Integrand size = 24, antiderivative size = 210 \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx=-\frac {2 (b c-a d)^3 \sqrt {e+f x}}{d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}} \]
2/3*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)^(3/2)/d^3/f^3-2/5*b^2*(-3*a*d*f+b*c*f+2*b*d*e)*(f*x+e)^(5/2)/d^2/f^3 +2/7*b^3*(f*x+e)^(7/2)/d/f^3+2*(-a*d+b*c)^3*arctanh(d^(1/2)*(f*x+e)^(1/2)/ (-c*f+d*e)^(1/2))*(-c*f+d*e)^(1/2)/d^(9/2)-2*(-a*d+b*c)^3*(f*x+e)^(1/2)/d^ 4
Time = 0.33 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (105 a^3 d^3 f^3+105 a^2 b d^2 f^2 (-3 c f+d (e+f x))-21 a b^2 d f \left (-15 c^2 f^2+5 c d f (e+f x)+d^2 \left (2 e^2-e f x-3 f^2 x^2\right )\right )+b^3 \left (-105 c^3 f^3+35 c^2 d f^2 (e+f x)-7 c d^2 f \left (-2 e^2+e f x+3 f^2 x^2\right )+d^3 \left (8 e^3-4 e^2 f x+3 e f^2 x^2+15 f^3 x^3\right )\right )\right )}{105 d^4 f^3}-\frac {2 (-b c+a d)^3 \sqrt {-d e+c f} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{9/2}} \]
(2*Sqrt[e + f*x]*(105*a^3*d^3*f^3 + 105*a^2*b*d^2*f^2*(-3*c*f + d*(e + f*x )) - 21*a*b^2*d*f*(-15*c^2*f^2 + 5*c*d*f*(e + f*x) + d^2*(2*e^2 - e*f*x - 3*f^2*x^2)) + b^3*(-105*c^3*f^3 + 35*c^2*d*f^2*(e + f*x) - 7*c*d^2*f*(-2*e ^2 + e*f*x + 3*f^2*x^2) + d^3*(8*e^3 - 4*e^2*f*x + 3*e*f^2*x^2 + 15*f^3*x^ 3))))/(105*d^4*f^3) - (2*(-(b*c) + a*d)^3*Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[ d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(9/2)
Time = 0.35 (sec) , antiderivative size = 210, 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.
| \(\displaystyle \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {b \sqrt {e+f x} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{d^3 f^2}-\frac {b^2 (e+f x)^{3/2} (-3 a d f+b c f+2 b d e)}{d^2 f^2}+\frac {\sqrt {e+f x} (a d-b c)^3}{d^3 (c+d x)}+\frac {b^3 (e+f x)^{5/2}}{d f^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}-\frac {2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}-\frac {2 \sqrt {e+f x} (b c-a d)^3}{d^4}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}\) |
(-2*(b*c - a*d)^3*Sqrt[e + f*x])/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)^(3/2))/(3*d^3*f^3) - (2*b^2*(2*b*d*e + b*c*f - 3*a*d*f)*(e + f*x)^(5/2))/(5*d^2*f^3) + (2*b^ 3*(e + f*x)^(7/2))/(7*d*f^3) + (2*(b*c - a*d)^3*Sqrt[d*e - c*f]*ArcTanh[(S qrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(9/2)
3.18.84.3.1 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]))
Time = 1.07 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.14
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\left (\left (\frac {8 \left (f x +e \right ) \left (\frac {15}{8} f^{2} x^{2}-\frac {3}{2} e f x +e^{2}\right ) d^{3}}{105}+\frac {2 \left (-\frac {3 f x}{2}+e \right ) f \left (f x +e \right ) c \,d^{2}}{15}+\frac {c^{2} f^{2} \left (f x +e \right ) d}{3}-c^{3} f^{3}\right ) b^{3}+3 f d a \left (-\frac {2 \left (-\frac {3 f x}{2}+e \right ) \left (f x +e \right ) d^{2}}{15}-\frac {c f \left (f x +e \right ) d}{3}+c^{2} f^{2}\right ) b^{2}-3 \left (\frac {\left (-f x -e \right ) d}{3}+c f \right ) f^{2} d^{2} a^{2} b +a^{3} d^{3} f^{3}\right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}+f^{3} \left (a d -b c \right )^{3} \left (c f -d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, f^{3} d^{4}}\) | \(239\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {3 a \,b^{2} d^{3} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{2} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d^{3} e \left (f x +e \right )^{\frac {5}{2}}}{5}+a^{2} b \,d^{3} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c \,d^{2} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} d^{3} e f \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{2} d \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} c \,d^{2} e f \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} d^{3} e^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d^{3} f^{3} \sqrt {f x +e}-3 a^{2} b c \,d^{2} f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{2} d \,f^{3} \sqrt {f x +e}-b^{3} c^{3} f^{3} \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{3} \left (a^{3} c \,d^{3} f -a^{3} e \,d^{4}-3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -c^{4} b^{3} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(390\) |
| default | \(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {3 a \,b^{2} d^{3} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{2} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d^{3} e \left (f x +e \right )^{\frac {5}{2}}}{5}+a^{2} b \,d^{3} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c \,d^{2} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} d^{3} e f \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{2} d \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} c \,d^{2} e f \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} d^{3} e^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d^{3} f^{3} \sqrt {f x +e}-3 a^{2} b c \,d^{2} f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{2} d \,f^{3} \sqrt {f x +e}-b^{3} c^{3} f^{3} \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{3} \left (a^{3} c \,d^{3} f -a^{3} e \,d^{4}-3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -c^{4} b^{3} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(390\) |
| risch | \(\frac {2 \left (15 b^{3} d^{3} f^{3} x^{3}+63 a \,b^{2} d^{3} f^{3} x^{2}-21 x^{2} b^{3} c \,d^{2} f^{3}+3 x^{2} b^{3} d^{3} e \,f^{2}+105 a^{2} b \,d^{3} f^{3} x -105 x a \,b^{2} c \,d^{2} f^{3}+21 x a \,b^{2} d^{3} e \,f^{2}+35 x \,b^{3} c^{2} d \,f^{3}-7 b^{3} c \,d^{2} e \,f^{2} x -4 x \,b^{3} d^{3} e^{2} f +105 a^{3} d^{3} f^{3}-315 a^{2} b c \,d^{2} f^{3}+105 a^{2} b \,d^{3} e \,f^{2}+315 a \,b^{2} c^{2} d \,f^{3}-105 a \,b^{2} c \,d^{2} e \,f^{2}-42 a \,b^{2} d^{3} e^{2} f -105 b^{3} c^{3} f^{3}+35 b^{3} c^{2} d e \,f^{2}+14 b^{3} c \,d^{2} e^{2} f +8 b^{3} d^{3} e^{3}\right ) \sqrt {f x +e}}{105 f^{3} d^{4}}-\frac {2 \left (a^{3} c \,d^{3} f -a^{3} e \,d^{4}-3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -c^{4} b^{3} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}\) | \(407\) |
-2*(-((8/105*(f*x+e)*(15/8*f^2*x^2-3/2*e*f*x+e^2)*d^3+2/15*(-3/2*f*x+e)*f* (f*x+e)*c*d^2+1/3*c^2*f^2*(f*x+e)*d-c^3*f^3)*b^3+3*f*d*a*(-2/15*(-3/2*f*x+ e)*(f*x+e)*d^2-1/3*c*f*(f*x+e)*d+c^2*f^2)*b^2-3*(1/3*(-f*x-e)*d+c*f)*f^2*d ^2*a^2*b+a^3*d^3*f^3)*((c*f-d*e)*d)^(1/2)*(f*x+e)^(1/2)+f^3*(a*d-b*c)^3*(c *f-d*e)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))/((c*f-d*e)*d)^(1/2)/f ^3/d^4
Time = 0.24 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.36 \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx=\left [-\frac {105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt {\frac {d e - c f}{d}} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, \sqrt {f x + e} d \sqrt {\frac {d e - c f}{d}}}{d x + c}\right ) - 2 \, {\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \, {\left (b^{3} d^{3} e f^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} - {\left (4 \, b^{3} d^{3} e^{2} f + 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}}{105 \, d^{4} f^{3}}, \frac {2 \, {\left (105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt {-\frac {d e - c f}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {-\frac {d e - c f}{d}}}{d e - c f}\right ) + {\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \, {\left (b^{3} d^{3} e f^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} - {\left (4 \, b^{3} d^{3} e^{2} f + 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{3}}\right ] \]
[-1/105*(105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3*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*(15*b^3*d^3*f^3*x^3 + 8*b^3*d^3*e^3 + 14*(b^3*c*d^2 - 3*a*b^2*d^3)*e^2*f + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)*e*f^2 - 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3 + 3*(b^3*d^3 *e*f^2 - 7*(b^3*c*d^2 - 3*a*b^2*d^3)*f^3)*x^2 - (4*b^3*d^3*e^2*f + 7*(b^3* c*d^2 - 3*a*b^2*d^3)*e*f^2 - 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b*d^3)* f^3)*x)*sqrt(f*x + e))/(d^4*f^3), 2/105*(105*(b^3*c^3 - 3*a*b^2*c^2*d + 3* a^2*b*c*d^2 - a^3*d^3)*f^3*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*sq rt(-(d*e - c*f)/d)/(d*e - c*f)) + (15*b^3*d^3*f^3*x^3 + 8*b^3*d^3*e^3 + 14 *(b^3*c*d^2 - 3*a*b^2*d^3)*e^2*f + 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3*a^2*b *d^3)*e*f^2 - 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*f^3 + 3*(b^3*d^3*e*f^2 - 7*(b^3*c*d^2 - 3*a*b^2*d^3)*f^3)*x^2 - (4*b^3*d^3*e^2 *f + 7*(b^3*c*d^2 - 3*a*b^2*d^3)*e*f^2 - 35*(b^3*c^2*d - 3*a*b^2*c*d^2 + 3 *a^2*b*d^3)*f^3)*x)*sqrt(f*x + e))/(d^4*f^3)]
Time = 4.54 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} \left (e + f x\right )^{\frac {7}{2}}}{7 d f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{5 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \cdot \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{3 d^{3} f^{2}} + \frac {\sqrt {e + f x} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{d^{4}} - \frac {f \left (a d - b c\right )^{3} \left (c f - d e\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\\sqrt {e} \left (\frac {b^{3} x^{3}}{3 d} + \frac {x^{2} \cdot \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac {x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac {\left (a d - b c\right )^{3} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**3*(e + f*x)**(7/2)/(7*d*f**2) + (e + f*x)**(5/2)*(3*a*b** 2*d*f - b**3*c*f - 2*b**3*d*e)/(5*d**2*f**2) + (e + f*x)**(3/2)*(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)/(3*d**3*f**2) + sqrt(e + f*x)*(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)/d**4 - f*(a*d - b*c)**3* (c*f - d*e)*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(d**5*sqrt((c*f - d*e) /d)))/f, Ne(f, 0)), (sqrt(e)*(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. \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx=\text {Exception raised: ValueError} \]
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. 413 vs. \(2 (186) = 372\).
Time = 0.28 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx=-\frac {2 \, {\left (b^{3} c^{3} d e - 3 \, a b^{2} c^{2} d^{2} e + 3 \, a^{2} b c d^{3} e - a^{3} d^{4} e - b^{3} c^{4} f + 3 \, a b^{2} c^{3} d f - 3 \, a^{2} b c^{2} d^{2} f + a^{3} c d^{3} f\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{4}} + \frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{6} f^{18} - 42 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{6} e f^{18} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{6} e^{2} f^{18} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c d^{5} f^{19} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} d^{6} f^{19} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c d^{5} e f^{19} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d^{6} e f^{19} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} f^{20} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c d^{5} f^{20} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d^{6} f^{20} - 105 \, \sqrt {f x + e} b^{3} c^{3} d^{3} f^{21} + 315 \, \sqrt {f x + e} a b^{2} c^{2} d^{4} f^{21} - 315 \, \sqrt {f x + e} a^{2} b c d^{5} f^{21} + 105 \, \sqrt {f x + e} a^{3} d^{6} f^{21}\right )}}{105 \, d^{7} f^{21}} \]
-2*(b^3*c^3*d*e - 3*a*b^2*c^2*d^2*e + 3*a^2*b*c*d^3*e - a^3*d^4*e - b^3*c^ 4*f + 3*a*b^2*c^3*d*f - 3*a^2*b*c^2*d^2*f + a^3*c*d^3*f)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^4) + 2/105*(15*(f*x + e)^(7/2)*b^3*d^6*f^18 - 42*(f*x + e)^(5/2)*b^3*d^6*e*f^18 + 35*(f*x + e)^( 3/2)*b^3*d^6*e^2*f^18 - 21*(f*x + e)^(5/2)*b^3*c*d^5*f^19 + 63*(f*x + e)^( 5/2)*a*b^2*d^6*f^19 + 35*(f*x + e)^(3/2)*b^3*c*d^5*e*f^19 - 105*(f*x + e)^ (3/2)*a*b^2*d^6*e*f^19 + 35*(f*x + e)^(3/2)*b^3*c^2*d^4*f^20 - 105*(f*x + e)^(3/2)*a*b^2*c*d^5*f^20 + 105*(f*x + e)^(3/2)*a^2*b*d^6*f^20 - 105*sqrt( f*x + e)*b^3*c^3*d^3*f^21 + 315*sqrt(f*x + e)*a*b^2*c^2*d^4*f^21 - 315*sqr t(f*x + e)*a^2*b*c*d^5*f^21 + 105*sqrt(f*x + e)*a^3*d^6*f^21)/(d^7*f^21)
Time = 0.16 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x)^3 \sqrt {e+f x}}{c+d x} \, dx={\left (e+f\,x\right )}^{3/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}+\frac {2\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )-{\left (e+f\,x\right )}^{5/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{5\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d^2\,f^6}\right )+\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\sqrt {d\,e-c\,f}\,1{}\mathrm {i}}{-f\,a^3\,c\,d^3+e\,a^3\,d^4+3\,f\,a^2\,b\,c^2\,d^2-3\,e\,a^2\,b\,c\,d^3-3\,f\,a\,b^2\,c^3\,d+3\,e\,a\,b^2\,c^2\,d^2+f\,b^3\,c^4-e\,b^3\,c^3\,d}\right )\,{\left (a\,d-b\,c\right )}^3\,\sqrt {d\,e-c\,f}\,2{}\mathrm {i}}{d^{9/2}} \]
(e + f*x)^(3/2)*((((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))/(3*d*f^3) + (2*b*(a*f - b*e)^2)/(d*f^3)) - (e + f*x)^(5/2)*((6*b^3*e - 6*a*b^2*f)/(5*d*f^3) + (2*b^3*(c*f^4 - d*e*f ^3))/(5*d^2*f^6)) + (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) ) + (2*b^3*(e + f*x)^(7/2))/(7*d*f^3) + (atan((d^(1/2)*(e + f*x)^(1/2)*(a* d - b*c)^3*(d*e - c*f)^(1/2)*1i)/(a^3*d^4*e + b^3*c^4*f - a^3*c*d^3*f - b^ 3*c^3*d*e - 3*a^2*b*c*d^3*e - 3*a*b^2*c^3*d*f + 3*a*b^2*c^2*d^2*e + 3*a^2* b*c^2*d^2*f))*(a*d - b*c)^3*(d*e - c*f)^(1/2)*2i)/d^(9/2)