1/2))-1/2/a^8/x^2+16/3*b/a^9/x^ (3/2)-36*b^2/a^10/x+240*b^3/a^11/x^(1/2)-660*b^4*ln(a+b*x^(1/2))/a^12+330* b^4*ln(x)/a^12
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (185) = 370\).
Time = 0.26 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.13 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=-\frac {13860 \, a^{2} b^{16} x^{8} - 90090 \, a^{4} b^{14} x^{7} + 247170 \, a^{6} b^{12} x^{6} - 368445 \, a^{8} b^{10} x^{5} + 318087 \, a^{10} b^{8} x^{4} - 154532 \, a^{12} b^{6} x^{3} + 36104 \, a^{14} b^{4} x^{2} - 1365 \, a^{16} b^{2} x - 21 \, a^{18} + 27720 \, {\left (b^{18} x^{9} - 7 \, a^{2} b^{16} x^{8} + 21 \, a^{4} b^{14} x^{7} - 35 \, a^{6} b^{12} x^{6} + 35 \, a^{8} b^{10} x^{5} - 21 \, a^{10} b^{8} x^{4} + 7 \, a^{12} b^{6} x^{3} - a^{14} b^{4} x^{2}\right )} \log \left (b \sqrt {x} + a\right ) - 27720 \, {\left (b^{18} x^{9} - 7 \, a^{2} b^{16} x^{8} + 21 \, a^{4} b^{14} x^{7} - 35 \, a^{6} b^{12} x^{6} + 35 \, a^{8} b^{10} x^{5} - 21 \, a^{10} b^{8} x^{4} + 7 \, a^{12} b^{6} x^{3} - a^{14} b^{4} x^{2}\right )} \log \left (\sqrt {x}\right ) - 8 \, {\left (3465 \, a b^{17} x^{8} - 23100 \, a^{3} b^{15} x^{7} + 65373 \, a^{5} b^{13} x^{6} - 101376 \, a^{7} b^{11} x^{5} + 92323 \, a^{9} b^{9} x^{4} - 48580 \, a^{11} b^{7} x^{3} + 13083 \, a^{13} b^{5} x^{2} - 1064 \, a^{15} b^{3} x - 28 \, a^{17} b\right )} \sqrt {x}}{42 \, {\left (a^{12} b^{14} x^{9} - 7 \, a^{14} b^{12} x^{8} + 21 \, a^{16} b^{10} x^{7} - 35 \, a^{18} b^{8} x^{6} + 35 \, a^{20} b^{6} x^{5} - 21 \, a^{22} b^{4} x^{4} + 7 \, a^{24} b^{2} x^{3} - a^{26} x^{2}\right )}} \] Input:
integrate(1/(a+b*x^(1/2))^8/x^3,x, algorithm="fricas")
Output:
-1/42*(13860*a^2*b^16*x^8 - 90090*a^4*b^14*x^7 + 247170*a^6*b^12*x^6 - 368 445*a^8*b^10*x^5 + 318087*a^10*b^8*x^4 - 154532*a^12*b^6*x^3 + 36104*a^14* b^4*x^2 - 1365*a^16*b^2*x - 21*a^18 + 27720*(b^18*x^9 - 7*a^2*b^16*x^8 + 2 1*a^4*b^14*x^7 - 35*a^6*b^12*x^6 + 35*a^8*b^10*x^5 - 21*a^10*b^8*x^4 + 7*a ^12*b^6*x^3 - a^14*b^4*x^2)*log(b*sqrt(x) + a) - 27720*(b^18*x^9 - 7*a^2*b ^16*x^8 + 21*a^4*b^14*x^7 - 35*a^6*b^12*x^6 + 35*a^8*b^10*x^5 - 21*a^10*b^ 8*x^4 + 7*a^12*b^6*x^3 - a^14*b^4*x^2)*log(sqrt(x)) - 8*(3465*a*b^17*x^8 - 23100*a^3*b^15*x^7 + 65373*a^5*b^13*x^6 - 101376*a^7*b^11*x^5 + 92323*a^9 *b^9*x^4 - 48580*a^11*b^7*x^3 + 13083*a^13*b^5*x^2 - 1064*a^15*b^3*x - 28* a^17*b)*sqrt(x))/(a^12*b^14*x^9 - 7*a^14*b^12*x^8 + 21*a^16*b^10*x^7 - 35* a^18*b^8*x^6 + 35*a^20*b^6*x^5 - 21*a^22*b^4*x^4 + 7*a^24*b^2*x^3 - a^26*x ^2)
Leaf count of result is larger than twice the leaf count of optimal. 3077 vs. \(2 (212) = 424\).
Time = 9.62 (sec) , antiderivative size = 3077, normalized size of antiderivative = 14.18 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*x**(1/2))**8/x**3,x)
Output:
Piecewise((zoo/x**6, Eq(a, 0) & Eq(b, 0)), (-1/(2*a**8*x**2), Eq(b, 0)), ( -1/(6*b**8*x**6), Eq(a, 0)), (-21*a**11*sqrt(x)/(42*a**19*x**(5/2) + 294*a **18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15* b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**6*x**(11/2) + 42*a**12* b**7*x**6) + 77*a**10*b*x/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**1 7*b**2*x**(7/2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a* *14*b**5*x**5 + 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) - 385*a**9* b**2*x**(3/2)/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7 /2) + 1470*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x** 5 + 294*a**13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 3465*a**8*b**3*x**2/( 42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**1 6*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b **6*x**(11/2) + 42*a**12*b**7*x**6) + 13860*a**7*b**4*x**(5/2)*log(x)/(42* a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16*b **3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a**13*b**6 *x**(11/2) + 42*a**12*b**7*x**6) - 27720*a**7*b**4*x**(5/2)*log(a/b + sqrt (x))/(42*a**19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 147 0*a**16*b**3*x**4 + 1470*a**15*b**4*x**(9/2) + 882*a**14*b**5*x**5 + 294*a **13*b**6*x**(11/2) + 42*a**12*b**7*x**6) + 71874*a**7*b**4*x**(5/2)/(42*a **19*x**(5/2) + 294*a**18*b*x**3 + 882*a**17*b**2*x**(7/2) + 1470*a**16...
Time = 0.05 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=\frac {27720 \, b^{10} x^{5} + 180180 \, a b^{9} x^{\frac {9}{2}} + 494340 \, a^{2} b^{8} x^{4} + 736890 \, a^{3} b^{7} x^{\frac {7}{2}} + 636174 \, a^{4} b^{6} x^{3} + 309078 \, a^{5} b^{5} x^{\frac {5}{2}} + 71874 \, a^{6} b^{4} x^{2} + 3465 \, a^{7} b^{3} x^{\frac {3}{2}} - 385 \, a^{8} b^{2} x + 77 \, a^{9} b \sqrt {x} - 21 \, a^{10}}{42 \, {\left (a^{11} b^{7} x^{\frac {11}{2}} + 7 \, a^{12} b^{6} x^{5} + 21 \, a^{13} b^{5} x^{\frac {9}{2}} + 35 \, a^{14} b^{4} x^{4} + 35 \, a^{15} b^{3} x^{\frac {7}{2}} + 21 \, a^{16} b^{2} x^{3} + 7 \, a^{17} b x^{\frac {5}{2}} + a^{18} x^{2}\right )}} - \frac {660 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{12}} + \frac {330 \, b^{4} \log \left (x\right )}{a^{12}} \] Input:
integrate(1/(a+b*x^(1/2))^8/x^3,x, algorithm="maxima")
Output:
1/42*(27720*b^10*x^5 + 180180*a*b^9*x^(9/2) + 494340*a^2*b^8*x^4 + 736890* a^3*b^7*x^(7/2) + 636174*a^4*b^6*x^3 + 309078*a^5*b^5*x^(5/2) + 71874*a^6* b^4*x^2 + 3465*a^7*b^3*x^(3/2) - 385*a^8*b^2*x + 77*a^9*b*sqrt(x) - 21*a^1 0)/(a^11*b^7*x^(11/2) + 7*a^12*b^6*x^5 + 21*a^13*b^5*x^(9/2) + 35*a^14*b^4 *x^4 + 35*a^15*b^3*x^(7/2) + 21*a^16*b^2*x^3 + 7*a^17*b*x^(5/2) + a^18*x^2 ) - 660*b^4*log(b*sqrt(x) + a)/a^12 + 330*b^4*log(x)/a^12
Time = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=-\frac {660 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{12}} + \frac {330 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac {27720 \, a b^{10} x^{5} + 180180 \, a^{2} b^{9} x^{\frac {9}{2}} + 494340 \, a^{3} b^{8} x^{4} + 736890 \, a^{4} b^{7} x^{\frac {7}{2}} + 636174 \, a^{5} b^{6} x^{3} + 309078 \, a^{6} b^{5} x^{\frac {5}{2}} + 71874 \, a^{7} b^{4} x^{2} + 3465 \, a^{8} b^{3} x^{\frac {3}{2}} - 385 \, a^{9} b^{2} x + 77 \, a^{10} b \sqrt {x} - 21 \, a^{11}}{42 \, {\left (b \sqrt {x} + a\right )}^{7} a^{12} x^{2}} \] Input:
integrate(1/(a+b*x^(1/2))^8/x^3,x, algorithm="giac")
Output:
-660*b^4*log(abs(b*sqrt(x) + a))/a^12 + 330*b^4*log(abs(x))/a^12 + 1/42*(2 7720*a*b^10*x^5 + 180180*a^2*b^9*x^(9/2) + 494340*a^3*b^8*x^4 + 736890*a^4 *b^7*x^(7/2) + 636174*a^5*b^6*x^3 + 309078*a^6*b^5*x^(5/2) + 71874*a^7*b^4 *x^2 + 3465*a^8*b^3*x^(3/2) - 385*a^9*b^2*x + 77*a^10*b*sqrt(x) - 21*a^11) /((b*sqrt(x) + a)^7*a^12*x^2)
Time = 0.90 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=\frac {\frac {11\,b\,\sqrt {x}}{6\,a^2}-\frac {1}{2\,a}-\frac {55\,b^2\,x}{6\,a^3}+\frac {11979\,b^4\,x^2}{7\,a^5}+\frac {165\,b^3\,x^{3/2}}{2\,a^4}+\frac {15147\,b^6\,x^3}{a^7}+\frac {7359\,b^5\,x^{5/2}}{a^6}+\frac {11770\,b^8\,x^4}{a^9}+\frac {17545\,b^7\,x^{7/2}}{a^8}+\frac {660\,b^{10}\,x^5}{a^{11}}+\frac {4290\,b^9\,x^{9/2}}{a^{10}}}{a^7\,x^2+b^7\,x^{11/2}+7\,a\,b^6\,x^5+7\,a^6\,b\,x^{5/2}+21\,a^5\,b^2\,x^3+35\,a^3\,b^4\,x^4+35\,a^4\,b^3\,x^{7/2}+21\,a^2\,b^5\,x^{9/2}}-\frac {1320\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^{12}} \] Input:
int(1/(x^3*(a + b*x^(1/2))^8),x)
Output:
((11*b*x^(1/2))/(6*a^2) - 1/(2*a) - (55*b^2*x)/(6*a^3) + (11979*b^4*x^2)/( 7*a^5) + (165*b^3*x^(3/2))/(2*a^4) + (15147*b^6*x^3)/a^7 + (7359*b^5*x^(5/ 2))/a^6 + (11770*b^8*x^4)/a^9 + (17545*b^7*x^(7/2))/a^8 + (660*b^10*x^5)/a ^11 + (4290*b^9*x^(9/2))/a^10)/(a^7*x^2 + b^7*x^(11/2) + 7*a*b^6*x^5 + 7*a ^6*b*x^(5/2) + 21*a^5*b^2*x^3 + 35*a^3*b^4*x^4 + 35*a^4*b^3*x^(7/2) + 21*a ^2*b^5*x^(9/2)) - (1320*b^4*atanh((2*b*x^(1/2))/a + 1))/a^12
Time = 0.21 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.13 \[ \int \frac {1}{\left (a+b \sqrt {x}\right )^8 x^3} \, dx=\frac {-27720 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) b^{11} x^{5}+27720 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) b^{11} x^{5}-970200 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{3} b^{8} x^{4}-194040 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a \,b^{10} x^{5}+27720 \,\mathrm {log}\left (\sqrt {x}\right ) a^{7} b^{4} x^{2}+582120 \,\mathrm {log}\left (\sqrt {x}\right ) a^{5} b^{6} x^{3}+970200 \,\mathrm {log}\left (\sqrt {x}\right ) a^{3} b^{8} x^{4}+194040 \,\mathrm {log}\left (\sqrt {x}\right ) a \,b^{10} x^{5}+77 \sqrt {x}\, a^{10} b -3960 \sqrt {x}\, b^{11} x^{5}-385 a^{9} b^{2} x +67914 a^{7} b^{4} x^{2}+355740 a^{3} b^{8} x^{4}-194040 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{6} b^{5} x^{2}-970200 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{4} b^{7} x^{3}-582120 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{2} b^{9} x^{4}+194040 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{6} b^{5} x^{2}+970200 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{4} b^{7} x^{3}+582120 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) a^{2} b^{9} x^{4}+3465 \sqrt {x}\, a^{8} b^{3} x +281358 \sqrt {x}\, a^{6} b^{5} x^{2}+598290 \sqrt {x}\, a^{4} b^{7} x^{3}+97020 \sqrt {x}\, a^{2} b^{9} x^{4}-27720 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{7} b^{4} x^{2}-582120 \,\mathrm {log}\left (\sqrt {x}\, b +a \right ) a^{5} b^{6} x^{3}-21 a^{11}+553014 a^{5} b^{6} x^{3}}{42 a^{12} x^{2} \left (7 \sqrt {x}\, a^{6} b +35 \sqrt {x}\, a^{4} b^{3} x +21 \sqrt {x}\, a^{2} b^{5} x^{2}+\sqrt {x}\, b^{7} x^{3}+a^{7}+21 a^{5} b^{2} x +35 a^{3} b^{4} x^{2}+7 a \,b^{6} x^{3}\right )} \] Input:
int(1/(a+b*x^(1/2))^8/x^3,x)
Output:
( - 194040*sqrt(x)*log(sqrt(x)*b + a)*a**6*b**5*x**2 - 970200*sqrt(x)*log( sqrt(x)*b + a)*a**4*b**7*x**3 - 582120*sqrt(x)*log(sqrt(x)*b + a)*a**2*b** 9*x**4 - 27720*sqrt(x)*log(sqrt(x)*b + a)*b**11*x**5 + 194040*sqrt(x)*log( sqrt(x))*a**6*b**5*x**2 + 970200*sqrt(x)*log(sqrt(x))*a**4*b**7*x**3 + 582 120*sqrt(x)*log(sqrt(x))*a**2*b**9*x**4 + 27720*sqrt(x)*log(sqrt(x))*b**11 *x**5 + 77*sqrt(x)*a**10*b + 3465*sqrt(x)*a**8*b**3*x + 281358*sqrt(x)*a** 6*b**5*x**2 + 598290*sqrt(x)*a**4*b**7*x**3 + 97020*sqrt(x)*a**2*b**9*x**4 - 3960*sqrt(x)*b**11*x**5 - 27720*log(sqrt(x)*b + a)*a**7*b**4*x**2 - 582 120*log(sqrt(x)*b + a)*a**5*b**6*x**3 - 970200*log(sqrt(x)*b + a)*a**3*b** 8*x**4 - 194040*log(sqrt(x)*b + a)*a*b**10*x**5 + 27720*log(sqrt(x))*a**7* b**4*x**2 + 582120*log(sqrt(x))*a**5*b**6*x**3 + 970200*log(sqrt(x))*a**3* b**8*x**4 + 194040*log(sqrt(x))*a*b**10*x**5 - 21*a**11 - 385*a**9*b**2*x + 67914*a**7*b**4*x**2 + 553014*a**5*b**6*x**3 + 355740*a**3*b**8*x**4)/(4 2*a**12*x**2*(7*sqrt(x)*a**6*b + 35*sqrt(x)*a**4*b**3*x + 21*sqrt(x)*a**2* b**5*x**2 + sqrt(x)*b**7*x**3 + a**7 + 21*a**5*b**2*x + 35*a**3*b**4*x**2 + 7*a*b**6*x**3))