Integrand size = 24, antiderivative size = 280 \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx=-\frac {2 (b c-a d)^3 (d e-c f)^2 \sqrt {e+f x}}{d^6}-\frac {2 (b c-a d)^3 (d e-c f) (e+f x)^{3/2}}{3 d^5}-\frac {2 (b c-a d)^3 (e+f x)^{5/2}}{5 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{7/2}}{7 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{9/2}}{9 d^2 f^3}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}} \]
-2/3*(-a*d+b*c)^3*(-c*f+d*e)*(f*x+e)^(3/2)/d^5-2/5*(-a*d+b*c)^3*(f*x+e)^(5 /2)/d^4+2/7*b*(3*a^2*d^2*f^2-3*a*b*d*f*(c*f+d*e)+b^2*(c^2*f^2+c*d*e*f+d^2* e^2))*(f*x+e)^(7/2)/d^3/f^3-2/9*b^2*(-3*a*d*f+b*c*f+2*b*d*e)*(f*x+e)^(9/2) /d^2/f^3+2/11*b^3*(f*x+e)^(11/2)/d/f^3+2*(-a*d+b*c)^3*(-c*f+d*e)^(5/2)*arc tanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(13/2)-2*(-a*d+b*c)^3*(-c*f +d*e)^2*(f*x+e)^(1/2)/d^6
Time = 0.59 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.55 \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (231 a^3 d^3 f^3 \left (15 c^2 f^2-5 c d f (7 e+f x)+d^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+99 a^2 b d^2 f^2 \left (-105 c^3 f^3+15 d^3 (e+f x)^3+35 c^2 d f^2 (7 e+f x)-7 c d^2 f \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )-33 a b^2 d f \left (-315 c^4 f^4+45 c d^3 f (e+f x)^3+5 d^4 (2 e-7 f x) (e+f x)^3+105 c^3 d f^3 (7 e+f x)-21 c^2 d^2 f^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+b^3 \left (-3465 c^5 f^5+495 c^2 d^3 f^2 (e+f x)^3+55 c d^4 f (2 e-7 f x) (e+f x)^3+1155 c^4 d f^4 (7 e+f x)-231 c^3 d^2 f^3 \left (23 e^2+11 e f x+3 f^2 x^2\right )+5 d^5 (e+f x)^3 \left (8 e^2-28 e f x+63 f^2 x^2\right )\right )\right )}{3465 d^6 f^3}-\frac {2 (-b c+a d)^3 (-d e+c f)^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{13/2}} \]
(2*Sqrt[e + f*x]*(231*a^3*d^3*f^3*(15*c^2*f^2 - 5*c*d*f*(7*e + f*x) + d^2* (23*e^2 + 11*e*f*x + 3*f^2*x^2)) + 99*a^2*b*d^2*f^2*(-105*c^3*f^3 + 15*d^3 *(e + f*x)^3 + 35*c^2*d*f^2*(7*e + f*x) - 7*c*d^2*f*(23*e^2 + 11*e*f*x + 3 *f^2*x^2)) - 33*a*b^2*d*f*(-315*c^4*f^4 + 45*c*d^3*f*(e + f*x)^3 + 5*d^4*( 2*e - 7*f*x)*(e + f*x)^3 + 105*c^3*d*f^3*(7*e + f*x) - 21*c^2*d^2*f^2*(23* e^2 + 11*e*f*x + 3*f^2*x^2)) + b^3*(-3465*c^5*f^5 + 495*c^2*d^3*f^2*(e + f *x)^3 + 55*c*d^4*f*(2*e - 7*f*x)*(e + f*x)^3 + 1155*c^4*d*f^4*(7*e + f*x) - 231*c^3*d^2*f^3*(23*e^2 + 11*e*f*x + 3*f^2*x^2) + 5*d^5*(e + f*x)^3*(8*e ^2 - 28*e*f*x + 63*f^2*x^2))))/(3465*d^6*f^3) - (2*(-(b*c) + a*d)^3*(-(d*e ) + c*f)^(5/2)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(13/2 )
Time = 0.46 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 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.
| \(\displaystyle \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^2}-\frac {b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{d^2 f^2}+\frac {(e+f x)^{5/2} (a d-b c)^3}{d^3 (c+d x)}+\frac {b^3 (e+f x)^{9/2}}{d f^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b (e+f x)^{7/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{7 d^3 f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{13/2}}-\frac {2 b^2 (e+f x)^{9/2} (-3 a d f+b c f+2 b d e)}{9 d^2 f^3}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)^2}{d^6}-\frac {2 (e+f x)^{3/2} (b c-a d)^3 (d e-c f)}{3 d^5}-\frac {2 (e+f x)^{5/2} (b c-a d)^3}{5 d^4}+\frac {2 b^3 (e+f x)^{11/2}}{11 d f^3}\) |
(-2*(b*c - a*d)^3*(d*e - c*f)^2*Sqrt[e + f*x])/d^6 - (2*(b*c - a*d)^3*(d*e - c*f)*(e + f*x)^(3/2))/(3*d^5) - (2*(b*c - a*d)^3*(e + f*x)^(5/2))/(5*d^ 4) + (2*b*(3*a^2*d^2*f^2 - 3*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + c*d*e*f + c^2*f^2))*(e + f*x)^(7/2))/(7*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d *f)*(e + f*x)^(9/2))/(9*d^2*f^3) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + ( 2*(b*c - a*d)^3*(d*e - c*f)^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(13/2)
3.18.82.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Leaf count of result is larger than twice the leaf count of optimal. \(552\) vs. \(2(248)=496\).
Time = 1.29 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.98
| method | result | size |
| pseudoelliptic | \(\frac {-2 f^{3} \left (c f -d e \right )^{3} \left (a d -b c \right )^{3} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+2 \left (\left (\frac {x^{2} \left (\frac {5}{11} b^{3} x^{3}+\frac {5}{3} a \,b^{2} x^{2}+\frac {15}{7} a^{2} b x +a^{3}\right ) d^{5}}{5}-\frac {x \left (\frac {1}{3} b^{3} x^{3}+\frac {9}{7} a \,b^{2} x^{2}+\frac {9}{5} a^{2} b x +a^{3}\right ) c \,d^{4}}{3}+c^{2} \left (\frac {3}{5} a \,b^{2} x^{2}+a^{2} b x +a^{3}+\frac {1}{7} b^{3} x^{3}\right ) d^{3}-3 \left (\frac {1}{15} b^{2} x^{2}+\frac {1}{3} a b x +a^{2}\right ) b \,c^{3} d^{2}+3 b^{2} \left (\frac {b x}{9}+a \right ) c^{4} d -b^{3} c^{5}\right ) f^{5}-\frac {7 e d \left (-\frac {11 x \left (\frac {115}{363} b^{3} x^{3}+\frac {95}{77} a \,b^{2} x^{2}+\frac {135}{77} a^{2} b x +a^{3}\right ) d^{4}}{35}+c \left (\frac {19}{147} b^{3} x^{3}+\frac {27}{49} a \,b^{2} x^{2}+\frac {33}{35} a^{2} b x +a^{3}\right ) d^{3}-3 \left (\frac {3}{49} b^{2} x^{2}+\frac {11}{35} a b x +a^{2}\right ) b \,c^{2} d^{2}+3 b^{2} \left (\frac {11 b x}{105}+a \right ) c^{3} d -c^{4} b^{3}\right ) f^{4}}{3}+\frac {23 e^{2} \left (\left (\frac {565}{5313} b^{3} x^{3}+\frac {75}{161} a \,b^{2} x^{2}+\frac {135}{161} a^{2} b x +a^{3}\right ) d^{3}-3 \left (\frac {25}{483} b^{2} x^{2}+\frac {45}{161} a b x +a^{2}\right ) b c \,d^{2}+3 \left (\frac {15 b x}{161}+a \right ) b^{2} c^{2} d -b^{3} c^{3}\right ) d^{2} f^{3}}{15}+\frac {3 e^{3} \left (\left (\frac {1}{99} b^{2} x^{2}+\frac {1}{9} a b x +a^{2}\right ) d^{2}-b \left (\frac {b x}{27}+a \right ) c d +\frac {b^{2} c^{2}}{3}\right ) b \,d^{3} f^{2}}{7}-\frac {2 e^{4} b^{2} \left (\left (\frac {2 b x}{33}+a \right ) d -\frac {b c}{3}\right ) d^{4} f}{21}+\frac {8 b^{3} d^{5} e^{5}}{693}\right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}}{f^{3} d^{6} \sqrt {\left (c f -d e \right ) d}}\) | \(553\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {11}{2}} d^{5}}{11}+\frac {\left (\left (a d f -b c f \right ) b^{2} d^{4}+b d \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (a d f -b c f \right ) \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )+b d \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {\left (\left (a d f -b c f \right ) \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )+b d \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )+b d \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right ) \sqrt {f x +e}\right )}{d^{6}}-\frac {2 f^{3} \left (a^{3} c^{3} d^{3} f^{3}-3 a^{3} c^{2} d^{4} e \,f^{2}+3 a^{3} c \,d^{5} e^{2} f -a^{3} e^{3} d^{6}-3 a^{2} b \,c^{4} d^{2} f^{3}+9 a^{2} b \,c^{3} d^{3} e \,f^{2}-9 a^{2} b \,c^{2} d^{4} e^{2} f +3 a^{2} b c \,d^{5} e^{3}+3 a \,b^{2} c^{5} d \,f^{3}-9 a \,b^{2} c^{4} d^{2} e \,f^{2}+9 a \,b^{2} c^{3} d^{3} e^{2} f -3 a \,b^{2} c^{2} d^{4} e^{3}-c^{6} b^{3} f^{3}+3 b^{3} c^{5} d e \,f^{2}-3 b^{3} c^{4} d^{2} e^{2} f +b^{3} c^{3} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{6} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(964\) |
| default | \(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {11}{2}} d^{5}}{11}+\frac {\left (\left (a d f -b c f \right ) b^{2} d^{4}+b d \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )\right ) \left (f x +e \right )^{\frac {9}{2}}}{9}+\frac {\left (\left (a d f -b c f \right ) \left (2 a b \,d^{4} f -2 b^{2} d^{4} e \right )+b d \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )\right ) \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {\left (\left (a d f -b c f \right ) \left (a^{2} d^{4} f^{2}-a b c \,d^{3} f^{2}-a b \,d^{4} e f +b^{2} c^{2} d^{2} f^{2}-b^{2} c \,d^{3} e f +b^{2} d^{4} e^{2}\right )+b d \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a^{2} c \,d^{3} f^{3}+a^{2} d^{4} e \,f^{2}+a b \,c^{2} d^{2} f^{3}-a b \,d^{4} e^{2} f -b^{2} c^{2} d^{2} e \,f^{2}+b^{2} c \,d^{3} e^{2} f \right )+b d \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a^{2} c^{2} d^{2} f^{4}-2 a^{2} c \,d^{3} e \,f^{3}+a^{2} d^{4} e^{2} f^{2}-2 a b \,c^{3} d \,f^{4}+4 a b \,c^{2} d^{2} e \,f^{3}-2 a b c \,d^{3} e^{2} f^{2}+b^{2} c^{4} f^{4}-2 b^{2} c^{3} d e \,f^{3}+b^{2} c^{2} d^{2} e^{2} f^{2}\right ) \sqrt {f x +e}\right )}{d^{6}}-\frac {2 f^{3} \left (a^{3} c^{3} d^{3} f^{3}-3 a^{3} c^{2} d^{4} e \,f^{2}+3 a^{3} c \,d^{5} e^{2} f -a^{3} e^{3} d^{6}-3 a^{2} b \,c^{4} d^{2} f^{3}+9 a^{2} b \,c^{3} d^{3} e \,f^{2}-9 a^{2} b \,c^{2} d^{4} e^{2} f +3 a^{2} b c \,d^{5} e^{3}+3 a \,b^{2} c^{5} d \,f^{3}-9 a \,b^{2} c^{4} d^{2} e \,f^{2}+9 a \,b^{2} c^{3} d^{3} e^{2} f -3 a \,b^{2} c^{2} d^{4} e^{3}-c^{6} b^{3} f^{3}+3 b^{3} c^{5} d e \,f^{2}-3 b^{3} c^{4} d^{2} e^{2} f +b^{3} c^{3} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{6} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(964\) |
| risch | \(\text {Expression too large to display}\) | \(1083\) |
2/((c*f-d*e)*d)^(1/2)*(-f^3*(c*f-d*e)^3*(a*d-b*c)^3*arctan(d*(f*x+e)^(1/2) /((c*f-d*e)*d)^(1/2))+((1/5*x^2*(5/11*b^3*x^3+5/3*a*b^2*x^2+15/7*a^2*b*x+a ^3)*d^5-1/3*x*(1/3*b^3*x^3+9/7*a*b^2*x^2+9/5*a^2*b*x+a^3)*c*d^4+c^2*(3/5*a *b^2*x^2+a^2*b*x+a^3+1/7*b^3*x^3)*d^3-3*(1/15*b^2*x^2+1/3*a*b*x+a^2)*b*c^3 *d^2+3*b^2*(1/9*b*x+a)*c^4*d-b^3*c^5)*f^5-7/3*e*d*(-11/35*x*(115/363*b^3*x ^3+95/77*a*b^2*x^2+135/77*a^2*b*x+a^3)*d^4+c*(19/147*b^3*x^3+27/49*a*b^2*x ^2+33/35*a^2*b*x+a^3)*d^3-3*(3/49*b^2*x^2+11/35*a*b*x+a^2)*b*c^2*d^2+3*b^2 *(11/105*b*x+a)*c^3*d-c^4*b^3)*f^4+23/15*e^2*((565/5313*b^3*x^3+75/161*a*b ^2*x^2+135/161*a^2*b*x+a^3)*d^3-3*(25/483*b^2*x^2+45/161*a*b*x+a^2)*b*c*d^ 2+3*(15/161*b*x+a)*b^2*c^2*d-b^3*c^3)*d^2*f^3+3/7*e^3*((1/99*b^2*x^2+1/9*a *b*x+a^2)*d^2-b*(1/27*b*x+a)*c*d+1/3*b^2*c^2)*b*d^3*f^2-2/21*e^4*b^2*((2/3 3*b*x+a)*d-1/3*b*c)*d^4*f+8/693*b^3*d^5*e^5)*((c*f-d*e)*d)^(1/2)*(f*x+e)^( 1/2))/f^3/d^6
Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (248) = 496\).
Time = 0.25 (sec) , antiderivative size = 1741, normalized size of antiderivative = 6.22 \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx=\text {Too large to display} \]
[-1/3465*(3465*((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)* e^2*f^3 - 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*e* f^4 + (b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5)*sqrt( (d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqrt((d*e - c* f)/d))/(d*x + c)) - 2*(315*b^3*d^5*f^5*x^5 + 40*b^3*d^5*e^5 + 110*(b^3*c*d ^4 - 3*a*b^2*d^5)*e^4*f + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)* e^3*f^2 - 5313*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e ^2*f^3 + 8085*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)* e*f^4 - 3465*(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*f^5 + 35*(23*b^3*d^5*e*f^4 - 11*(b^3*c*d^4 - 3*a*b^2*d^5)*f^5)*x^4 + 5*(113*b ^3*d^5*e^2*f^3 - 209*(b^3*c*d^4 - 3*a*b^2*d^5)*e*f^4 + 99*(b^3*c^2*d^3 - 3 *a*b^2*c*d^4 + 3*a^2*b*d^5)*f^5)*x^3 + 3*(5*b^3*d^5*e^3*f^2 - 275*(b^3*c*d ^4 - 3*a*b^2*d^5)*e^2*f^3 + 495*(b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5 )*e*f^4 - 231*(b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*f^ 5)*x^2 - (20*b^3*d^5*e^4*f + 55*(b^3*c*d^4 - 3*a*b^2*d^5)*e^3*f^2 - 1485*( b^3*c^2*d^3 - 3*a*b^2*c*d^4 + 3*a^2*b*d^5)*e^2*f^3 + 2541*(b^3*c^3*d^2 - 3 *a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e*f^4 - 1155*(b^3*c^4*d - 3*a*b^ 2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*f^5)*x)*sqrt(f*x + e))/(d^6*f^3), 2/3465*(3465*((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*e ^2*f^3 - 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*...
Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (267) = 534\).
Time = 5.10 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.46 \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} \left (e + f x\right )^{\frac {11}{2}}}{11 d f^{2}} + \frac {\left (e + f x\right )^{\frac {9}{2}} \cdot \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{9 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{7 d^{3} f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{5 d^{4}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- a^{3} c d^{3} f^{2} + a^{3} d^{4} e f + 3 a^{2} b c^{2} d^{2} f^{2} - 3 a^{2} b c d^{3} e f - 3 a b^{2} c^{3} d f^{2} + 3 a b^{2} c^{2} d^{2} e f + b^{3} c^{4} f^{2} - b^{3} c^{3} d e f\right )}{3 d^{5}} + \frac {\sqrt {e + f x} \left (a^{3} c^{2} d^{3} f^{3} - 2 a^{3} c d^{4} e f^{2} + a^{3} d^{5} e^{2} f - 3 a^{2} b c^{3} d^{2} f^{3} + 6 a^{2} b c^{2} d^{3} e f^{2} - 3 a^{2} b c d^{4} e^{2} f + 3 a b^{2} c^{4} d f^{3} - 6 a b^{2} c^{3} d^{2} e f^{2} + 3 a b^{2} c^{2} d^{3} e^{2} f - b^{3} c^{5} f^{3} + 2 b^{3} c^{4} d e f^{2} - b^{3} c^{3} d^{2} e^{2} f\right )}{d^{6}} - \frac {f \left (a d - b c\right )^{3} \left (c f - d e\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{7} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {5}{2}} \left (\frac {b^{3} x^{3}}{3 d} + \frac {x^{2} \cdot \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac {x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac {\left (a d - b c\right )^{3} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**3*(e + f*x)**(11/2)/(11*d*f**2) + (e + f*x)**(9/2)*(3*a*b **2*d*f - b**3*c*f - 2*b**3*d*e)/(9*d**2*f**2) + (e + f*x)**(7/2)*(3*a**2* b*d**2*f**2 - 3*a*b**2*c*d*f**2 - 3*a*b**2*d**2*e*f + b**3*c**2*f**2 + b** 3*c*d*e*f + b**3*d**2*e**2)/(7*d**3*f**2) + (e + f*x)**(5/2)*(a**3*d**3*f - 3*a**2*b*c*d**2*f + 3*a*b**2*c**2*d*f - b**3*c**3*f)/(5*d**4) + (e + f*x )**(3/2)*(-a**3*c*d**3*f**2 + a**3*d**4*e*f + 3*a**2*b*c**2*d**2*f**2 - 3* a**2*b*c*d**3*e*f - 3*a*b**2*c**3*d*f**2 + 3*a*b**2*c**2*d**2*e*f + b**3*c **4*f**2 - b**3*c**3*d*e*f)/(3*d**5) + sqrt(e + f*x)*(a**3*c**2*d**3*f**3 - 2*a**3*c*d**4*e*f**2 + a**3*d**5*e**2*f - 3*a**2*b*c**3*d**2*f**3 + 6*a* *2*b*c**2*d**3*e*f**2 - 3*a**2*b*c*d**4*e**2*f + 3*a*b**2*c**4*d*f**3 - 6* a*b**2*c**3*d**2*e*f**2 + 3*a*b**2*c**2*d**3*e**2*f - b**3*c**5*f**3 + 2*b **3*c**4*d*e*f**2 - b**3*c**3*d**2*e**2*f)/d**6 - f*(a*d - b*c)**3*(c*f - d*e)**3*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**7*sqrt((c*f - d*e)/d)) )/f, Ne(f, 0)), (e**(5/2)*(b**3*x**3/(3*d) + x**2*(3*a*b**2*d - b**3*c)/(2 *d**2) + x*(3*a**2*b*d**2 - 3*a*b**2*c*d + b**3*c**2)/d**3 + (a*d - b*c)** 3*Piecewise((x/c, Eq(d, 0)), (log(c + d*x)/d, True))/d**3), True))
Exception generated. \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (248) = 496\).
Time = 0.30 (sec) , antiderivative size = 989, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx=-\frac {2 \, {\left (b^{3} c^{3} d^{3} e^{3} - 3 \, a b^{2} c^{2} d^{4} e^{3} + 3 \, a^{2} b c d^{5} e^{3} - a^{3} d^{6} e^{3} - 3 \, b^{3} c^{4} d^{2} e^{2} f + 9 \, a b^{2} c^{3} d^{3} e^{2} f - 9 \, a^{2} b c^{2} d^{4} e^{2} f + 3 \, a^{3} c d^{5} e^{2} f + 3 \, b^{3} c^{5} d e f^{2} - 9 \, a b^{2} c^{4} d^{2} e f^{2} + 9 \, a^{2} b c^{3} d^{3} e f^{2} - 3 \, a^{3} c^{2} d^{4} e f^{2} - b^{3} c^{6} f^{3} + 3 \, a b^{2} c^{5} d f^{3} - 3 \, a^{2} b c^{4} d^{2} f^{3} + a^{3} c^{3} d^{3} f^{3}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{6}} + \frac {2 \, {\left (315 \, {\left (f x + e\right )}^{\frac {11}{2}} b^{3} d^{10} f^{30} - 770 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} d^{10} e f^{30} + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{10} e^{2} f^{30} - 385 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} c d^{9} f^{31} + 1155 \, {\left (f x + e\right )}^{\frac {9}{2}} a b^{2} d^{10} f^{31} + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c d^{9} e f^{31} - 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} d^{10} e f^{31} + 495 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c^{2} d^{8} f^{32} - 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} c d^{9} f^{32} + 1485 \, {\left (f x + e\right )}^{\frac {7}{2}} a^{2} b d^{10} f^{32} - 693 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c^{3} d^{7} f^{33} + 2079 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} c^{2} d^{8} f^{33} - 2079 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} b c d^{9} f^{33} + 693 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{3} d^{10} f^{33} - 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{3} d^{7} e f^{33} + 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{8} e f^{33} - 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c d^{9} e f^{33} + 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} d^{10} e f^{33} - 3465 \, \sqrt {f x + e} b^{3} c^{3} d^{7} e^{2} f^{33} + 10395 \, \sqrt {f x + e} a b^{2} c^{2} d^{8} e^{2} f^{33} - 10395 \, \sqrt {f x + e} a^{2} b c d^{9} e^{2} f^{33} + 3465 \, \sqrt {f x + e} a^{3} d^{10} e^{2} f^{33} + 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{4} d^{6} f^{34} - 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{3} d^{7} f^{34} + 3465 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c^{2} d^{8} f^{34} - 1155 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} c d^{9} f^{34} + 6930 \, \sqrt {f x + e} b^{3} c^{4} d^{6} e f^{34} - 20790 \, \sqrt {f x + e} a b^{2} c^{3} d^{7} e f^{34} + 20790 \, \sqrt {f x + e} a^{2} b c^{2} d^{8} e f^{34} - 6930 \, \sqrt {f x + e} a^{3} c d^{9} e f^{34} - 3465 \, \sqrt {f x + e} b^{3} c^{5} d^{5} f^{35} + 10395 \, \sqrt {f x + e} a b^{2} c^{4} d^{6} f^{35} - 10395 \, \sqrt {f x + e} a^{2} b c^{3} d^{7} f^{35} + 3465 \, \sqrt {f x + e} a^{3} c^{2} d^{8} f^{35}\right )}}{3465 \, d^{11} f^{33}} \]
-2*(b^3*c^3*d^3*e^3 - 3*a*b^2*c^2*d^4*e^3 + 3*a^2*b*c*d^5*e^3 - a^3*d^6*e^ 3 - 3*b^3*c^4*d^2*e^2*f + 9*a*b^2*c^3*d^3*e^2*f - 9*a^2*b*c^2*d^4*e^2*f + 3*a^3*c*d^5*e^2*f + 3*b^3*c^5*d*e*f^2 - 9*a*b^2*c^4*d^2*e*f^2 + 9*a^2*b*c^ 3*d^3*e*f^2 - 3*a^3*c^2*d^4*e*f^2 - b^3*c^6*f^3 + 3*a*b^2*c^5*d*f^3 - 3*a^ 2*b*c^4*d^2*f^3 + a^3*c^3*d^3*f^3)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c* d*f))/(sqrt(-d^2*e + c*d*f)*d^6) + 2/3465*(315*(f*x + e)^(11/2)*b^3*d^10*f ^30 - 770*(f*x + e)^(9/2)*b^3*d^10*e*f^30 + 495*(f*x + e)^(7/2)*b^3*d^10*e ^2*f^30 - 385*(f*x + e)^(9/2)*b^3*c*d^9*f^31 + 1155*(f*x + e)^(9/2)*a*b^2* d^10*f^31 + 495*(f*x + e)^(7/2)*b^3*c*d^9*e*f^31 - 1485*(f*x + e)^(7/2)*a* b^2*d^10*e*f^31 + 495*(f*x + e)^(7/2)*b^3*c^2*d^8*f^32 - 1485*(f*x + e)^(7 /2)*a*b^2*c*d^9*f^32 + 1485*(f*x + e)^(7/2)*a^2*b*d^10*f^32 - 693*(f*x + e )^(5/2)*b^3*c^3*d^7*f^33 + 2079*(f*x + e)^(5/2)*a*b^2*c^2*d^8*f^33 - 2079* (f*x + e)^(5/2)*a^2*b*c*d^9*f^33 + 693*(f*x + e)^(5/2)*a^3*d^10*f^33 - 115 5*(f*x + e)^(3/2)*b^3*c^3*d^7*e*f^33 + 3465*(f*x + e)^(3/2)*a*b^2*c^2*d^8* e*f^33 - 3465*(f*x + e)^(3/2)*a^2*b*c*d^9*e*f^33 + 1155*(f*x + e)^(3/2)*a^ 3*d^10*e*f^33 - 3465*sqrt(f*x + e)*b^3*c^3*d^7*e^2*f^33 + 10395*sqrt(f*x + e)*a*b^2*c^2*d^8*e^2*f^33 - 10395*sqrt(f*x + e)*a^2*b*c*d^9*e^2*f^33 + 34 65*sqrt(f*x + e)*a^3*d^10*e^2*f^33 + 1155*(f*x + e)^(3/2)*b^3*c^4*d^6*f^34 - 3465*(f*x + e)^(3/2)*a*b^2*c^3*d^7*f^34 + 3465*(f*x + e)^(3/2)*a^2*b*c^ 2*d^8*f^34 - 1155*(f*x + e)^(3/2)*a^3*c*d^9*f^34 + 6930*sqrt(f*x + e)*b...
Time = 0.17 (sec) , antiderivative size = 897, normalized size of antiderivative = 3.20 \[ \int \frac {(a+b x)^3 (e+f x)^{5/2}}{c+d x} \, dx={\left (e+f\,x\right )}^{7/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{7\,d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{7\,d\,f^3}\right )-{\left (e+f\,x\right )}^{9/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{9\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{9\,d^2\,f^6}\right )+{\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{5\,d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{11/2}}{11\,d\,f^3}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{5/2}}{-a^3\,c^3\,d^3\,f^3+3\,a^3\,c^2\,d^4\,e\,f^2-3\,a^3\,c\,d^5\,e^2\,f+a^3\,d^6\,e^3+3\,a^2\,b\,c^4\,d^2\,f^3-9\,a^2\,b\,c^3\,d^3\,e\,f^2+9\,a^2\,b\,c^2\,d^4\,e^2\,f-3\,a^2\,b\,c\,d^5\,e^3-3\,a\,b^2\,c^5\,d\,f^3+9\,a\,b^2\,c^4\,d^2\,e\,f^2-9\,a\,b^2\,c^3\,d^3\,e^2\,f+3\,a\,b^2\,c^2\,d^4\,e^3+b^3\,c^6\,f^3-3\,b^3\,c^5\,d\,e\,f^2+3\,b^3\,c^4\,d^2\,e^2\,f-b^3\,c^3\,d^3\,e^3}\right )\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{5/2}}{d^{13/2}}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,{\left (c\,f^4-d\,e\,f^3\right )}^2}{d^2\,f^6} \]
(e + f*x)^(7/2)*((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3 ))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(7*d*f^3) + (6*b*(a*f - b*e)^2)/(7*d*f^3) ) - (e + f*x)^(9/2)*((6*b^3*e - 6*a*b^2*f)/(9*d*f^3) + (2*b^3*(c*f^4 - d*e *f^3))/(9*d^2*f^6)) + (e + f*x)^(5/2)*((2*(a*f - b*e)^3)/(5*d*f^3) - ((((( 6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^2)/(d*f^3))*(c*f^4 - d*e*f^3))/(5* d*f^3)) + (2*b^3*(e + f*x)^(11/2))/(11*d*f^3) + (2*atan((d^(1/2)*(e + f*x) ^(1/2)*(a*d - b*c)^3*(c*f - d*e)^(5/2))/(a^3*d^6*e^3 + b^3*c^6*f^3 - a^3*c ^3*d^3*f^3 - b^3*c^3*d^3*e^3 - 3*a^2*b*c*d^5*e^3 - 3*a*b^2*c^5*d*f^3 - 3*a ^3*c*d^5*e^2*f - 3*b^3*c^5*d*e*f^2 + 3*a*b^2*c^2*d^4*e^3 + 3*a^2*b*c^4*d^2 *f^3 + 3*a^3*c^2*d^4*e*f^2 + 3*b^3*c^4*d^2*e^2*f - 9*a*b^2*c^3*d^3*e^2*f + 9*a*b^2*c^4*d^2*e*f^2 + 9*a^2*b*c^2*d^4*e^2*f - 9*a^2*b*c^3*d^3*e*f^2))*( a*d - b*c)^3*(c*f - d*e)^(5/2))/d^(13/2) - ((e + f*x)^(3/2)*((2*(a*f - b*e )^3)/(d*f^3) - (((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c*f^4 - d*e*f^3 ))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^2)/(d*f^3))*(c *f^4 - d*e*f^3))/(d*f^3))*(c*f^4 - d*e*f^3))/(3*d*f^3) + ((e + f*x)^(1/2)* ((2*(a*f - b*e)^3)/(d*f^3) - (((((6*b^3*e - 6*a*b^2*f)/(d*f^3) + (2*b^3*(c *f^4 - d*e*f^3))/(d^2*f^6))*(c*f^4 - d*e*f^3))/(d*f^3) + (6*b*(a*f - b*e)^ 2)/(d*f^3))*(c*f^4 - d*e*f^3))/(d*f^3))*(c*f^4 - d*e*f^3)^2)/(d^2*f^6)