Integrand size = 17, antiderivative size = 512 \[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=-\frac {(c+d x)^{4/5}}{b (a+b x)}-\frac {2 \sqrt {2 \left (5+\sqrt {5}\right )} d \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{5 b^{9/5} \sqrt [5]{b c-a d}}+\frac {2 \sqrt {2 \left (5-\sqrt {5}\right )} d \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}+\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{5 b^{9/5} \sqrt [5]{b c-a d}}+\frac {4 d \log \left (\sqrt [5]{b c-a d}-\sqrt [5]{b} \sqrt [5]{c+d x}\right )}{5 b^{9/5} \sqrt [5]{b c-a d}}-\frac {\left (1-\sqrt {5}\right ) d \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}-\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{5 b^{9/5} \sqrt [5]{b c-a d}}-\frac {\left (1+\sqrt {5}\right ) d \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{5 b^{9/5} \sqrt [5]{b c-a d}} \] Output:
-(d*x+c)^(4/5)/b/(b*x+a)+2/5*(10+2*5^(1/2))^(1/2)*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))/b^(9/5)/(-a*d+b*c)^(1/5)+2/5*(10-2*5^(1/2))^(1/2)*d*arctan(1/5*(25+1 0*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))/b^(9/5)/(-a*d+b*c)^(1/5)+4/5*d*ln((-a*d+b*c)^(1/5)-b^(1/5)*(d*x+c )^(1/5))/b^(9/5)/(-a*d+b*c)^(1/5)-1/5*(-5^(1/2)+1)*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))/b^(9/5)/(-a*d+b*c)^(1/5)-1/5*(5^(1/2 )+1)*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))/b^(9/5)/ (-a*d+b*c)^(1/5)
Time = 1.60 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.82 \[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=\frac {-5 b^{4/5} \sqrt [5]{-b c+a d} (c+d x)^{4/5}-2 \sqrt {10-2 \sqrt {5}} d (a+b x) \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \left (5+\sqrt {5}-\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} d (a+b x) \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \left (5-\sqrt {5}+\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )-4 d (a+b x) \log \left (\sqrt [5]{-b c+a d}+\sqrt [5]{b} \sqrt [5]{c+d x}\right )-\left (-1+\sqrt {5}\right ) d (a+b x) \log \left (2 (-b c+a d)^{2/5}+\left (-1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )+\left (1+\sqrt {5}\right ) d (a+b x) \log \left (2 (-b c+a d)^{2/5}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{5 b^{9/5} \sqrt [5]{-b c+a d} (a+b x)} \] Input:
Integrate[(c + d*x)^(4/5)/(a + b*x)^2,x]
Output:
(-5*b^(4/5)*(-(b*c) + a*d)^(1/5)*(c + d*x)^(4/5) - 2*Sqrt[10 - 2*Sqrt[5]]* d*(a + b*x)*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] + 2*Sqrt[2*(5 + Sqrt[5])]*d* (a + b*x)*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] - 4*d*(a + b*x)*Log[(-(b*c) + a*d)^(1/5) + b^(1/5)*(c + d*x)^(1/5)] - (-1 + Sqrt[5])*d*(a + b*x)*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)] + (1 + Sqrt[5])*d*(a + b*x)*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)])/(5*b^(9/5)*(-(b*c) + a*d)^(1/5)*(a + b* x))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.17 (sec) , antiderivative size = 72, 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 = {51, 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.
| \(\displaystyle \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {4 d \int \frac {1}{(a+b x) \sqrt [5]{c+d x}}dx}{5 b}-\frac {(c+d x)^{4/5}}{b (a+b x)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle -\frac {d (c+d x)^{4/5} \operatorname {Hypergeometric2F1}\left (\frac {4}{5},1,\frac {9}{5},\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac {(c+d x)^{4/5}}{b (a+b x)}\) |
Input:
Int[(c + d*x)^(4/5)/(a + b*x)^2,x]
Output:
-((c + d*x)^(4/5)/(b*(a + b*x))) - (d*(c + d*x)^(4/5)*Hypergeometric2F1[4/ 5, 1, 9/5, (b*(c + d*x))/(b*c - a*d)])/(b*(b*c - a*d))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[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 = 1.41 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(-\frac {2 \sqrt {5}\, \left (-\frac {d \sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}+1\right ) \left (b x +a \right ) \ln \left (2 \left (\frac {a d -b c}{b}\right )^{\frac {2}{5}}+\left (-\sqrt {5}-1\right ) \left (x d +c \right )^{\frac {1}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+2 \left (x d +c \right )^{\frac {2}{5}}\right )}{4}+\frac {d \sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}-1\right ) \left (b x +a \right ) \ln \left (2 \left (\frac {a d -b c}{b}\right )^{\frac {2}{5}}+\left (\sqrt {5}-1\right ) \left (x d +c \right )^{\frac {1}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+2 \left (x d +c \right )^{\frac {2}{5}}\right )}{4}-\frac {\sqrt {2}\, d \sqrt {5+\sqrt {5}}\, \left (\sqrt {5}-5\right ) \left (b x +a \right ) \arctan \left (\frac {\sqrt {2}\, \left (\left (\sqrt {5}+1\right ) \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}-4 \left (x d +c \right )^{\frac {1}{5}}\right )}{2 \sqrt {5-\sqrt {5}}\, \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )}{2}+\left (-\frac {\sqrt {2}\, d \left (5+\sqrt {5}\right ) \left (b x +a \right ) \arctan \left (\frac {\left (\left (\sqrt {5}-1\right ) \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+4 \left (x d +c \right )^{\frac {1}{5}}\right ) \sqrt {2}}{2 \sqrt {5+\sqrt {5}}\, \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )}{2}+\left (d \left (b x +a \right ) \ln \left (\left (x d +c \right )^{\frac {1}{5}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}\right )+\frac {5 \left (x d +c \right )^{\frac {4}{5}} b \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}{4}\right ) \sqrt {5+\sqrt {5}}\right ) \sqrt {5-\sqrt {5}}\right )}{25 \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}} \left (b x +a \right ) b^{2}}\) | \(428\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1313\) |
| default | \(\text {Expression too large to display}\) | \(1313\) |
Input:
int((d*x+c)^(4/5)/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-2/25/((a*d-b*c)/b)^(1/5)*5^(1/2)*(-1/4*d*(5+5^(1/2))^(1/2)*(5-5^(1/2))^(1 /2)*(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*(5+5^(1/2))^(1/2)*(5-5^(1/2)) ^(1/2)*(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))-1/2*2^(1/2)*d*(5+5^(1/2))^(1/2)*( 5^(1/2)-5)*(b*x+a)*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*(5+5^ (1/2))*(b*x+a)*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))+(d*(b*x+a)*ln((d*x+c)^(1/ 5)+((a*d-b*c)/b)^(1/5))+5/4*(d*x+c)^(4/5)*b*((a*d-b*c)/b)^(1/5))*(5+5^(1/2 ))^(1/2))*(5-5^(1/2))^(1/2))/(b*x+a)/b^2
Timed out. \[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)^(4/5)/(b*x+a)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{\frac {4}{5}}}{\left (a + b x\right )^{2}}\, dx \] Input:
integrate((d*x+c)**(4/5)/(b*x+a)**2,x)
Output:
Integral((c + d*x)**(4/5)/(a + b*x)**2, x)
Time = 0.17 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(4/5)/(b*x+a)^2,x, algorithm="maxima")
Output:
1/5*(2*d^2*(sqrt(5)*(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)^( 1/5)*b^(4/5)*sqrt(2*sqrt(5) - 10)) + sqrt(5)*(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)^(1/5)*b^(4/5)*sqrt(-2*sqrt(5) - 10)) + (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)^(1/5)*b^(4/5)*( sqrt(5) + 1)) + (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)^(1/5)*b^(4/5)*(sqrt(5) - 1)) - 2*log((d*x + c)^(1/5)*b^(1/5) + (- b*c + a*d)^(1/5))/((-b*c + a*d)^(1/5)*b^(4/5)))/b - 5*(d*x + c)^(4/5)*d^2/ ((d*x + c)*b^2 - b^2*c + a*b*d))/d
Time = 0.40 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=-\frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} d {\left (\sqrt {5} + 1\right )} \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 )}{5 \, {\left (b^{6} c - a b^{5} d\right )}} + \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} d {\left (\sqrt {5} - 1\right )} \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 )}{5 \, {\left (b^{6} c - a b^{5} d\right )}} + \frac {4 \, {\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} d \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 )}{5 \, {\left (b^{6} c {\left (\sqrt {5} - 1\right )} - a b^{5} d {\left (\sqrt {5} - 1\right )}\right )}} + \frac {4 \, {\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} d \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 )}{5 \, {\left (b^{6} c {\left (\sqrt {5} + 1\right )} - a b^{5} d {\left (\sqrt {5} + 1\right )}\right )}} + \frac {4 \, d \left (\frac {b c - a d}{b}\right )^{\frac {4}{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 )}{5 \, {\left (b^{2} c - a b d\right )}} - \frac {{\left (d x + c\right )}^{\frac {4}{5}} d}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b} \] Input:
integrate((d*x+c)^(4/5)/(b*x+a)^2,x, algorithm="giac")
Output:
-1/5*(b^5*c - a*b^4*d)^(4/5)*d*(sqrt(5) + 1)*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^6*c - a*b^5*d) + 1/5*(b^5*c - a*b^4*d)^(4/5)*d*(sqr t(5) - 1)*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^6*c - a*b^5 *d) + 4/5*(b^5*c - a*b^4*d)^(4/5)*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^6*c*(sqrt(5) - 1) - a*b^5*d*(sqrt(5) - 1)) + 4/5* (b^5*c - a*b^4*d)^(4/5)*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^6*c*(sqrt(5) + 1) - a*b^5*d*(sqrt(5) + 1)) + 4/5*d*((b*c - a *d)/b)^(4/5)*log(abs((d*x + c)^(1/5) - ((b*c - a*d)/b)^(1/5)))/(b^2*c - a* b*d) - (d*x + c)^(4/5)*d/(((d*x + c)*b - b*c + a*d)*b)
Time = 0.59 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.89 \[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=\frac {\ln \left (\frac {320\,d^3\,{\left (a\,d-b\,c\right )}^2}{b}-\frac {80\,d^3\,{\left (a\,d-b\,c\right )}^{9/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}-\sqrt {5}+1\right )}{b^{4/5}}\right )\,\left (d-\sqrt {5}\,d+d\,\sqrt {-2\,\sqrt {5}-10}\right )}{5\,b^{9/5}\,{\left (a\,d-b\,c\right )}^{1/5}}-\frac {\ln \left (\frac {320\,d^3\,{\left (a\,d-b\,c\right )}^2}{b}+\frac {80\,d^3\,{\left (a\,d-b\,c\right )}^{9/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\sqrt {2}\,\sqrt {-\sqrt {5}-5}+\sqrt {5}-1\right )}{b^{4/5}}\right )\,\left (\sqrt {5}\,d-d+d\,\sqrt {-2\,\sqrt {5}-10}\right )}{5\,b^{9/5}\,{\left (a\,d-b\,c\right )}^{1/5}}-\frac {d\,{\left (c+d\,x\right )}^{4/5}}{b\,\left (a\,d-b\,c+b\,\left (c+d\,x\right )\right )}-\frac {4\,d\,\ln \left ({\left (a\,d-b\,c\right )}^{1/5}+b^{1/5}\,{\left (c+d\,x\right )}^{1/5}\right )}{5\,b^{9/5}\,{\left (a\,d-b\,c\right )}^{1/5}}+\frac {\ln \left (\frac {320\,d^3\,{\left (a\,d-b\,c\right )}^2}{b}-\frac {80\,d^3\,{\left (a\,d-b\,c\right )}^{9/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\sqrt {5}+\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}{b^{4/5}}\right )\,\left (d+\sqrt {5}\,d+d\,\sqrt {2\,\sqrt {5}-10}\right )}{5\,b^{9/5}\,{\left (a\,d-b\,c\right )}^{1/5}}+\frac {\ln \left (\frac {320\,d^3\,{\left (a\,d-b\,c\right )}^2}{b}-\frac {80\,d^3\,{\left (a\,d-b\,c\right )}^{9/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\sqrt {5}-\sqrt {2}\,\sqrt {\sqrt {5}-5}+1\right )}{b^{4/5}}\right )\,\left (d+\sqrt {5}\,d-d\,\sqrt {2\,\sqrt {5}-10}\right )}{5\,b^{9/5}\,{\left (a\,d-b\,c\right )}^{1/5}} \] Input:
int((c + d*x)^(4/5)/(a + b*x)^2,x)
Output:
(log((320*d^3*(a*d - b*c)^2)/b - (80*d^3*(a*d - b*c)^(9/5)*(c + d*x)^(1/5) *(2^(1/2)*(- 5^(1/2) - 5)^(1/2) - 5^(1/2) + 1))/b^(4/5))*(d - 5^(1/2)*d + d*(- 2*5^(1/2) - 10)^(1/2)))/(5*b^(9/5)*(a*d - b*c)^(1/5)) - (log((320*d^3 *(a*d - b*c)^2)/b + (80*d^3*(a*d - b*c)^(9/5)*(c + d*x)^(1/5)*(2^(1/2)*(- 5^(1/2) - 5)^(1/2) + 5^(1/2) - 1))/b^(4/5))*(5^(1/2)*d - d + d*(- 2*5^(1/2 ) - 10)^(1/2)))/(5*b^(9/5)*(a*d - b*c)^(1/5)) - (d*(c + d*x)^(4/5))/(b*(a* d - b*c + b*(c + d*x))) - (4*d*log((a*d - b*c)^(1/5) + b^(1/5)*(c + d*x)^( 1/5)))/(5*b^(9/5)*(a*d - b*c)^(1/5)) + (log((320*d^3*(a*d - b*c)^2)/b - (8 0*d^3*(a*d - b*c)^(9/5)*(c + d*x)^(1/5)*(5^(1/2) + 2^(1/2)*(5^(1/2) - 5)^( 1/2) + 1))/b^(4/5))*(d + 5^(1/2)*d + d*(2*5^(1/2) - 10)^(1/2)))/(5*b^(9/5) *(a*d - b*c)^(1/5)) + (log((320*d^3*(a*d - b*c)^2)/b - (80*d^3*(a*d - b*c) ^(9/5)*(c + d*x)^(1/5)*(5^(1/2) - 2^(1/2)*(5^(1/2) - 5)^(1/2) + 1))/b^(4/5 ))*(d + 5^(1/2)*d - d*(2*5^(1/2) - 10)^(1/2)))/(5*b^(9/5)*(a*d - b*c)^(1/5 ))
\[ \int \frac {(c+d x)^{4/5}}{(a+b x)^2} \, dx=\text {too large to display} \] Input:
int((d*x+c)^(4/5)/(b*x+a)^2,x)
Output:
(4*(c + d*x)**(1/5)*int(x/((c + d*x)**(1/5)*a**4*c*d**2 + (c + d*x)**(1/5) *a**4*d**3*x + 22*(c + d*x)**(1/5)*a**3*b*c**2*d + 24*(c + d*x)**(1/5)*a** 3*b*c*d**2*x + 2*(c + d*x)**(1/5)*a**3*b*d**3*x**2 + 121*(c + d*x)**(1/5)* a**2*b**2*c**3 + 165*(c + d*x)**(1/5)*a**2*b**2*c**2*d*x + 45*(c + d*x)**( 1/5)*a**2*b**2*c*d**2*x**2 + (c + d*x)**(1/5)*a**2*b**2*d**3*x**3 + 242*(c + d*x)**(1/5)*a*b**3*c**3*x + 264*(c + d*x)**(1/5)*a*b**3*c**2*d*x**2 + 2 2*(c + d*x)**(1/5)*a*b**3*c*d**2*x**3 + 121*(c + d*x)**(1/5)*b**4*c**3*x** 2 + 121*(c + d*x)**(1/5)*b**4*c**2*d*x**3),x)*a**6*d**6 + 100*(c + d*x)**( 1/5)*int(x/((c + d*x)**(1/5)*a**4*c*d**2 + (c + d*x)**(1/5)*a**4*d**3*x + 22*(c + d*x)**(1/5)*a**3*b*c**2*d + 24*(c + d*x)**(1/5)*a**3*b*c*d**2*x + 2*(c + d*x)**(1/5)*a**3*b*d**3*x**2 + 121*(c + d*x)**(1/5)*a**2*b**2*c**3 + 165*(c + d*x)**(1/5)*a**2*b**2*c**2*d*x + 45*(c + d*x)**(1/5)*a**2*b**2* c*d**2*x**2 + (c + d*x)**(1/5)*a**2*b**2*d**3*x**3 + 242*(c + d*x)**(1/5)* a*b**3*c**3*x + 264*(c + d*x)**(1/5)*a*b**3*c**2*d*x**2 + 22*(c + d*x)**(1 /5)*a*b**3*c*d**2*x**3 + 121*(c + d*x)**(1/5)*b**4*c**3*x**2 + 121*(c + d* x)**(1/5)*b**4*c**2*d*x**3),x)*a**5*b*c*d**5 + 4*(c + d*x)**(1/5)*int(x/(( c + d*x)**(1/5)*a**4*c*d**2 + (c + d*x)**(1/5)*a**4*d**3*x + 22*(c + d*x)* *(1/5)*a**3*b*c**2*d + 24*(c + d*x)**(1/5)*a**3*b*c*d**2*x + 2*(c + d*x)** (1/5)*a**3*b*d**3*x**2 + 121*(c + d*x)**(1/5)*a**2*b**2*c**3 + 165*(c + d* x)**(1/5)*a**2*b**2*c**2*d*x + 45*(c + d*x)**(1/5)*a**2*b**2*c*d**2*x**...