Integrand size = 17, antiderivative size = 498 \[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\frac {5 (c+d x)^{4/5}}{4 b}-\frac {\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} (b c-a d)^{4/5} \arctan \left (\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}-\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{b^{9/5}}+\frac {\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} (b c-a d)^{4/5} \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}+\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{b c-a d}}\right )}{b^{9/5}}+\frac {(b c-a d)^{4/5} \log \left (\sqrt [5]{b c-a d}-\sqrt [5]{b} \sqrt [5]{c+d x}\right )}{b^{9/5}}-\frac {\left (1-\sqrt {5}\right ) (b c-a d)^{4/5} \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}-\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{4 b^{9/5}}-\frac {\left (1+\sqrt {5}\right ) (b c-a d)^{4/5} \log \left (2 (b c-a d)^{2/5}+\sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+\sqrt {5} \sqrt [5]{b} \sqrt [5]{b c-a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{4 b^{9/5}} \] Output:
5/4*(d*x+c)^(4/5)/b+1/2*(10+2*5^(1/2))^(1/2)*(-a*d+b*c)^(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))/b^(9/5)+1/2*(10-2*5^(1/2))^(1/2)*(-a*d+b*c)^(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))/b^(9/5)+(-a*d+b*c)^(4/5)*ln((-a*d+b*c)^(1/5)-b^(1/5)*(d*x +c)^(1/5))/b^(9/5)-1/4*(-5^(1/2)+1)*(-a*d+b*c)^(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))/b^(9/5)-1/4*(5^(1/2)+1)*(-a*d+b*c)^( 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))/b^(9/5)
Time = 1.14 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\frac {5 b^{4/5} (c+d x)^{4/5}+2 \sqrt {10-2 \sqrt {5}} (-b c+a d)^{4/5} \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \left (5+\sqrt {5}-\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )-2 \sqrt {2 \left (5+\sqrt {5}\right )} (-b c+a d)^{4/5} \arctan \left (\frac {1}{10} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \left (5-\sqrt {5}+\frac {4 \sqrt {5} \sqrt [5]{b} \sqrt [5]{c+d x}}{\sqrt [5]{-b c+a d}}\right )\right )+4 (-b c+a d)^{4/5} \log \left (\sqrt [5]{-b c+a d}+\sqrt [5]{b} \sqrt [5]{c+d x}\right )+\left (-1+\sqrt {5}\right ) (-b c+a d)^{4/5} \log \left (2 (-b c+a d)^{2/5}+\left (-1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )-\left (1+\sqrt {5}\right ) (-b c+a d)^{4/5} \log \left (2 (-b c+a d)^{2/5}-\left (1+\sqrt {5}\right ) \sqrt [5]{b} \sqrt [5]{-b c+a d} \sqrt [5]{c+d x}+2 b^{2/5} (c+d x)^{2/5}\right )}{4 b^{9/5}} \] Input:
Integrate[(c + d*x)^(4/5)/(a + b*x),x]
Output:
(5*b^(4/5)*(c + d*x)^(4/5) + 2*Sqrt[10 - 2*Sqrt[5]]*(-(b*c) + a*d)^(4/5)*A rcTan[(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])]*(-(b*c) + a*d)^ (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] + 4*(-(b*c) + a*d)^(4/5)*Log[(-(b* c) + a*d)^(1/5) + b^(1/5)*(c + d*x)^(1/5)] + (-1 + Sqrt[5])*(-(b*c) + a*d) ^(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)] - (1 + Sqrt[5])*(-(b*c) + a*d)^(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)])/(4*b^(9/5))
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.
| \(\displaystyle \int \frac {(c+d x)^{4/5}}{a+b x} \, dx\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle -\frac {5 (c+d x)^{9/5} \operatorname {Hypergeometric2F1}\left (1,\frac {9}{5},\frac {14}{5},\frac {b (c+d x)}{b c-a d}\right )}{9 (b c-a d)}\) |
Input:
Int[(c + d*x)^(4/5)/(a + b*x),x]
Output:
(-5*(c + d*x)^(9/5)*Hypergeometric2F1[1, 9/5, 14/5, (b*(c + d*x))/(b*c - a *d)])/(9*(b*c - a*d))
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 = 3.19 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {\left (-\frac {\sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}+1\right ) \left (a d -b c \right ) \ln \left (-2 \left (\frac {a d -b c}{b}\right )^{\frac {2}{5}}+\left (\sqrt {5}+1\right ) \left (x d +c \right )^{\frac {1}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}-2 \left (x d +c \right )^{\frac {2}{5}}\right )}{4}+\frac {\sqrt {5+\sqrt {5}}\, \sqrt {5-\sqrt {5}}\, \left (\sqrt {5}-1\right ) \left (a d -b c \right ) \ln \left (2 \left (\frac {a d -b c}{b}\right )^{\frac {2}{5}}+\left (\sqrt {5}-1\right ) \left (x d +c \right )^{\frac {1}{5}} \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+2 \left (x d +c \right )^{\frac {2}{5}}\right )}{4}-\frac {\sqrt {2}\, \sqrt {5+\sqrt {5}}\, \left (\sqrt {5}-5\right ) \left (a d -b c \right ) \arctan \left (\frac {\sqrt {2}\, \left (\left (\sqrt {5}+1\right ) \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}-4 \left (x d +c \right )^{\frac {1}{5}}\right )}{2 \sqrt {5-\sqrt {5}}\, \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )}{2}+\left (-\frac {\sqrt {2}\, \left (5+\sqrt {5}\right ) \left (a d -b c \right ) \arctan \left (\frac {\left (\left (\sqrt {5}-1\right ) \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}+4 \left (x d +c \right )^{\frac {1}{5}}\right ) \sqrt {2}}{2 \sqrt {5+\sqrt {5}}\, \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}\right )}{2}+\sqrt {5+\sqrt {5}}\, \left (\left (a d -b c \right ) \ln \left (\left (x d +c \right )^{\frac {1}{5}}+\left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}\right )+\frac {5 \left (x d +c \right )^{\frac {4}{5}} b \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}}}{4}\right )\right ) \sqrt {5-\sqrt {5}}\right ) \sqrt {5}}{10 \left (\frac {a d -b c}{b}\right )^{\frac {1}{5}} b^{2}}\) | \(429\) |
| risch | \(\text {Expression too large to display}\) | \(1298\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1301\) |
| default | \(\text {Expression too large to display}\) | \(1301\) |
Input:
int((d*x+c)^(4/5)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/10/((a*d-b*c)/b)^(1/5)*(-1/4*(5+5^(1/2))^(1/2)*(5-5^(1/2))^(1/2)*(5^(1/2 )+1)*(a*d-b*c)*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)*(5-5^(1/2))^(1/2)*(5^( 1/2)-1)*(a*d-b*c)*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)*(5^(1/2)-5)* (a*d-b*c)*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)*(5+5^(1/2))*(a*d -b*c)*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)*((a*d-b*c)*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^(1/2)/b^2
Result contains complex when optimal does not.
Time = 2.53 (sec) , antiderivative size = 93460, normalized size of antiderivative = 187.67 \[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(4/5)/(b*x+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\int \frac {\left (c + d x\right )^{\frac {4}{5}}}{a + b x}\, dx \] Input:
integrate((d*x+c)**(4/5)/(b*x+a),x)
Output:
Integral((c + d*x)**(4/5)/(a + b*x), x)
Time = 0.13 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(4/5)/(b*x+a),x, algorithm="maxima")
Output:
1/4*(2*(b*c*d - a*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)*sq rt(-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/b)/d
Time = 0.33 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\frac {{\left (b^{4} c \left (\frac {b c - a d}{b}\right )^{\frac {3}{5}} - a b^{3} d \left (\frac {b c - a d}{b}\right )^{\frac {3}{5}}\right )} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} \log \left ({\left | {\left (d x + c\right )}^{\frac {1}{5}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} \right |}\right )}{b^{5} c - a b^{4} d} + \frac {5 \, {\left (d x + c\right )}^{\frac {4}{5}}}{4 \, b} - \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} {\left (\sqrt {5} + 1\right )} \log \left (\frac {1}{2} \, {\left (d x + c\right )}^{\frac {1}{5}} {\left (\sqrt {5} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} + \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}\right )} + {\left (d x + c\right )}^{\frac {2}{5}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{5}}\right )}{4 \, b^{5}} + \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} {\left (\sqrt {5} - 1\right )} \log \left (-\frac {1}{2} \, {\left (d x + c\right )}^{\frac {1}{5}} {\left (\sqrt {5} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} - \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}\right )} + {\left (d x + c\right )}^{\frac {2}{5}} + \left (\frac {b c - a d}{b}\right )^{\frac {2}{5}}\right )}{4 \, b^{5}} + \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {{\left (\sqrt {5} - 1\right )} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} - 4 \, {\left (d x + c\right )}^{\frac {1}{5}}}{\sqrt {2 \, \sqrt {5} + 10} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}}\right )}{2 \, b^{5}} + \frac {{\left (b^{5} c - a b^{4} d\right )}^{\frac {4}{5}} \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\sqrt {5} + 1\right )} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}} + 4 \, {\left (d x + c\right )}^{\frac {1}{5}}}{\sqrt {-2 \, \sqrt {5} + 10} \left (\frac {b c - a d}{b}\right )^{\frac {1}{5}}}\right )}{2 \, b^{5}} \] Input:
integrate((d*x+c)^(4/5)/(b*x+a),x, algorithm="giac")
Output:
(b^4*c*((b*c - a*d)/b)^(3/5) - a*b^3*d*((b*c - a*d)/b)^(3/5))*((b*c - a*d) /b)^(1/5)*log(abs((d*x + c)^(1/5) - ((b*c - a*d)/b)^(1/5)))/(b^5*c - a*b^4 *d) + 5/4*(d*x + c)^(4/5)/b - 1/4*(b^5*c - a*b^4*d)^(4/5)*(sqrt(5) + 1)*lo g(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^5 + 1/4*(b^5*c - a*b^4*d) ^(4/5)*(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^5 + 1/2*(b^5*c - a*b^4*d)^(4/5)*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^5 + 1/2*(b^5*c - a*b^4*d)^(4/5)*sqrt(-2*sqrt(5) + 10)*ar ctan(((sqrt(5) + 1)*((b*c - a*d)/b)^(1/5) + 4*(d*x + c)^(1/5))/(sqrt(-2*sq rt(5) + 10)*((b*c - a*d)/b)^(1/5)))/b^5
Time = 0.66 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\frac {5\,{\left (c+d\,x\right )}^{4/5}}{4\,b}+\frac {\ln \left (a\,d-b\,c+b^{1/5}\,{\left (a\,d-b\,c\right )}^{4/5}\,{\left (c+d\,x\right )}^{1/5}\right )\,{\left (a\,d-b\,c\right )}^{4/5}}{b^{9/5}}-\frac {\ln \left (\frac {625\,{\left (a\,d-b\,c\right )}^5}{b}-\frac {625\,{\left (a\,d-b\,c\right )}^{24/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\frac {\sqrt {5}}{4}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{4}+\frac {1}{4}\right )}{b^{4/5}}\right )\,{\left (a\,d-b\,c\right )}^{4/5}\,\left (\frac {\sqrt {5}}{4}-\frac {\sqrt {2\,\sqrt {5}-10}}{4}+\frac {1}{4}\right )}{b^{9/5}}-\frac {\ln \left (\frac {625\,{\left (a\,d-b\,c\right )}^5}{b}-\frac {625\,{\left (a\,d-b\,c\right )}^{24/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\frac {\sqrt {5}}{4}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{4}+\frac {1}{4}\right )}{b^{4/5}}\right )\,{\left (a\,d-b\,c\right )}^{4/5}\,\left (\frac {\sqrt {5}}{4}+\frac {\sqrt {2\,\sqrt {5}-10}}{4}+\frac {1}{4}\right )}{b^{9/5}}-\frac {\ln \left (\frac {625\,{\left (a\,d-b\,c\right )}^5}{b}-\frac {625\,{\left (a\,d-b\,c\right )}^{24/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{4}-\frac {\sqrt {5}}{4}+\frac {1}{4}\right )}{b^{4/5}}\right )\,{\left (a\,d-b\,c\right )}^{4/5}\,\left (\frac {\sqrt {-2\,\sqrt {5}-10}}{4}-\frac {\sqrt {5}}{4}+\frac {1}{4}\right )}{b^{9/5}}+\frac {\ln \left (\frac {625\,{\left (a\,d-b\,c\right )}^5}{b}+\frac {625\,{\left (a\,d-b\,c\right )}^{24/5}\,{\left (c+d\,x\right )}^{1/5}\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{4}+\frac {\sqrt {5}}{4}-\frac {1}{4}\right )}{b^{4/5}}\right )\,{\left (a\,d-b\,c\right )}^{4/5}\,\left (\frac {\sqrt {5}}{4}+\frac {\sqrt {-2\,\sqrt {5}-10}}{4}-\frac {1}{4}\right )}{b^{9/5}} \] Input:
int((c + d*x)^(4/5)/(a + b*x),x)
Output:
(5*(c + d*x)^(4/5))/(4*b) + (log(a*d - b*c + b^(1/5)*(a*d - b*c)^(4/5)*(c + d*x)^(1/5))*(a*d - b*c)^(4/5))/b^(9/5) - (log((625*(a*d - b*c)^5)/b - (6 25*(a*d - b*c)^(24/5)*(c + d*x)^(1/5)*(5^(1/2)/4 - (2^(1/2)*(5^(1/2) - 5)^ (1/2))/4 + 1/4))/b^(4/5))*(a*d - b*c)^(4/5)*(5^(1/2)/4 - (2*5^(1/2) - 10)^ (1/2)/4 + 1/4))/b^(9/5) - (log((625*(a*d - b*c)^5)/b - (625*(a*d - b*c)^(2 4/5)*(c + d*x)^(1/5)*(5^(1/2)/4 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/4 + 1/4))/ b^(4/5))*(a*d - b*c)^(4/5)*(5^(1/2)/4 + (2*5^(1/2) - 10)^(1/2)/4 + 1/4))/b ^(9/5) - (log((625*(a*d - b*c)^5)/b - (625*(a*d - b*c)^(24/5)*(c + d*x)^(1 /5)*((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/4 - 5^(1/2)/4 + 1/4))/b^(4/5))*(a*d - b*c)^(4/5)*((- 2*5^(1/2) - 10)^(1/2)/4 - 5^(1/2)/4 + 1/4))/b^(9/5) + (log ((625*(a*d - b*c)^5)/b + (625*(a*d - b*c)^(24/5)*(c + d*x)^(1/5)*((2^(1/2) *(- 5^(1/2) - 5)^(1/2))/4 + 5^(1/2)/4 - 1/4))/b^(4/5))*(a*d - b*c)^(4/5)*( 5^(1/2)/4 + (- 2*5^(1/2) - 10)^(1/2)/4 - 1/4))/b^(9/5)
\[ \int \frac {(c+d x)^{4/5}}{a+b x} \, dx=\text {too large to display} \] Input:
int((d*x+c)^(4/5)/(b*x+a),x)
Output:
( - 4*(c + d*x)**(1/5)*int(x/((c + d*x)**(1/5)*a**3*c*d**2 + (c + d*x)**(1 /5)*a**3*d**3*x + 2*(c + d*x)**(1/5)*a**2*b*c**2*d + 3*(c + d*x)**(1/5)*a* *2*b*c*d**2*x + (c + d*x)**(1/5)*a**2*b*d**3*x**2 + (c + d*x)**(1/5)*a*b** 2*c**3 + 3*(c + d*x)**(1/5)*a*b**2*c**2*d*x + 2*(c + d*x)**(1/5)*a*b**2*c* d**2*x**2 + (c + d*x)**(1/5)*b**3*c**3*x + (c + d*x)**(1/5)*b**3*c**2*d*x* *2),x)*a**5*d**6 + 8*(c + d*x)**(1/5)*int(x/((c + d*x)**(1/5)*a**3*c*d**2 + (c + d*x)**(1/5)*a**3*d**3*x + 2*(c + d*x)**(1/5)*a**2*b*c**2*d + 3*(c + d*x)**(1/5)*a**2*b*c*d**2*x + (c + d*x)**(1/5)*a**2*b*d**3*x**2 + (c + d* x)**(1/5)*a*b**2*c**3 + 3*(c + d*x)**(1/5)*a*b**2*c**2*d*x + 2*(c + d*x)** (1/5)*a*b**2*c*d**2*x**2 + (c + d*x)**(1/5)*b**3*c**3*x + (c + d*x)**(1/5) *b**3*c**2*d*x**2),x)*a**3*b**2*c**2*d**4 - 4*(c + d*x)**(1/5)*int(x/((c + d*x)**(1/5)*a**3*c*d**2 + (c + d*x)**(1/5)*a**3*d**3*x + 2*(c + d*x)**(1/ 5)*a**2*b*c**2*d + 3*(c + d*x)**(1/5)*a**2*b*c*d**2*x + (c + d*x)**(1/5)*a **2*b*d**3*x**2 + (c + d*x)**(1/5)*a*b**2*c**3 + 3*(c + d*x)**(1/5)*a*b**2 *c**2*d*x + 2*(c + d*x)**(1/5)*a*b**2*c*d**2*x**2 + (c + d*x)**(1/5)*b**3* c**3*x + (c + d*x)**(1/5)*b**3*c**2*d*x**2),x)*a*b**4*c**4*d**2 + 4*(c + d *x)**(1/5)*int(1/((c + d*x)**(1/5)*a**3*c*d**2 + (c + d*x)**(1/5)*a**3*d** 3*x + 2*(c + d*x)**(1/5)*a**2*b*c**2*d + 3*(c + d*x)**(1/5)*a**2*b*c*d**2* x + (c + d*x)**(1/5)*a**2*b*d**3*x**2 + (c + d*x)**(1/5)*a*b**2*c**3 + 3*( c + d*x)**(1/5)*a*b**2*c**2*d*x + 2*(c + d*x)**(1/5)*a*b**2*c*d**2*x**2...