Integrand size = 17, antiderivative size = 505 \[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx=\frac {5}{4 (b c-a d) (c+d x)^{4/5}}+\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} b^{4/5} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{(b c-a d)^{9/5}}-\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} b^{4/5} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}+\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{(b c-a d)^{9/5}}+\frac {b^{4/5} \log \left (\sqrt [5]{b c-a d}-\sqrt [5]{b} \sqrt [5]{c+d x}\right )}{(b c-a d)^{9/5}}-\frac {\left (1-\sqrt {5}\right ) b^{4/5} \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}-\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{4 (b c-a d)^{9/5}}-\frac {\left (1+\sqrt {5}\right ) b^{4/5} \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{4 (b c-a d)^{9/5}} \] Output:
5/4/(-a*d+b*c)/(d*x+c)^(4/5)-1/2*(10+2*5^(1/2))^(1/2)*b^(4/5)*arctan(-1/5* (25-10*5^(1/2))^(1/2)+2*2^(1/2)/(5+5^(1/2))^(1/2)*b^(1/5)*(d*x+c)^(1/5)/(- a*d+b*c)^(1/5))/(-a*d+b*c)^(9/5)-1/2*(10-2*5^(1/2))^(1/2)*b^(4/5)*arctan(1 /5*(25+10*5^(1/2))^(1/2)+1/5*(50+10*5^(1/2))^(1/2)*b^(1/5)*(d*x+c)^(1/5)/( -a*d+b*c)^(1/5))/(-a*d+b*c)^(9/5)+b^(4/5)*ln((-a*d+b*c)^(1/5)-b^(1/5)*(d*x +c)^(1/5))/(-a*d+b*c)^(9/5)-1/4*(-5^(1/2)+1)*b^(4/5)*ln(2*(-a*d+b*c)^(2/5) +b^(1/5)*(-a*d+b*c)^(1/5)*(d*x+c)^(1/5)-5^(1/2)*b^(1/5)*(-a*d+b*c)^(1/5)*( d*x+c)^(1/5)+2*b^(2/5)*(d*x+c)^(2/5))/(-a*d+b*c)^(9/5)-1/4*(5^(1/2)+1)*b^( 4/5)*ln(2*(-a*d+b*c)^(2/5)+b^(1/5)*(-a*d+b*c)^(1/5)*(d*x+c)^(1/5)+5^(1/2)* b^(1/5)*(-a*d+b*c)^(1/5)*(d*x+c)^(1/5)+2*b^(2/5)*(d*x+c)^(2/5))/(-a*d+b*c) ^(9/5)
Time = 1.20 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx=\frac {1}{4} \left (\frac {5}{(b c-a d) (c+d x)^{4/5}}+\frac {2 \sqrt {10-2 \sqrt {5}} b^{4/5} \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \left (5+\sqrt {5}-\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )}{(-b c+a d)^{9/5}}-\frac {2 \sqrt {2 \left (5+\sqrt {5}\right )} b^{4/5} \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \left (5-\sqrt {5}+\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )}{(-b c+a d)^{9/5}}-\frac {4 b^{4/5} \log \left (\sqrt [5]{-b c+a d}+\sqrt [5]{b} \sqrt [5]{c+d x}\right )}{(-b c+a d)^{9/5}}-\frac {\left (-1+\sqrt {5}\right ) b^{4/5} \log \left (2 (-b c+a d)^{2/5}+\left (-1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{(-b c+a d)^{9/5}}+\frac {\left (1+\sqrt {5}\right ) b^{4/5} \log \left (2 (-b c+a d)^{2/5}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{(-b c+a d)^{9/5}}\right ) \] Input:
Integrate[1/((a + b*x)*(c + d*x)^(9/5)),x]
Output:
(5/((b*c - a*d)*(c + d*x)^(4/5)) + (2*Sqrt[10 - 2*Sqrt[5]]*b^(4/5)*ArcTan[ (Sqrt[(5 + Sqrt[5])/2]*(5 + Sqrt[5] - (4*Sqrt[5]*b^(1/5)*(c + d*x)^(1/5))/ (-(b*c) + a*d)^(1/5)))/10])/(-(b*c) + a*d)^(9/5) - (2*Sqrt[2*(5 + Sqrt[5]) ]*b^(4/5)*ArcTan[(Sqrt[(5 - Sqrt[5])/2]*(5 - Sqrt[5] + (4*Sqrt[5]*b^(1/5)* (c + d*x)^(1/5))/(-(b*c) + a*d)^(1/5)))/10])/(-(b*c) + a*d)^(9/5) - (4*b^( 4/5)*Log[(-(b*c) + a*d)^(1/5) + b^(1/5)*(c + d*x)^(1/5)])/(-(b*c) + a*d)^( 9/5) - ((-1 + Sqrt[5])*b^(4/5)*Log[2*(-(b*c) + a*d)^(2/5) + (-1 + Sqrt[5]) *b^(1/5)*(-(b*c) + a*d)^(1/5)*(c + d*x)^(1/5) + 2*b^(2/5)*(c + d*x)^(2/5)] )/(-(b*c) + a*d)^(9/5) + ((1 + Sqrt[5])*b^(4/5)*Log[2*(-(b*c) + a*d)^(2/5) - (1 + Sqrt[5])*b^(1/5)*(-(b*c) + a*d)^(1/5)*(c + d*x)^(1/5) + 2*b^(2/5)* (c + d*x)^(2/5)])/(-(b*c) + a*d)^(9/5))/4
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.10, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {78}
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.
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {5 \operatorname {Hypergeometric2F1}\left (-\frac {4}{5},1,\frac {1}{5},\frac {b (c+d x)}{b c-a d}\right )}{4 (c+d x)^{4/5} (b c-a d)}\) |
Input:
Int[1/((a + b*x)*(c + d*x)^(9/5)),x]
Output:
(5*Hypergeometric2F1[-4/5, 1, 1/5, (b*(c + d*x))/(b*c - a*d)])/(4*(b*c - a *d)*(c + d*x)^(4/5))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Time = 1.31 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (-\frac {\sqrt {5+\sqrt {5}}\, \left (x d +c \right )^{\frac {4}{5}} \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}+1\right ) \ln \left (-2 \left (\frac {a d -b c}{b}\right )^{\frac {2}{5}}+\left (\sqrt {5}+1\right ) \left (x d +c \right )^{\frac {1}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}-2 \left (x d +c \right )^{\frac {2}{5}}\right )}{4}+\frac {\sqrt {5+\sqrt {5}}\, \left (x d +c \right )^{\frac {4}{5}} \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}-1\right ) \ln \left (2 \left (\frac {a d -b c}{b}\right )^{\frac {2}{5}}+\left (\sqrt {5}-1\right ) \left (x d +c \right )^{\frac {1}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+2 \left (x d +c \right )^{\frac {2}{5}}\right )}{4}+\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \left (x d +c \right )^{\frac {4}{5}} \left (\sqrt {5}-5\right ) \arctan \left (\frac {\sqrt {2}\, \left (\left (\sqrt {5}+1\right ) \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}-4 \left (x d +c \right )^{\frac {1}{5}}\right )}{2 \sqrt {5-\sqrt {5}}\, \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )}{2}+\left (\frac {\sqrt {2}\, \left (x d +c \right )^{\frac {4}{5}} \left (5+\sqrt {5}\right ) \arctan \left (\frac {\left (\left (\sqrt {5}-1\right ) \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+4 \left (x d +c \right )^{\frac {1}{5}}\right ) \sqrt {2}}{2 \sqrt {5+\sqrt {5}}\, \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )}{2}+\sqrt {5+\sqrt {5}}\, \left (\left (x d +c \right )^{\frac {4}{5}} \ln \left (\left (x d +c \right )^{\frac {1}{5}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}\right )+\frac {5 \left (\frac {a d -b c}{b}\right )^{\frac {4}{5}}}{4}\right )\right ) \sqrt {5-\sqrt {5}}\right ) \sqrt {5}}{\left (x d +c \right )^{\frac {4}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {4}{5}} \left (40 a d -40 b c \right )}\) | \(431\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1354\) |
| default | \(\text {Expression too large to display}\) | \(1354\) |
Input:
int(1/(b*x+a)/(d*x+c)^(9/5),x,method=_RETURNVERBOSE)
Output:
-4/(d*x+c)^(4/5)*(-1/4*(5+5^(1/2))^(1/2)*(d*x+c)^(4/5)*(5-5^(1/2))^(1/2)*( 5^(1/2)+1)*ln(-2*((a*d-b*c)/b)^(2/5)+(5^(1/2)+1)*(d*x+c)^(1/5)*((a*d-b*c)/ b)^(1/5)-2*(d*x+c)^(2/5))+1/4*(5+5^(1/2))^(1/2)*(d*x+c)^(4/5)*(5-5^(1/2))^ (1/2)*(5^(1/2)-1)*ln(2*((a*d-b*c)/b)^(2/5)+(5^(1/2)-1)*(d*x+c)^(1/5)*((a*d -b*c)/b)^(1/5)+2*(d*x+c)^(2/5))+1/2*2^(1/2)*(5+5^(1/2))^(1/2)*(d*x+c)^(4/5 )*(5^(1/2)-5)*arctan(1/2/(5-5^(1/2))^(1/2)*2^(1/2)/((a*d-b*c)/b)^(1/5)*((5 ^(1/2)+1)*((a*d-b*c)/b)^(1/5)-4*(d*x+c)^(1/5)))+(1/2*2^(1/2)*(d*x+c)^(4/5) *(5+5^(1/2))*arctan(1/2*((5^(1/2)-1)*((a*d-b*c)/b)^(1/5)+4*(d*x+c)^(1/5))/ (5+5^(1/2))^(1/2)/((a*d-b*c)/b)^(1/5)*2^(1/2))+(5+5^(1/2))^(1/2)*((d*x+c)^ (4/5)*ln((d*x+c)^(1/5)+((a*d-b*c)/b)^(1/5))+5/4*((a*d-b*c)/b)^(4/5)))*(5-5 ^(1/2))^(1/2))*5^(1/2)/((a*d-b*c)/b)^(4/5)/(40*a*d-40*b*c)
Result contains complex when optimal does not.
Time = 5.65 (sec) , antiderivative size = 216715, normalized size of antiderivative = 429.14 \[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(9/5),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {9}{5}}}\, dx \] Input:
integrate(1/(b*x+a)/(d*x+c)**(9/5),x)
Output:
Integral(1/((a + b*x)*(c + d*x)**(9/5)), x)
Time = 0.15 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx =\text {Too large to display} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(9/5),x, algorithm="maxima")
Output:
1/4*(2*(sqrt(5)*d*(sqrt(5) - 1)*log(((-b*c + a*d)^(1/5)*b^(1/5)*(sqrt(5) + 1) + (-b*c + a*d)^(1/5)*b^(1/5)*sqrt(2*sqrt(5) - 10) - 4*(d*x + c)^(1/5)* b^(2/5))/((-b*c + a*d)^(1/5)*b^(1/5)*(sqrt(5) + 1) - (-b*c + a*d)^(1/5)*b^ (1/5)*sqrt(2*sqrt(5) - 10) - 4*(d*x + c)^(1/5)*b^(2/5)))/((-b*c + a*d)^(4/ 5)*b^(1/5)*sqrt(2*sqrt(5) - 10)) + sqrt(5)*d*(sqrt(5) + 1)*log(((-b*c + a* d)^(1/5)*b^(1/5)*(sqrt(5) - 1) - (-b*c + a*d)^(1/5)*b^(1/5)*sqrt(-2*sqrt(5 ) - 10) + 4*(d*x + c)^(1/5)*b^(2/5))/((-b*c + a*d)^(1/5)*b^(1/5)*(sqrt(5) - 1) + (-b*c + a*d)^(1/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10) + 4*(d*x + c)^(1/5 )*b^(2/5)))/((-b*c + a*d)^(4/5)*b^(1/5)*sqrt(-2*sqrt(5) - 10)) - d*(sqrt(5 ) + 3)*log(-(-b*c + a*d)^(1/5)*(d*x + c)^(1/5)*b^(1/5)*(sqrt(5) + 1) + 2*( d*x + c)^(2/5)*b^(2/5) + 2*(-b*c + a*d)^(2/5))/((-b*c + a*d)^(4/5)*b^(1/5) *(sqrt(5) + 1)) - d*(sqrt(5) - 3)*log((-b*c + a*d)^(1/5)*(d*x + c)^(1/5)*b ^(1/5)*(sqrt(5) - 1) + 2*(d*x + c)^(2/5)*b^(2/5) + 2*(-b*c + a*d)^(2/5))/( (-b*c + a*d)^(4/5)*b^(1/5)*(sqrt(5) - 1)) + 2*d*log((d*x + c)^(1/5)*b^(1/5 ) + (-b*c + a*d)^(1/5))/((-b*c + a*d)^(4/5)*b^(1/5)))*b/(b*c - a*d) + 5*d/ ((b*c - a*d)*(d*x + c)^(4/5)))/d
Time = 0.36 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx=\frac {b \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{5}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} \right |}\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {1}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {{\left (\sqrt {5} - 1\right )} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} - 4 \, {\left (d x + c\right )}^{\frac {1}{5}}}{\sqrt {2 \, \sqrt {5} + 10} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {1}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\sqrt {5} + 1\right )} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} + 4 \, {\left (d x + c\right )}^{\frac {1}{5}}}{\sqrt {-2 \, \sqrt {5} + 10} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {1}{5}} \log \left (\frac {1}{2} \, {\left (d x + c\right )}^{\frac {1}{5}} {\left (\sqrt {5} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}\right )} + {\left (d x + c\right )}^{\frac {2}{5}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{5}}\right )}{b^{2} c^{2} {\left (\sqrt {5} - 1\right )} - 2 \, a b c d {\left (\sqrt {5} - 1\right )} + a^{2} d^{2} {\left (\sqrt {5} - 1\right )}} + \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {1}{5}} \log \left (-\frac {1}{2} \, {\left (d x + c\right )}^{\frac {1}{5}} {\left (\sqrt {5} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}\right )} + {\left (d x + c\right )}^{\frac {2}{5}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{5}}\right )}{b^{2} c^{2} {\left (\sqrt {5} + 1\right )} - 2 \, a b c d {\left (\sqrt {5} + 1\right )} + a^{2} d^{2} {\left (\sqrt {5} + 1\right )}} + \frac {5}{4 \, {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {4}{5}}} \] Input:
integrate(1/(b*x+a)/(d*x+c)^(9/5),x, algorithm="giac")
Output:
b*((b*c - a*d)/b)^(1/5)*log(abs((d*x + c)^(1/5) - ((b*c - a*d)/b)^(1/5)))/ (b^2*c^2 - 2*a*b*c*d + a^2*d^2) - 1/2*(b^5*c - a*b^4*d)^(1/5)*sqrt(2*sqrt( 5) + 10)*arctan(-((sqrt(5) - 1)*((b*c - a*d)/b)^(1/5) - 4*(d*x + c)^(1/5)) /(sqrt(2*sqrt(5) + 10)*((b*c - a*d)/b)^(1/5)))/(b^2*c^2 - 2*a*b*c*d + a^2* d^2) - 1/2*(b^5*c - a*b^4*d)^(1/5)*sqrt(-2*sqrt(5) + 10)*arctan(((sqrt(5) + 1)*((b*c - a*d)/b)^(1/5) + 4*(d*x + c)^(1/5))/(sqrt(-2*sqrt(5) + 10)*((b *c - a*d)/b)^(1/5)))/(b^2*c^2 - 2*a*b*c*d + a^2*d^2) - (b^5*c - a*b^4*d)^( 1/5)*log(1/2*(d*x + c)^(1/5)*(sqrt(5)*((b*c - a*d)/b)^(1/5) + ((b*c - a*d) /b)^(1/5)) + (d*x + c)^(2/5) + ((b*c - a*d)/b)^(2/5))/(b^2*c^2*(sqrt(5) - 1) - 2*a*b*c*d*(sqrt(5) - 1) + a^2*d^2*(sqrt(5) - 1)) + (b^5*c - a*b^4*d)^ (1/5)*log(-1/2*(d*x + c)^(1/5)*(sqrt(5)*((b*c - a*d)/b)^(1/5) - ((b*c - a* d)/b)^(1/5)) + (d*x + c)^(2/5) + ((b*c - a*d)/b)^(2/5))/(b^2*c^2*(sqrt(5) + 1) - 2*a*b*c*d*(sqrt(5) + 1) + a^2*d^2*(sqrt(5) + 1)) + 5/4/((b*c - a*d) *(d*x + c)^(4/5))
Time = 0.13 (sec) , antiderivative size = 1059, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx =\text {Too large to display} \] Input:
int(1/((a + b*x)*(c + d*x)^(9/5)),x)
Output:
(b^(4/5)*log((c + d*x)^(1/5)*(625*b^11*c^4 + 625*a^4*b^7*d^4 - 2500*a^3*b^ 8*c*d^3 + 3750*a^2*b^9*c^2*d^2 - 2500*a*b^10*c^3*d) - (b^(4/5)*(625*b^12*c ^6 + 625*a^6*b^6*d^6 - 3750*a^5*b^7*c*d^5 + 9375*a^2*b^10*c^4*d^2 - 12500* a^3*b^9*c^3*d^3 + 9375*a^4*b^8*c^2*d^4 - 3750*a*b^11*c^5*d))/(b*c - a*d)^( 9/5)))/(b*c - a*d)^(9/5) - 5/(4*(a*d - b*c)*(c + d*x)^(4/5)) - (b^(4/5)*lo g((c + d*x)^(1/5)*(625*b^11*c^4 + 625*a^4*b^7*d^4 - 2500*a^3*b^8*c*d^3 + 3 750*a^2*b^9*c^2*d^2 - 2500*a*b^10*c^3*d) + (b^(4/5)*((- 2*5^(1/2) - 10)^(1 /2)/4 - 5^(1/2)/4 + 1/4)*(625*b^12*c^6 + 625*a^6*b^6*d^6 - 3750*a^5*b^7*c* d^5 + 9375*a^2*b^10*c^4*d^2 - 12500*a^3*b^9*c^3*d^3 + 9375*a^4*b^8*c^2*d^4 - 3750*a*b^11*c^5*d))/(b*c - a*d)^(9/5))*((- 2*5^(1/2) - 10)^(1/2)/4 - 5^ (1/2)/4 + 1/4))/(b*c - a*d)^(9/5) - (b^(4/5)*log((c + d*x)^(1/5)*(625*b^11 *c^4 + 625*a^4*b^7*d^4 - 2500*a^3*b^8*c*d^3 + 3750*a^2*b^9*c^2*d^2 - 2500* a*b^10*c^3*d) + (b^(4/5)*(5^(1/2)/4 - (2*5^(1/2) - 10)^(1/2)/4 + 1/4)*(625 *b^12*c^6 + 625*a^6*b^6*d^6 - 3750*a^5*b^7*c*d^5 + 9375*a^2*b^10*c^4*d^2 - 12500*a^3*b^9*c^3*d^3 + 9375*a^4*b^8*c^2*d^4 - 3750*a*b^11*c^5*d))/(b*c - a*d)^(9/5))*(5^(1/2)/4 - (2*5^(1/2) - 10)^(1/2)/4 + 1/4))/(b*c - a*d)^(9/ 5) - (b^(4/5)*log((c + d*x)^(1/5)*(625*b^11*c^4 + 625*a^4*b^7*d^4 - 2500*a ^3*b^8*c*d^3 + 3750*a^2*b^9*c^2*d^2 - 2500*a*b^10*c^3*d) + (b^(4/5)*(5^(1/ 2)/4 + (2*5^(1/2) - 10)^(1/2)/4 + 1/4)*(625*b^12*c^6 + 625*a^6*b^6*d^6 - 3 750*a^5*b^7*c*d^5 + 9375*a^2*b^10*c^4*d^2 - 12500*a^3*b^9*c^3*d^3 + 937...
\[ \int \frac {1}{(a+b x) (c+d x)^{9/5}} \, dx=\int \frac {\left (d x +c \right )^{\frac {1}{5}}}{b \,d^{2} x^{3}+a \,d^{2} x^{2}+2 b c d \,x^{2}+2 a c d x +b \,c^{2} x +a \,c^{2}}d x \] Input:
int(1/(b*x+a)/(d*x+c)^(9/5),x)
Output:
int((c + d*x)**(1/5)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b*c**2*x + 2*b*c* d*x**2 + b*d**2*x**3),x)