Integrand size = 24, antiderivative size = 244 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx=-\frac {2 (b c-a d)^3 (d e-c f) \sqrt {e+f x}}{d^5}-\frac {2 (b c-a d)^3 (e+f x)^{3/2}}{3 d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{5/2}}{5 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{7/2}}{7 d^2 f^3}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}} \] Output:
-2*(-a*d+b*c)^3*(-c*f+d*e)*(f*x+e)^(1/2)/d^5-2/3*(-a*d+b*c)^3*(f*x+e)^(3/2 )/d^4+2/5*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)^(5/2)/d^3/f^3-2/7*b^2*(-3*a*d*f+b*c*f+2*b*d*e)*(f*x+e)^(7/2)/d ^2/f^3+2/9*b^3*(f*x+e)^(9/2)/d/f^3+2*(-a*d+b*c)^3*(-c*f+d*e)^(3/2)*arctanh (d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(11/2)
Time = 0.38 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (105 a^3 d^3 f^3 (4 d e-3 c f+d f x)+63 a^2 b d^2 f^2 \left (15 c^2 f^2+3 d^2 (e+f x)^2-5 c d f (4 e+f x)\right )-9 a b^2 d f \left (105 c^3 f^3+21 c d^2 f (e+f x)^2+3 d^3 (2 e-5 f x) (e+f x)^2-35 c^2 d f^2 (4 e+f x)\right )+b^3 \left (315 c^4 f^4+63 c^2 d^2 f^2 (e+f x)^2+9 c d^3 f (2 e-5 f x) (e+f x)^2-105 c^3 d f^3 (4 e+f x)+d^4 (e+f x)^2 \left (8 e^2-20 e f x+35 f^2 x^2\right )\right )\right )}{315 d^5 f^3}+\frac {2 (-b c+a d)^3 (-d e+c f)^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{11/2}} \] Input:
Integrate[((a + b*x)^3*(e + f*x)^(3/2))/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3*(4*d*e - 3*c*f + d*f*x) + 63*a^2*b*d^2*f ^2*(15*c^2*f^2 + 3*d^2*(e + f*x)^2 - 5*c*d*f*(4*e + f*x)) - 9*a*b^2*d*f*(1 05*c^3*f^3 + 21*c*d^2*f*(e + f*x)^2 + 3*d^3*(2*e - 5*f*x)*(e + f*x)^2 - 35 *c^2*d*f^2*(4*e + f*x)) + b^3*(315*c^4*f^4 + 63*c^2*d^2*f^2*(e + f*x)^2 + 9*c*d^3*f*(2*e - 5*f*x)*(e + f*x)^2 - 105*c^3*d*f^3*(4*e + f*x) + d^4*(e + f*x)^2*(8*e^2 - 20*e*f*x + 35*f^2*x^2))))/(315*d^5*f^3) + (2*(-(b*c) + a* d)^3*(-(d*e) + c*f)^(3/2)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f ]])/d^(11/2)
Time = 0.39 (sec) , antiderivative size = 244, 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.
| \(\displaystyle \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^2}-\frac {b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{d^2 f^2}+\frac {(e+f x)^{3/2} (a d-b c)^3}{d^3 (c+d x)}+\frac {b^3 (e+f x)^{7/2}}{d f^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b (e+f x)^{5/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{5 d^3 f^3}+\frac {2 (b c-a d)^3 (d e-c f)^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}}-\frac {2 b^2 (e+f x)^{7/2} (-3 a d f+b c f+2 b d e)}{7 d^2 f^3}-\frac {2 \sqrt {e+f x} (b c-a d)^3 (d e-c f)}{d^5}-\frac {2 (e+f x)^{3/2} (b c-a d)^3}{3 d^4}+\frac {2 b^3 (e+f x)^{9/2}}{9 d f^3}\) |
Input:
Int[((a + b*x)^3*(e + f*x)^(3/2))/(c + d*x),x]
Output:
(-2*(b*c - a*d)^3*(d*e - c*f)*Sqrt[e + f*x])/d^5 - (2*(b*c - a*d)^3*(e + f *x)^(3/2))/(3*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)^(5/2))/(5*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(7/2))/(7*d^2*f^3) + (2*b^3*(e + f*x)^(9/2)) /(9*d*f^3) + (2*(b*c - a*d)^3*(d*e - c*f)^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(11/2)
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]))
Time = 0.62 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {\left (c f -d e \right ) d}\, \left (\left (-\frac {8 \left (f x +e \right )^{2} \left (\frac {35}{8} f^{2} x^{2}-\frac {5}{2} e f x +e^{2}\right ) d^{4}}{315}-\frac {2 c \left (f x +e \right )^{2} f \left (-\frac {5 f x}{2}+e \right ) d^{3}}{35}-\frac {c^{2} f^{2} \left (f x +e \right )^{2} d^{2}}{5}+\frac {4 c^{3} \left (\frac {f x}{4}+e \right ) f^{3} d}{3}-c^{4} f^{4}\right ) b^{3}+3 a d \left (\frac {2 \left (f x +e \right )^{2} \left (-\frac {5 f x}{2}+e \right ) d^{3}}{35}+\frac {c f \left (f x +e \right )^{2} d^{2}}{5}-\frac {4 c^{2} \left (\frac {f x}{4}+e \right ) f^{2} d}{3}+c^{3} f^{3}\right ) f \,b^{2}-3 \left (\frac {\left (f x +e \right )^{2} d^{2}}{5}-\frac {4 c \left (\frac {f x}{4}+e \right ) f d}{3}+c^{2} f^{2}\right ) a^{2} d^{2} f^{2} b +a^{3} d^{3} \left (\frac {\left (-f x -4 e \right ) d}{3}+c f \right ) f^{3}\right ) \sqrt {f x +e}+2 f^{3} \left (c f -d e \right )^{2} \left (a d -b c \right )^{3} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{f^{3} d^{5} \sqrt {\left (c f -d e \right ) d}}\) | \(316\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {b^{3} \left (f x +e \right )^{\frac {9}{2}} d^{4}}{9}-\frac {3 a \,b^{2} d^{4} f \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {b^{3} c \,d^{3} f \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d^{4} e \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {3 a^{2} b \,d^{4} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} c \,d^{3} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} d^{4} e f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c^{2} d^{2} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{3} e f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} d^{4} e^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a^{3} d^{4} f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} b c \,d^{3} f^{3} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c^{2} d^{2} f^{3} \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{3} d \,f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} c \,d^{3} f^{4} \sqrt {f x +e}-a^{3} d^{4} e \,f^{3} \sqrt {f x +e}-3 a^{2} b \,c^{2} d^{2} f^{4} \sqrt {f x +e}+3 a^{2} b c \,d^{3} e \,f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{3} d \,f^{4} \sqrt {f x +e}-3 a \,b^{2} c^{2} d^{2} e \,f^{3} \sqrt {f x +e}-b^{3} c^{4} f^{4} \sqrt {f x +e}+b^{3} c^{3} d e \,f^{3} \sqrt {f x +e}\right )}{d^{5}}+\frac {2 f^{3} \left (a^{3} c^{2} d^{3} f^{2}-2 a^{3} c \,d^{4} e f +a^{3} e^{2} d^{5}-3 a^{2} b \,c^{3} d^{2} f^{2}+6 a^{2} b \,c^{2} d^{3} e f -3 a^{2} b c \,d^{4} e^{2}+3 a \,b^{2} c^{4} d \,f^{2}-6 a \,b^{2} c^{3} d^{2} e f +3 a \,b^{2} c^{2} d^{3} e^{2}-c^{5} b^{3} f^{2}+2 b^{3} c^{4} d e f -b^{3} c^{3} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(626\) |
| default | \(\frac {-\frac {2 \left (-\frac {b^{3} \left (f x +e \right )^{\frac {9}{2}} d^{4}}{9}-\frac {3 a \,b^{2} d^{4} f \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {b^{3} c \,d^{3} f \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {2 b^{3} d^{4} e \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {3 a^{2} b \,d^{4} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} c \,d^{3} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {3 a \,b^{2} d^{4} e f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c^{2} d^{2} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{3} e f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} d^{4} e^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a^{3} d^{4} f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} b c \,d^{3} f^{3} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c^{2} d^{2} f^{3} \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{3} d \,f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} c \,d^{3} f^{4} \sqrt {f x +e}-a^{3} d^{4} e \,f^{3} \sqrt {f x +e}-3 a^{2} b \,c^{2} d^{2} f^{4} \sqrt {f x +e}+3 a^{2} b c \,d^{3} e \,f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{3} d \,f^{4} \sqrt {f x +e}-3 a \,b^{2} c^{2} d^{2} e \,f^{3} \sqrt {f x +e}-b^{3} c^{4} f^{4} \sqrt {f x +e}+b^{3} c^{3} d e \,f^{3} \sqrt {f x +e}\right )}{d^{5}}+\frac {2 f^{3} \left (a^{3} c^{2} d^{3} f^{2}-2 a^{3} c \,d^{4} e f +a^{3} e^{2} d^{5}-3 a^{2} b \,c^{3} d^{2} f^{2}+6 a^{2} b \,c^{2} d^{3} e f -3 a^{2} b c \,d^{4} e^{2}+3 a \,b^{2} c^{4} d \,f^{2}-6 a \,b^{2} c^{3} d^{2} e f +3 a \,b^{2} c^{2} d^{3} e^{2}-c^{5} b^{3} f^{2}+2 b^{3} c^{4} d e f -b^{3} c^{3} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(626\) |
| risch | \(-\frac {2 \left (-35 b^{3} f^{4} d^{4} x^{4}-135 a \,b^{2} d^{4} f^{4} x^{3}+45 b^{3} c \,d^{3} f^{4} x^{3}-50 b^{3} d^{4} e \,f^{3} x^{3}-189 a^{2} b \,d^{4} f^{4} x^{2}+189 a \,b^{2} c \,d^{3} f^{4} x^{2}-216 a \,b^{2} d^{4} e \,f^{3} x^{2}-63 b^{3} c^{2} d^{2} f^{4} x^{2}+72 b^{3} c \,d^{3} e \,f^{3} x^{2}-3 b^{3} d^{4} e^{2} f^{2} x^{2}-105 a^{3} d^{4} f^{4} x +315 a^{2} b c \,d^{3} f^{4} x -378 a^{2} b \,d^{4} e \,f^{3} x -315 a \,b^{2} c^{2} d^{2} f^{4} x +378 a \,b^{2} c \,d^{3} e \,f^{3} x -27 a \,b^{2} d^{4} e^{2} f^{2} x +105 b^{3} c^{3} d \,f^{4} x -126 b^{3} c^{2} d^{2} e \,f^{3} x +9 b^{3} c \,d^{3} e^{2} f^{2} x +4 b^{3} d^{4} e^{3} f x +315 a^{3} c \,d^{3} f^{4}-420 a^{3} d^{4} e \,f^{3}-945 a^{2} b \,c^{2} d^{2} f^{4}+1260 a^{2} b c \,d^{3} e \,f^{3}-189 a^{2} b \,d^{4} e^{2} f^{2}+945 a \,b^{2} c^{3} d \,f^{4}-1260 a \,b^{2} c^{2} d^{2} e \,f^{3}+189 a \,b^{2} c \,d^{3} e^{2} f^{2}+54 a \,b^{2} d^{4} e^{3} f -315 b^{3} c^{4} f^{4}+420 b^{3} c^{3} d e \,f^{3}-63 b^{3} c^{2} d^{2} e^{2} f^{2}-18 b^{3} c \,d^{3} e^{3} f -8 b^{3} d^{4} e^{4}\right ) \sqrt {f x +e}}{315 f^{3} d^{5}}+\frac {2 \left (a^{3} c^{2} d^{3} f^{2}-2 a^{3} c \,d^{4} e f +a^{3} e^{2} d^{5}-3 a^{2} b \,c^{3} d^{2} f^{2}+6 a^{2} b \,c^{2} d^{3} e f -3 a^{2} b c \,d^{4} e^{2}+3 a \,b^{2} c^{4} d \,f^{2}-6 a \,b^{2} c^{3} d^{2} e f +3 a \,b^{2} c^{2} d^{3} e^{2}-c^{5} b^{3} f^{2}+2 b^{3} c^{4} d e f -b^{3} c^{3} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}\) | \(704\) |
Input:
int((b*x+a)^3*(f*x+e)^(3/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
2/((c*f-d*e)*d)^(1/2)*(-((c*f-d*e)*d)^(1/2)*((-8/315*(f*x+e)^2*(35/8*f^2*x ^2-5/2*e*f*x+e^2)*d^4-2/35*c*(f*x+e)^2*f*(-5/2*f*x+e)*d^3-1/5*c^2*f^2*(f*x +e)^2*d^2+4/3*c^3*(1/4*f*x+e)*f^3*d-c^4*f^4)*b^3+3*a*d*(2/35*(f*x+e)^2*(-5 /2*f*x+e)*d^3+1/5*c*f*(f*x+e)^2*d^2-4/3*c^2*(1/4*f*x+e)*f^2*d+c^3*f^3)*f*b ^2-3*(1/5*(f*x+e)^2*d^2-4/3*c*(1/4*f*x+e)*f*d+c^2*f^2)*a^2*d^2*f^2*b+a^3*d ^3*(1/3*(-f*x-4*e)*d+c*f)*f^3)*(f*x+e)^(1/2)+f^3*(c*f-d*e)^2*(a*d-b*c)^3*a rctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))/f^3/d^5
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (216) = 432\).
Time = 0.12 (sec) , antiderivative size = 1165, normalized size of antiderivative = 4.77 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^3*(f*x+e)^(3/2)/(d*x+c),x, algorithm="fricas")
Output:
[1/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4)*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*(35*b^3*d^4*f^4*x^4 + 8*b^3*d^4*e^4 + 18*(b^3*c*d^3 - 3*a *b^2*d^4)*e^3*f + 63*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e^2*f^2 - 420*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 + 315*( b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4 + 5*(10*b^3*d^4 *e*f^3 - 9*(b^3*c*d^3 - 3*a*b^2*d^4)*f^4)*x^3 + 3*(b^3*d^4*e^2*f^2 - 24*(b ^3*c*d^3 - 3*a*b^2*d^4)*e*f^3 + 21*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b* d^4)*f^4)*x^2 - (4*b^3*d^4*e^3*f + 9*(b^3*c*d^3 - 3*a*b^2*d^4)*e^2*f^2 - 1 26*(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e*f^3 + 105*(b^3*c^3*d - 3* a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*f^4)*x)*sqrt(f*x + e))/(d^5*f^3), 2/315*(315*((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 - (b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c*f)) + (35 *b^3*d^4*f^4*x^4 + 8*b^3*d^4*e^4 + 18*(b^3*c*d^3 - 3*a*b^2*d^4)*e^3*f + 63 *(b^3*c^2*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*e^2*f^2 - 420*(b^3*c^3*d - 3* a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*e*f^3 + 315*(b^3*c^4 - 3*a*b^2*c^ 3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*f^4 + 5*(10*b^3*d^4*e*f^3 - 9*(b^3*c*d^ 3 - 3*a*b^2*d^4)*f^4)*x^3 + 3*(b^3*d^4*e^2*f^2 - 24*(b^3*c*d^3 - 3*a*b^...
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (235) = 470\).
Time = 5.80 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} \left (e + f x\right )^{\frac {9}{2}}}{9 d f^{2}} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{7 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{5 d^{3} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{3 d^{4}} + \frac {\sqrt {e + f x} \left (- a^{3} c d^{3} f^{2} + a^{3} d^{4} e f + 3 a^{2} b c^{2} d^{2} f^{2} - 3 a^{2} b c d^{3} e f - 3 a b^{2} c^{3} d f^{2} + 3 a b^{2} c^{2} d^{2} e f + b^{3} c^{4} f^{2} - b^{3} c^{3} d e f\right )}{d^{5}} + \frac {f \left (a d - b c\right )^{3} \left (c f - d e\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{6} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (\frac {b^{3} x^{3}}{3 d} + \frac {x^{2} \cdot \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac {x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac {\left (a d - b c\right )^{3} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**3*(f*x+e)**(3/2)/(d*x+c),x)
Output:
Piecewise((2*(b**3*(e + f*x)**(9/2)/(9*d*f**2) + (e + f*x)**(7/2)*(3*a*b** 2*d*f - b**3*c*f - 2*b**3*d*e)/(7*d**2*f**2) + (e + f*x)**(5/2)*(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)/(5*d**3*f**2) + (e + f*x)**(3/2)*(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)/(3*d**4) + sqrt(e + f *x)*(-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)/d**5 + f*(a*d - b*c)**3*(c*f - d*e)**2*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**6*sqrt((c*f - d*e)/d)))/f, Ne(f, 0)), (e** (3/2)*(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. \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^3*(f*x+e)^(3/2)/(d*x+c),x, algorithm="maxima")
Output:
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. 652 vs. \(2 (216) = 432\).
Time = 0.14 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.67 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx=-\frac {2 \, {\left (b^{3} c^{3} d^{2} e^{2} - 3 \, a b^{2} c^{2} d^{3} e^{2} + 3 \, a^{2} b c d^{4} e^{2} - a^{3} d^{5} e^{2} - 2 \, b^{3} c^{4} d e f + 6 \, a b^{2} c^{3} d^{2} e f - 6 \, a^{2} b c^{2} d^{3} e f + 2 \, a^{3} c d^{4} e f + b^{3} c^{5} f^{2} - 3 \, a b^{2} c^{4} d f^{2} + 3 \, a^{2} b c^{3} d^{2} f^{2} - a^{3} c^{2} d^{3} f^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{5}} + \frac {2 \, {\left (35 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{3} d^{8} f^{24} - 90 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{8} e f^{24} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{8} e^{2} f^{24} - 45 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} c d^{7} f^{25} + 135 \, {\left (f x + e\right )}^{\frac {7}{2}} a b^{2} d^{8} f^{25} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c d^{7} e f^{25} - 189 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} d^{8} e f^{25} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c^{2} d^{6} f^{26} - 189 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} c d^{7} f^{26} + 189 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} b d^{8} f^{26} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{3} d^{5} f^{27} + 315 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c^{2} d^{6} f^{27} - 315 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b c d^{7} f^{27} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{3} d^{8} f^{27} - 315 \, \sqrt {f x + e} b^{3} c^{3} d^{5} e f^{27} + 945 \, \sqrt {f x + e} a b^{2} c^{2} d^{6} e f^{27} - 945 \, \sqrt {f x + e} a^{2} b c d^{7} e f^{27} + 315 \, \sqrt {f x + e} a^{3} d^{8} e f^{27} + 315 \, \sqrt {f x + e} b^{3} c^{4} d^{4} f^{28} - 945 \, \sqrt {f x + e} a b^{2} c^{3} d^{5} f^{28} + 945 \, \sqrt {f x + e} a^{2} b c^{2} d^{6} f^{28} - 315 \, \sqrt {f x + e} a^{3} c d^{7} f^{28}\right )}}{315 \, d^{9} f^{27}} \] Input:
integrate((b*x+a)^3*(f*x+e)^(3/2)/(d*x+c),x, algorithm="giac")
Output:
-2*(b^3*c^3*d^2*e^2 - 3*a*b^2*c^2*d^3*e^2 + 3*a^2*b*c*d^4*e^2 - a^3*d^5*e^ 2 - 2*b^3*c^4*d*e*f + 6*a*b^2*c^3*d^2*e*f - 6*a^2*b*c^2*d^3*e*f + 2*a^3*c* d^4*e*f + b^3*c^5*f^2 - 3*a*b^2*c^4*d*f^2 + 3*a^2*b*c^3*d^2*f^2 - a^3*c^2* d^3*f^2)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f )*d^5) + 2/315*(35*(f*x + e)^(9/2)*b^3*d^8*f^24 - 90*(f*x + e)^(7/2)*b^3*d ^8*e*f^24 + 63*(f*x + e)^(5/2)*b^3*d^8*e^2*f^24 - 45*(f*x + e)^(7/2)*b^3*c *d^7*f^25 + 135*(f*x + e)^(7/2)*a*b^2*d^8*f^25 + 63*(f*x + e)^(5/2)*b^3*c* d^7*e*f^25 - 189*(f*x + e)^(5/2)*a*b^2*d^8*e*f^25 + 63*(f*x + e)^(5/2)*b^3 *c^2*d^6*f^26 - 189*(f*x + e)^(5/2)*a*b^2*c*d^7*f^26 + 189*(f*x + e)^(5/2) *a^2*b*d^8*f^26 - 105*(f*x + e)^(3/2)*b^3*c^3*d^5*f^27 + 315*(f*x + e)^(3/ 2)*a*b^2*c^2*d^6*f^27 - 315*(f*x + e)^(3/2)*a^2*b*c*d^7*f^27 + 105*(f*x + e)^(3/2)*a^3*d^8*f^27 - 315*sqrt(f*x + e)*b^3*c^3*d^5*e*f^27 + 945*sqrt(f* x + e)*a*b^2*c^2*d^6*e*f^27 - 945*sqrt(f*x + e)*a^2*b*c*d^7*e*f^27 + 315*s qrt(f*x + e)*a^3*d^8*e*f^27 + 315*sqrt(f*x + e)*b^3*c^4*d^4*f^28 - 945*sqr t(f*x + e)*a*b^2*c^3*d^5*f^28 + 945*sqrt(f*x + e)*a^2*b*c^2*d^6*f^28 - 315 *sqrt(f*x + e)*a^3*c*d^7*f^28)/(d^9*f^27)
Time = 1.12 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.75 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx={\left (e+f\,x\right )}^{5/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{5\,d\,f^3}\right )-{\left (e+f\,x\right )}^{7/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{7\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{7\,d^2\,f^6}\right )+{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{3\,d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{9/2}}{9\,d\,f^3}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{3/2}}{a^3\,c^2\,d^3\,f^2-2\,a^3\,c\,d^4\,e\,f+a^3\,d^5\,e^2-3\,a^2\,b\,c^3\,d^2\,f^2+6\,a^2\,b\,c^2\,d^3\,e\,f-3\,a^2\,b\,c\,d^4\,e^2+3\,a\,b^2\,c^4\,d\,f^2-6\,a\,b^2\,c^3\,d^2\,e\,f+3\,a\,b^2\,c^2\,d^3\,e^2-b^3\,c^5\,f^2+2\,b^3\,c^4\,d\,e\,f-b^3\,c^3\,d^2\,e^2}\right )\,{\left (a\,d-b\,c\right )}^3\,{\left (c\,f-d\,e\right )}^{3/2}}{d^{11/2}}-\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3} \] Input:
int(((e + f*x)^(3/2)*(a + b*x)^3)/(c + d*x),x)
Output:
(e + f*x)^(5/2)*((((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))/(5*d*f^3) + (6*b*(a*f - b*e)^2)/(5*d*f^3) ) - (e + f*x)^(7/2)*((6*b^3*e - 6*a*b^2*f)/(7*d*f^3) + (2*b^3*(c*f^4 - d*e *f^3))/(7*d^2*f^6)) + (e + f*x)^(3/2)*((2*(a*f - b*e)^3)/(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))/(3* d*f^3)) + (2*b^3*(e + f*x)^(9/2))/(9*d*f^3) + (2*atan((d^(1/2)*(e + f*x)^( 1/2)*(a*d - b*c)^3*(c*f - d*e)^(3/2))/(a^3*d^5*e^2 - b^3*c^5*f^2 + a^3*c^2 *d^3*f^2 - b^3*c^3*d^2*e^2 - 2*a^3*c*d^4*e*f + 2*b^3*c^4*d*e*f - 3*a^2*b*c *d^4*e^2 + 3*a*b^2*c^4*d*f^2 + 3*a*b^2*c^2*d^3*e^2 - 3*a^2*b*c^3*d^2*f^2 - 6*a*b^2*c^3*d^2*e*f + 6*a^2*b*c^2*d^3*e*f))*(a*d - b*c)^3*(c*f - d*e)^(3/ 2))/d^(11/2) - ((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))/(d*f^3)
Time = 0.16 (sec) , antiderivative size = 1098, normalized size of antiderivative = 4.50 \[ \int \frac {(a+b x)^3 (e+f x)^{3/2}}{c+d x} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(f*x+e)^(3/2)/(d*x+c),x)
Output:
(2*(315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**3*f**4 - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x) *d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**4*e*f**3 - 945*sqrt(d)*sqrt(c*f - d *e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**2*f** 4 + 945*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c*d**3*e*f**3 + 945*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c**3*d*f**4 - 945*sqrt(d)*sqrt(c *f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c**2*d* *2*e*f**3 - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sq rt(c*f - d*e)))*b**3*c**4*f**4 + 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**3*c**3*d*e*f**3 - 315*sqrt(e + f*x )*a**3*c*d**4*f**4 + 420*sqrt(e + f*x)*a**3*d**5*e*f**3 + 105*sqrt(e + f*x )*a**3*d**5*f**4*x + 945*sqrt(e + f*x)*a**2*b*c**2*d**3*f**4 - 1260*sqrt(e + f*x)*a**2*b*c*d**4*e*f**3 - 315*sqrt(e + f*x)*a**2*b*c*d**4*f**4*x + 18 9*sqrt(e + f*x)*a**2*b*d**5*e**2*f**2 + 378*sqrt(e + f*x)*a**2*b*d**5*e*f* *3*x + 189*sqrt(e + f*x)*a**2*b*d**5*f**4*x**2 - 945*sqrt(e + f*x)*a*b**2* c**3*d**2*f**4 + 1260*sqrt(e + f*x)*a*b**2*c**2*d**3*e*f**3 + 315*sqrt(e + f*x)*a*b**2*c**2*d**3*f**4*x - 189*sqrt(e + f*x)*a*b**2*c*d**4*e**2*f**2 - 378*sqrt(e + f*x)*a*b**2*c*d**4*e*f**3*x - 189*sqrt(e + f*x)*a*b**2*c*d* *4*f**4*x**2 - 54*sqrt(e + f*x)*a*b**2*d**5*e**3*f + 27*sqrt(e + f*x)*a...