Integrand size = 17, antiderivative size = 547 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx=-\frac {9 d}{4 (b c-a d)^2 (c+d x)^{4/5}}-\frac {1}{(b c-a d) (a+b x) (c+d x)^{4/5}}-\frac {9 \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} b^{4/5} d \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{5 (b c-a d)^{14/5}}+\frac {9 \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} b^{4/5} d \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}+\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{5 (b c-a d)^{14/5}}-\frac {9 b^{4/5} d \log \left (\sqrt [5]{b c-a d}-\sqrt [5]{b} \sqrt [5]{c+d x}\right )}{5 (b c-a d)^{14/5}}+\frac {9 \left (1-\sqrt {5}\right ) b^{4/5} d \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}-\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{20 (b c-a d)^{14/5}}+\frac {9 \left (1+\sqrt {5}\right ) b^{4/5} d \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{20 (b c-a d)^{14/5}} \] Output:
-9/4*d/(-a*d+b*c)^2/(d*x+c)^(4/5)-1/(-a*d+b*c)/(b*x+a)/(d*x+c)^(4/5)+9/10* (10+2*5^(1/2))^(1/2)*b^(4/5)*d*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)^(14/ 5)+9/10*(10-2*5^(1/2))^(1/2)*b^(4/5)*d*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) ^(14/5)-9/5*b^(4/5)*d*ln((-a*d+b*c)^(1/5)-b^(1/5)*(d*x+c)^(1/5))/(-a*d+b*c )^(14/5)+9/20*(-5^(1/2)+1)*b^(4/5)*d*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)^(14/5)+9/20*(5^(1/2)+1)*b^(4/5)*d*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)^(14/5)
Time = 3.09 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx=\frac {1}{20} d \left (-\frac {5 (4 b c+5 a d+9 b d x)}{d (b c-a d)^2 (a+b x) (c+d x)^{4/5}}+\frac {18 \sqrt {10-2 \sqrt {5}} b^{4/5} \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \left (5+\sqrt {5}-\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )}{(-b c+a d)^{14/5}}-\frac {18 \sqrt {2 \left (5+\sqrt {5}\right )} b^{4/5} \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \left (5-\sqrt {5}+\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )}{(-b c+a d)^{14/5}}-\frac {36 b^{4/5} \log \left (\sqrt [5]{-b c+a d}+\sqrt [5]{b} \sqrt [5]{c+d x}\right )}{(-b c+a d)^{14/5}}-\frac {9 \left (-1+\sqrt {5}\right ) b^{4/5} \log \left (2 (-b c+a d)^{2/5}+\left (-1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{(-b c+a d)^{14/5}}+\frac {9 \left (1+\sqrt {5}\right ) b^{4/5} \log \left (2 (-b c+a d)^{2/5}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{(-b c+a d)^{14/5}}\right ) \] Input:
Integrate[1/((a + b*x)^2*(c + d*x)^(9/5)),x]
Output:
(d*((-5*(4*b*c + 5*a*d + 9*b*d*x))/(d*(b*c - a*d)^2*(a + b*x)*(c + d*x)^(4 /5)) + (18*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)^(14/5) - (18*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)^(14/5) - (36*b^(4/5)*Log[(-(b*c) + a*d) ^(1/5) + b^(1/5)*(c + d*x)^(1/5)])/(-(b*c) + a*d)^(14/5) - (9*(-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)^(14 /5) + (9*(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)^(14/5)))/20
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {52, 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.
| \(\displaystyle \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {9 d \int \frac {1}{(a+b x) (c+d x)^{9/5}}dx}{5 (b c-a d)}-\frac {1}{(a+b x) (c+d x)^{4/5} (b c-a d)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle -\frac {9 d \operatorname {Hypergeometric2F1}\left (-\frac {4}{5},1,\frac {1}{5},\frac {b (c+d x)}{b c-a d}\right )}{4 (c+d x)^{4/5} (b c-a d)^2}-\frac {1}{(a+b x) (c+d x)^{4/5} (b c-a d)}\) |
Input:
Int[1/((a + b*x)^2*(c + d*x)^(9/5)),x]
Output:
-(1/((b*c - a*d)*(a + b*x)*(c + d*x)^(4/5))) - (9*d*Hypergeometric2F1[-4/5 , 1, 1/5, (b*(c + d*x))/(b*c - a*d)])/(4*(b*c - a*d)^2*(c + d*x)^(4/5))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 1.86 (sec) , antiderivative size = 487, normalized size of antiderivative = 0.89
| method | result | size |
| pseudoelliptic | \(-\frac {9 \left (-\frac {\left (x d +c \right )^{\frac {4}{5}} \left (5+\sqrt {5}\right )^{\frac {3}{2}} \left (5-\sqrt {5}\right )^{\frac {3}{2}} d \left (\sqrt {5}+1\right ) \left (b x +a \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 {\left (x d +c \right )^{\frac {4}{5}} \left (5+\sqrt {5}\right )^{\frac {3}{2}} \left (5-\sqrt {5}\right )^{\frac {3}{2}} d \left (\sqrt {5}-1\right ) \left (b x +a \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}+5 \left (5-\sqrt {5}\right )^{\frac {3}{2}} \left (b x +a \right ) d \left (x d +c \right )^{\frac {4}{5}} \left (3+\sqrt {5}\right ) \sqrt {2}\, \arctan \left (\frac {5 \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 ) \left (3+\sqrt {5}\right ) \sqrt {2}}{\left (5+\sqrt {5}\right )^{\frac {5}{2}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )+\left (5+\sqrt {5}\right )^{\frac {3}{2}} \left (5 \left (b x +a \right ) d \left (x d +c \right )^{\frac {4}{5}} \left (\sqrt {5}-3\right ) \sqrt {2}\, \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 )+\left (\left (b x +a \right ) d \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 {25 \left (\left (\frac {9 b x}{5}+a \right ) d +\frac {4 b c}{5}\right ) \left (\frac {a d -b c}{b}\right )^{\frac {4}{5}}}{36}\right ) \left (5-\sqrt {5}\right )^{\frac {3}{2}}\right )\right ) \sqrt {5}}{1000 \left (x d +c \right )^{\frac {4}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {4}{5}} \left (b x +a \right ) \left (a d -b c \right )^{2}}\) | \(487\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1386\) |
| default | \(\text {Expression too large to display}\) | \(1386\) |
Input:
int(1/(b*x+a)^2/(d*x+c)^(9/5),x,method=_RETURNVERBOSE)
Output:
-9/1000/(d*x+c)^(4/5)*(-1/4*(d*x+c)^(4/5)*(5+5^(1/2))^(3/2)*(5-5^(1/2))^(3 /2)*d*(5^(1/2)+1)*(b*x+a)*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*(d*x+c)^(4/5)*(5+5^(1/2))^(3/ 2)*(5-5^(1/2))^(3/2)*d*(5^(1/2)-1)*(b*x+a)*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))+5*(5-5^(1/2))^(3/ 2)*(b*x+a)*d*(d*x+c)^(4/5)*(3+5^(1/2))*2^(1/2)*arctan(5*((5^(1/2)-1)*((a*d -b*c)/b)^(1/5)+4*(d*x+c)^(1/5))/(5+5^(1/2))^(5/2)/((a*d-b*c)/b)^(1/5)*(3+5 ^(1/2))*2^(1/2))+(5+5^(1/2))^(3/2)*(5*(b*x+a)*d*(d*x+c)^(4/5)*(5^(1/2)-3)* 2^(1/2)*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)))+((b*x+a)*d*(d*x+c)^(4/5)*ln((d*x +c)^(1/5)+((a*d-b*c)/b)^(1/5))+25/36*((9/5*b*x+a)*d+4/5*b*c)*((a*d-b*c)/b) ^(4/5))*(5-5^(1/2))^(3/2)))*5^(1/2)/((a*d-b*c)/b)^(4/5)/(b*x+a)/(a*d-b*c)^ 2
Timed out. \[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^(9/5),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \left (c + d x\right )^{\frac {9}{5}}}\, dx \] Input:
integrate(1/(b*x+a)**2/(d*x+c)**(9/5),x)
Output:
Integral(1/((a + b*x)**2*(c + d*x)**(9/5)), x)
Time = 0.14 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx =\text {Too large to display} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^(9/5),x, algorithm="maxima")
Output:
-1/20*(18*(sqrt(5)*d^2*(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^2*(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)*(s qrt(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^ 2*(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^2*(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^2*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^2* c^2 - 2*a*b*c*d + a^2*d^2) + 5*(9*(d*x + c)*b*d^2 - 5*b*c*d^2 + 5*a*d^3)/( (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*(d*x + c)^(9/5) - (b^3*c^3 - 3*a*b^2*c ^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x + c)^(4/5)))/d
Time = 0.45 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx =\text {Too large to display} \] Input:
integrate(1/(b*x+a)^2/(d*x+c)^(9/5),x, algorithm="giac")
Output:
-9/5*b*d*((b*c - a*d)/b)^(1/5)*log(abs((d*x + c)^(1/5) - ((b*c - a*d)/b)^( 1/5)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 9/10*(b^5*c - a*b^4*d)^(1/5)*d*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^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 9/10*(b^5*c - a*b ^4*d)^(1/5)*d*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^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 9/5*(b^5*c - a*b^4*d )^(1/5)*d*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^3*c^3*(sqrt( 5) - 1) - 3*a*b^2*c^2*d*(sqrt(5) - 1) + 3*a^2*b*c*d^2*(sqrt(5) - 1) - a^3* d^3*(sqrt(5) - 1)) - 9/5*(b^5*c - a*b^4*d)^(1/5)*d*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^3*c^3*(sqrt(5) + 1) - 3*a*b^2*c^2*d*(sqrt(5) + 1) + 3*a^2*b*c*d^2*(sqrt(5) + 1) - a^3*d^3*(sqrt(5) + 1)) - (d*x + c)^( 1/5)*b*d/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*((d*x + c)*b - b*c + a*d)) - 5/4 *d/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(d*x + c)^(4/5))
Time = 0.37 (sec) , antiderivative size = 1799, normalized size of antiderivative = 3.29 \[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx=\text {Too large to display} \] Input:
int(1/((a + b*x)^2*(c + d*x)^(9/5)),x)
Output:
(9*b^(4/5)*d*log((c + d*x)^(1/5)*(3645*a^8*b^7*d^11 + 3645*b^15*c^8*d^3 - 29160*a*b^14*c^7*d^4 - 29160*a^7*b^8*c*d^10 + 102060*a^2*b^13*c^6*d^5 - 20 4120*a^3*b^12*c^5*d^6 + 255150*a^4*b^11*c^4*d^7 - 204120*a^5*b^10*c^3*d^8 + 102060*a^6*b^9*c^2*d^9) - (9*b^(4/5)*d*((- 2*5^(1/2) - 10)^(1/2)/4 - 5^( 1/2)/4 + 1/4)*(2025*a^11*b^6*d^13 - 2025*b^17*c^11*d^2 + 22275*a*b^16*c^10 *d^3 - 22275*a^10*b^7*c*d^12 - 111375*a^2*b^15*c^9*d^4 + 334125*a^3*b^14*c ^8*d^5 - 668250*a^4*b^13*c^7*d^6 + 935550*a^5*b^12*c^6*d^7 - 935550*a^6*b^ 11*c^5*d^8 + 668250*a^7*b^10*c^4*d^9 - 334125*a^8*b^9*c^3*d^10 + 111375*a^ 9*b^8*c^2*d^11))/(5*(a*d - b*c)^(14/5)))*((- 2*5^(1/2) - 10)^(1/2)/4 - 5^( 1/2)/4 + 1/4))/(5*(a*d - b*c)^(14/5)) - (9*b^(4/5)*d*log((c + d*x)^(1/5)*( 3645*a^8*b^7*d^11 + 3645*b^15*c^8*d^3 - 29160*a*b^14*c^7*d^4 - 29160*a^7*b ^8*c*d^10 + 102060*a^2*b^13*c^6*d^5 - 204120*a^3*b^12*c^5*d^6 + 255150*a^4 *b^11*c^4*d^7 - 204120*a^5*b^10*c^3*d^8 + 102060*a^6*b^9*c^2*d^9) + (9*b^( 4/5)*d*(2025*a^11*b^6*d^13 - 2025*b^17*c^11*d^2 + 22275*a*b^16*c^10*d^3 - 22275*a^10*b^7*c*d^12 - 111375*a^2*b^15*c^9*d^4 + 334125*a^3*b^14*c^8*d^5 - 668250*a^4*b^13*c^7*d^6 + 935550*a^5*b^12*c^6*d^7 - 935550*a^6*b^11*c^5* d^8 + 668250*a^7*b^10*c^4*d^9 - 334125*a^8*b^9*c^3*d^10 + 111375*a^9*b^8*c ^2*d^11))/(5*(a*d - b*c)^(14/5))))/(5*(a*d - b*c)^(14/5)) - ((5*d)/(4*(a*d - b*c)) + (9*b*d*(c + d*x))/(4*(a*d - b*c)^2))/(b*(c + d*x)^(9/5) + (a*d - b*c)*(c + d*x)^(4/5)) - (9*b^(4/5)*d*log((c + d*x)^(1/5)*(3645*a^8*b^...
\[ \int \frac {1}{(a+b x)^2 (c+d x)^{9/5}} \, dx=\int \frac {\left (d x +c \right )^{\frac {1}{5}}}{b^{2} d^{2} x^{4}+2 a b \,d^{2} x^{3}+2 b^{2} c d \,x^{3}+a^{2} d^{2} x^{2}+4 a b c d \,x^{2}+b^{2} c^{2} x^{2}+2 a^{2} c d x +2 a b \,c^{2} x +a^{2} c^{2}}d x \] Input:
int(1/(b*x+a)^2/(d*x+c)^(9/5),x)
Output:
int((c + d*x)**(1/5)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 + 2*a*b*c* *2*x + 4*a*b*c*d*x**2 + 2*a*b*d**2*x**3 + b**2*c**2*x**2 + 2*b**2*c*d*x**3 + b**2*d**2*x**4),x)