Integrand size = 24, antiderivative size = 260 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=-\frac {2 (b e-a f)^3}{7 f^3 (d e-c f) (e+f x)^{7/2}}+\frac {2 (b e-a f)^2 (2 b d e-3 b c f+a d f)}{5 f^3 (d e-c f)^2 (e+f x)^{5/2}}-\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )\right )}{3 f^3 (d e-c f)^3 (e+f x)^{3/2}}-\frac {2 (b c-a d)^3}{(d e-c f)^4 \sqrt {e+f x}}+\frac {2 \sqrt {d} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}} \]
-2/7*(-a*f+b*e)^3/f^3/(-c*f+d*e)/(f*x+e)^(7/2)+2/5*(-a*f+b*e)^2*(a*d*f-3*b *c*f+2*b*d*e)/f^3/(-c*f+d*e)^2/(f*x+e)^(5/2)-2/3*(-a*f+b*e)*(a^2*d^2*f^2+a *b*d*f*(-3*c*f+d*e)+b^2*(3*c^2*f^2-3*c*d*e*f+d^2*e^2))/f^3/(-c*f+d*e)^3/(f *x+e)^(3/2)+2*(-a*d+b*c)^3*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2)) *d^(1/2)/(-c*f+d*e)^(9/2)-2*(-a*d+b*c)^3/(-c*f+d*e)^4/(f*x+e)^(1/2)
Time = 0.80 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=-\frac {2 \left (3 a b^2 f \left (3 d^3 e^4 (2 e+7 f x)-3 c d^2 e^3 f (13 e+28 f x)+c^3 f^3 \left (8 e^2+28 e f x+35 f^2 x^2\right )-5 c^2 d f^2 \left (16 e^3+56 e^2 f x+70 e f^2 x^2+21 f^3 x^3\right )\right )+b^3 \left (d^3 e^4 \left (8 e^2+28 e f x+35 f^2 x^2\right )+3 c^2 d e^2 f^2 \left (29 e^2+84 e f x+70 f^2 x^2\right )-c d^2 e^3 f \left (38 e^2+133 e f x+140 f^2 x^2\right )+3 c^3 f^3 \left (16 e^3+56 e^2 f x+70 e f^2 x^2+35 f^3 x^3\right )\right )+3 a^2 b f^2 \left (15 d^3 e^4+3 c^3 f^3 (2 e+7 f x)-c^2 d f^2 \left (32 e^2+112 e f x+35 f^2 x^2\right )+c d^2 f \left (116 e^3+406 e^2 f x+350 e f^2 x^2+105 f^3 x^3\right )\right )-a^3 f^3 \left (-15 c^3 f^3+3 c^2 d f^2 (22 e+7 f x)-c d^2 f \left (122 e^2+112 e f x+35 f^2 x^2\right )+d^3 \left (176 e^3+406 e^2 f x+350 e f^2 x^2+105 f^3 x^3\right )\right )\right )}{105 f^3 (d e-c f)^4 (e+f x)^{7/2}}+\frac {2 \sqrt {d} (-b c+a d)^3 \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{9/2}} \]
(-2*(3*a*b^2*f*(3*d^3*e^4*(2*e + 7*f*x) - 3*c*d^2*e^3*f*(13*e + 28*f*x) + c^3*f^3*(8*e^2 + 28*e*f*x + 35*f^2*x^2) - 5*c^2*d*f^2*(16*e^3 + 56*e^2*f*x + 70*e*f^2*x^2 + 21*f^3*x^3)) + b^3*(d^3*e^4*(8*e^2 + 28*e*f*x + 35*f^2*x ^2) + 3*c^2*d*e^2*f^2*(29*e^2 + 84*e*f*x + 70*f^2*x^2) - c*d^2*e^3*f*(38*e ^2 + 133*e*f*x + 140*f^2*x^2) + 3*c^3*f^3*(16*e^3 + 56*e^2*f*x + 70*e*f^2* x^2 + 35*f^3*x^3)) + 3*a^2*b*f^2*(15*d^3*e^4 + 3*c^3*f^3*(2*e + 7*f*x) - c ^2*d*f^2*(32*e^2 + 112*e*f*x + 35*f^2*x^2) + c*d^2*f*(116*e^3 + 406*e^2*f* x + 350*e*f^2*x^2 + 105*f^3*x^3)) - a^3*f^3*(-15*c^3*f^3 + 3*c^2*d*f^2*(22 *e + 7*f*x) - c*d^2*f*(122*e^2 + 112*e*f*x + 35*f^2*x^2) + d^3*(176*e^3 + 406*e^2*f*x + 350*e*f^2*x^2 + 105*f^3*x^3))))/(105*f^3*(d*e - c*f)^4*(e + f*x)^(7/2)) + (2*Sqrt[d]*(-(b*c) + a*d)^3*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/S qrt[-(d*e) + c*f]])/(-(d*e) + c*f)^(9/2)
Time = 0.48 (sec) , antiderivative size = 260, 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 = {98, 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.
| \(\displaystyle \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 98 |
| \(\displaystyle \int \left (\frac {-a^3 d^2 f^3+3 a^2 b c d f^3-3 a b^2 c^2 f^3+b^3 e \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )}{f^2 (e+f x)^{5/2} (d e-c f)^3}+\frac {(a f-b e)^2 (-a d f+3 b c f-2 b d e)}{f^2 (e+f x)^{7/2} (c f-d e)^2}+\frac {(a f-b e)^3}{f^2 (e+f x)^{9/2} (c f-d e)}+\frac {f (b c-a d)^3}{(e+f x)^{3/2} (c f-d e)^4}+\frac {d (a d-b c)^3}{(c+d x) \sqrt {e+f x} (d e-c f)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (b e-a f) \left (a^2 d^2 f^2+a b d f (d e-3 c f)+b^2 \left (3 c^2 f^2-3 c d e f+d^2 e^2\right )\right )}{3 f^3 (e+f x)^{3/2} (d e-c f)^3}+\frac {2 \sqrt {d} (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(d e-c f)^{9/2}}+\frac {2 (b e-a f)^2 (a d f-3 b c f+2 b d e)}{5 f^3 (e+f x)^{5/2} (d e-c f)^2}-\frac {2 (b e-a f)^3}{7 f^3 (e+f x)^{7/2} (d e-c f)}-\frac {2 (b c-a d)^3}{\sqrt {e+f x} (d e-c f)^4}\) |
(-2*(b*e - a*f)^3)/(7*f^3*(d*e - c*f)*(e + f*x)^(7/2)) + (2*(b*e - a*f)^2* (2*b*d*e - 3*b*c*f + a*d*f))/(5*f^3*(d*e - c*f)^2*(e + f*x)^(5/2)) - (2*(b *e - a*f)*(a^2*d^2*f^2 + a*b*d*f*(d*e - 3*c*f) + b^2*(d^2*e^2 - 3*c*d*e*f + 3*c^2*f^2)))/(3*f^3*(d*e - c*f)^3*(e + f*x)^(3/2)) - (2*(b*c - a*d)^3)/( (d*e - c*f)^4*Sqrt[e + f*x]) + (2*Sqrt[d]*(b*c - a*d)^3*ArcTanh[(Sqrt[d]*S qrt[e + f*x])/Sqrt[d*e - c*f]])/(d*e - c*f)^(9/2)
3.18.89.3.1 Defintions of rubi rules used
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x _)), x_] :> Int[ExpandIntegrand[(e + f*x)^FractionalPart[p], (c + d*x)^n*(( e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]
Time = 1.16 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {2 d \,f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}-\frac {2 \left (-a^{3} d \,f^{3}+3 a^{2} b c \,f^{3}-6 a \,b^{2} c e \,f^{2}+3 a \,b^{2} d \,e^{2} f +3 b^{3} c \,e^{2} f -2 b^{3} d \,e^{3}\right )}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}-\frac {2 \left (a^{3} d^{2} f^{3}-3 a^{2} b c d \,f^{3}+3 a \,b^{2} c^{2} f^{3}-3 b^{3} c^{2} e \,f^{2}+3 b^{3} c d \,e^{2} f -b^{3} d^{2} e^{3}\right )}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{\left (c f -d e \right )^{4} \sqrt {f x +e}}}{f^{3}}\) | \(371\) |
| default | \(\frac {\frac {2 d \,f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}{7 \left (c f -d e \right ) \left (f x +e \right )^{\frac {7}{2}}}-\frac {2 \left (-a^{3} d \,f^{3}+3 a^{2} b c \,f^{3}-6 a \,b^{2} c e \,f^{2}+3 a \,b^{2} d \,e^{2} f +3 b^{3} c \,e^{2} f -2 b^{3} d \,e^{3}\right )}{5 \left (c f -d e \right )^{2} \left (f x +e \right )^{\frac {5}{2}}}-\frac {2 \left (a^{3} d^{2} f^{3}-3 a^{2} b c d \,f^{3}+3 a \,b^{2} c^{2} f^{3}-3 b^{3} c^{2} e \,f^{2}+3 b^{3} c d \,e^{2} f -b^{3} d^{2} e^{3}\right )}{3 \left (c f -d e \right )^{3} \left (f x +e \right )^{\frac {3}{2}}}+\frac {2 f^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{\left (c f -d e \right )^{4} \sqrt {f x +e}}}{f^{3}}\) | \(371\) |
| pseudoelliptic | \(-\frac {2 \left (-7 d \,f^{3} \left (f x +e \right )^{\frac {7}{2}} \left (a d -b c \right )^{3} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (\left (-7 x^{3} a^{3} d^{3}+\frac {7 a^{2} c \,x^{2} \left (9 b x +a \right ) d^{2}}{3}-\frac {7 a \,c^{2} x \left (15 b^{2} x^{2}+5 a b x +a^{2}\right ) d}{5}+c^{3} \left (7 b^{3} x^{3}+a^{3}+7 a \,b^{2} x^{2}+\frac {21}{5} a^{2} b x \right )\right ) f^{6}-\frac {22 e \left (\frac {175 a^{3} d^{3} x^{2}}{33}-\frac {56 x c \,a^{2} \left (\frac {75 b x}{8}+a \right ) d^{2}}{33}+a \,c^{2} \left (\frac {175}{11} b^{2} x^{2}+\frac {56}{11} a b x +a^{2}\right ) d -\frac {3 b \left (\frac {35}{3} b^{2} x^{2}+\frac {14}{3} a b x +a^{2}\right ) c^{3}}{11}\right ) f^{5}}{5}+\frac {122 e^{2} \left (-\frac {203 a^{3} d^{3} x}{61}+a^{2} c \left (\frac {609 b x}{61}+a \right ) d^{2}-\frac {48 \left (-\frac {35}{16} b^{2} x^{2}+\frac {35}{4} a b x +a^{2}\right ) b \,c^{2} d}{61}+\frac {12 b^{2} c^{3} \left (7 b x +a \right )}{61}\right ) f^{4}}{15}-\frac {176 e^{3} \left (a^{3} d^{3}-\frac {87 \left (-\frac {35}{87} b^{2} x^{2}-\frac {21}{29} a b x +a^{2}\right ) b c \,d^{2}}{44}+\frac {15 b^{2} \left (-\frac {21 b x}{20}+a \right ) c^{2} d}{11}-\frac {3 b^{3} c^{3}}{11}\right ) f^{3}}{15}+3 e^{4} b d \left (\left (\frac {7}{9} b^{2} x^{2}+\frac {7}{5} a b x +a^{2}\right ) d^{2}-\frac {13 b \left (\frac {133 b x}{117}+a \right ) c d}{5}+\frac {29 b^{2} c^{2}}{15}\right ) f^{2}+\frac {6 \left (\left (\frac {14 b x}{9}+a \right ) d -\frac {19 b c}{9}\right ) e^{5} b^{2} d^{2} f}{5}+\frac {8 b^{3} d^{3} e^{6}}{15}\right ) \sqrt {\left (c f -d e \right ) d}\right )}{7 \sqrt {\left (c f -d e \right ) d}\, \left (f x +e \right )^{\frac {7}{2}} f^{3} \left (c f -d e \right )^{4}}\) | \(486\) |
2/f^3*(-1/7*(a^3*f^3-3*a^2*b*e*f^2+3*a*b^2*e^2*f-b^3*e^3)/(c*f-d*e)/(f*x+e )^(7/2)-1/5*(-a^3*d*f^3+3*a^2*b*c*f^3-6*a*b^2*c*e*f^2+3*a*b^2*d*e^2*f+3*b^ 3*c*e^2*f-2*b^3*d*e^3)/(c*f-d*e)^2/(f*x+e)^(5/2)-1/3*(a^3*d^2*f^3-3*a^2*b* c*d*f^3+3*a*b^2*c^2*f^3-3*b^3*c^2*e*f^2+3*b^3*c*d*e^2*f-b^3*d^2*e^3)/(c*f- d*e)^3/(f*x+e)^(3/2)+f^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(c* f-d*e)^4/(f*x+e)^(1/2)+d*f^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3) /(c*f-d*e)^4/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2 )))
Leaf count of result is larger than twice the leaf count of optimal. 1137 vs. \(2 (236) = 472\).
Time = 0.40 (sec) , antiderivative size = 2284, normalized size of antiderivative = 8.78 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=\text {Too large to display} \]
[-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^7*x^4 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e*f^6*x^3 + 6*(b^3 *c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^2*f^5*x^2 + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3*f^4*x + (b^3*c^3 - 3*a*b^2*c ^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^4*f^3)*sqrt(d/(d*e - c*f))*log((d*f*x + 2*d*e - c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x + c)) + 2*(8*b^3*d^3*e^6 + 15*a^3*c^3*f^6 + 105*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2 *b*c*d^2 - a^3*d^3)*f^6*x^3 - 2*(19*b^3*c*d^2 - 9*a*b^2*d^3)*e^5*f + 3*(29 *b^3*c^2*d - 39*a*b^2*c*d^2 + 15*a^2*b*d^3)*e^4*f^2 + 4*(12*b^3*c^3 - 60*a *b^2*c^2*d + 87*a^2*b*c*d^2 - 44*a^3*d^3)*e^3*f^3 + 2*(12*a*b^2*c^3 - 48*a ^2*b*c^2*d + 61*a^3*c*d^2)*e^2*f^4 + 6*(3*a^2*b*c^3 - 11*a^3*c^2*d)*e*f^5 + 35*(b^3*d^3*e^4*f^2 - 4*b^3*c*d^2*e^3*f^3 + 6*b^3*c^2*d*e^2*f^4 + 2*(3*b ^3*c^3 - 15*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 5*a^3*d^3)*e*f^5 + (3*a*b^2*c^3 - 3*a^2*b*c^2*d + a^3*c*d^2)*f^6)*x^2 + 7*(4*b^3*d^3*e^5*f - (19*b^3*c*d^ 2 - 9*a*b^2*d^3)*e^4*f^2 + 36*(b^3*c^2*d - a*b^2*c*d^2)*e^3*f^3 + 2*(12*b^ 3*c^3 - 60*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 29*a^3*d^3)*e^2*f^4 + 4*(3*a*b^2 *c^3 - 12*a^2*b*c^2*d + 4*a^3*c*d^2)*e*f^5 + 3*(3*a^2*b*c^3 - a^3*c^2*d)*f ^6)*x)*sqrt(f*x + e))/(d^4*e^8*f^3 - 4*c*d^3*e^7*f^4 + 6*c^2*d^2*e^6*f^5 - 4*c^3*d*e^5*f^6 + c^4*e^4*f^7 + (d^4*e^4*f^7 - 4*c*d^3*e^3*f^8 + 6*c^2*d^ 2*e^2*f^9 - 4*c^3*d*e*f^10 + c^4*f^11)*x^4 + 4*(d^4*e^5*f^6 - 4*c*d^3*e...
Time = 13.47 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=\begin {cases} \frac {2 \left (\frac {f \left (a d - b c\right )^{3}}{\sqrt {e + f x} \left (c f - d e\right )^{4}} + \frac {f \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (c f - d e\right )^{4}} - \frac {\left (a f - b e\right ) \left (a^{2} d^{2} f^{2} - 3 a b c d f^{2} + a b d^{2} e f + 3 b^{2} c^{2} f^{2} - 3 b^{2} c d e f + b^{2} d^{2} e^{2}\right )}{3 f^{2} \left (e + f x\right )^{\frac {3}{2}} \left (c f - d e\right )^{3}} + \frac {\left (a f - b e\right )^{2} \left (a d f - 3 b c f + 2 b d e\right )}{5 f^{2} \left (e + f x\right )^{\frac {5}{2}} \left (c f - d e\right )^{2}} - \frac {\left (a f - b e\right )^{3}}{7 f^{2} \left (e + f x\right )^{\frac {7}{2}} \left (c f - d e\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {\frac {b^{3} x^{3}}{3 d} + \frac {x^{2} \cdot \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac {x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac {\left (a d - b c\right )^{3} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{3}}}{e^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(f*(a*d - b*c)**3/(sqrt(e + f*x)*(c*f - d*e)**4) + f*(a*d - b *c)**3*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(sqrt((c*f - d*e)/d)*(c*f - d*e)**4) - (a*f - b*e)*(a**2*d**2*f**2 - 3*a*b*c*d*f**2 + a*b*d**2*e*f + 3*b**2*c**2*f**2 - 3*b**2*c*d*e*f + b**2*d**2*e**2)/(3*f**2*(e + f*x)**(3/ 2)*(c*f - d*e)**3) + (a*f - b*e)**2*(a*d*f - 3*b*c*f + 2*b*d*e)/(5*f**2*(e + f*x)**(5/2)*(c*f - d*e)**2) - (a*f - b*e)**3/(7*f**2*(e + f*x)**(7/2)*( c*f - d*e)))/f, Ne(f, 0)), ((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)/e**(9/2), Tru e))
Exception generated. \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=\text {Exception raised: ValueError} \]
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. 957 vs. \(2 (236) = 472\).
Time = 0.30 (sec) , antiderivative size = 957, normalized size of antiderivative = 3.68 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=-\frac {2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 6 \, c^{2} d^{2} e^{2} f^{2} - 4 \, c^{3} d e f^{3} + c^{4} f^{4}\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (35 \, {\left (f x + e\right )}^{2} b^{3} d^{3} e^{4} - 42 \, {\left (f x + e\right )} b^{3} d^{3} e^{5} + 15 \, b^{3} d^{3} e^{6} - 140 \, {\left (f x + e\right )}^{2} b^{3} c d^{2} e^{3} f + 147 \, {\left (f x + e\right )} b^{3} c d^{2} e^{4} f + 63 \, {\left (f x + e\right )} a b^{2} d^{3} e^{4} f - 45 \, b^{3} c d^{2} e^{5} f - 45 \, a b^{2} d^{3} e^{5} f + 210 \, {\left (f x + e\right )}^{2} b^{3} c^{2} d e^{2} f^{2} - 168 \, {\left (f x + e\right )} b^{3} c^{2} d e^{3} f^{2} - 252 \, {\left (f x + e\right )} a b^{2} c d^{2} e^{3} f^{2} + 45 \, b^{3} c^{2} d e^{4} f^{2} + 135 \, a b^{2} c d^{2} e^{4} f^{2} + 45 \, a^{2} b d^{3} e^{4} f^{2} + 105 \, {\left (f x + e\right )}^{3} b^{3} c^{3} f^{3} - 315 \, {\left (f x + e\right )}^{3} a b^{2} c^{2} d f^{3} + 315 \, {\left (f x + e\right )}^{3} a^{2} b c d^{2} f^{3} - 105 \, {\left (f x + e\right )}^{3} a^{3} d^{3} f^{3} - 105 \, {\left (f x + e\right )}^{2} b^{3} c^{3} e f^{3} - 105 \, {\left (f x + e\right )}^{2} a b^{2} c^{2} d e f^{3} + 105 \, {\left (f x + e\right )}^{2} a^{2} b c d^{2} e f^{3} - 35 \, {\left (f x + e\right )}^{2} a^{3} d^{3} e f^{3} + 63 \, {\left (f x + e\right )} b^{3} c^{3} e^{2} f^{3} + 315 \, {\left (f x + e\right )} a b^{2} c^{2} d e^{2} f^{3} + 63 \, {\left (f x + e\right )} a^{2} b c d^{2} e^{2} f^{3} - 21 \, {\left (f x + e\right )} a^{3} d^{3} e^{2} f^{3} - 15 \, b^{3} c^{3} e^{3} f^{3} - 135 \, a b^{2} c^{2} d e^{3} f^{3} - 135 \, a^{2} b c d^{2} e^{3} f^{3} - 15 \, a^{3} d^{3} e^{3} f^{3} + 105 \, {\left (f x + e\right )}^{2} a b^{2} c^{3} f^{4} - 105 \, {\left (f x + e\right )}^{2} a^{2} b c^{2} d f^{4} + 35 \, {\left (f x + e\right )}^{2} a^{3} c d^{2} f^{4} - 126 \, {\left (f x + e\right )} a b^{2} c^{3} e f^{4} - 126 \, {\left (f x + e\right )} a^{2} b c^{2} d e f^{4} + 42 \, {\left (f x + e\right )} a^{3} c d^{2} e f^{4} + 45 \, a b^{2} c^{3} e^{2} f^{4} + 135 \, a^{2} b c^{2} d e^{2} f^{4} + 45 \, a^{3} c d^{2} e^{2} f^{4} + 63 \, {\left (f x + e\right )} a^{2} b c^{3} f^{5} - 21 \, {\left (f x + e\right )} a^{3} c^{2} d f^{5} - 45 \, a^{2} b c^{3} e f^{5} - 45 \, a^{3} c^{2} d e f^{5} + 15 \, a^{3} c^{3} f^{6}\right )}}{105 \, {\left (d^{4} e^{4} f^{3} - 4 \, c d^{3} e^{3} f^{4} + 6 \, c^{2} d^{2} e^{2} f^{5} - 4 \, c^{3} d e f^{6} + c^{4} f^{7}\right )} {\left (f x + e\right )}^{\frac {7}{2}}} \]
-2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^4*e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^ 2 - 4*c^3*d*e*f^3 + c^4*f^4)*sqrt(-d^2*e + c*d*f)) - 2/105*(35*(f*x + e)^2 *b^3*d^3*e^4 - 42*(f*x + e)*b^3*d^3*e^5 + 15*b^3*d^3*e^6 - 140*(f*x + e)^2 *b^3*c*d^2*e^3*f + 147*(f*x + e)*b^3*c*d^2*e^4*f + 63*(f*x + e)*a*b^2*d^3* e^4*f - 45*b^3*c*d^2*e^5*f - 45*a*b^2*d^3*e^5*f + 210*(f*x + e)^2*b^3*c^2* d*e^2*f^2 - 168*(f*x + e)*b^3*c^2*d*e^3*f^2 - 252*(f*x + e)*a*b^2*c*d^2*e^ 3*f^2 + 45*b^3*c^2*d*e^4*f^2 + 135*a*b^2*c*d^2*e^4*f^2 + 45*a^2*b*d^3*e^4* f^2 + 105*(f*x + e)^3*b^3*c^3*f^3 - 315*(f*x + e)^3*a*b^2*c^2*d*f^3 + 315* (f*x + e)^3*a^2*b*c*d^2*f^3 - 105*(f*x + e)^3*a^3*d^3*f^3 - 105*(f*x + e)^ 2*b^3*c^3*e*f^3 - 105*(f*x + e)^2*a*b^2*c^2*d*e*f^3 + 105*(f*x + e)^2*a^2* b*c*d^2*e*f^3 - 35*(f*x + e)^2*a^3*d^3*e*f^3 + 63*(f*x + e)*b^3*c^3*e^2*f^ 3 + 315*(f*x + e)*a*b^2*c^2*d*e^2*f^3 + 63*(f*x + e)*a^2*b*c*d^2*e^2*f^3 - 21*(f*x + e)*a^3*d^3*e^2*f^3 - 15*b^3*c^3*e^3*f^3 - 135*a*b^2*c^2*d*e^3*f ^3 - 135*a^2*b*c*d^2*e^3*f^3 - 15*a^3*d^3*e^3*f^3 + 105*(f*x + e)^2*a*b^2* c^3*f^4 - 105*(f*x + e)^2*a^2*b*c^2*d*f^4 + 35*(f*x + e)^2*a^3*c*d^2*f^4 - 126*(f*x + e)*a*b^2*c^3*e*f^4 - 126*(f*x + e)*a^2*b*c^2*d*e*f^4 + 42*(f*x + e)*a^3*c*d^2*e*f^4 + 45*a*b^2*c^3*e^2*f^4 + 135*a^2*b*c^2*d*e^2*f^4 + 4 5*a^3*c*d^2*e^2*f^4 + 63*(f*x + e)*a^2*b*c^3*f^5 - 21*(f*x + e)*a^3*c^2*d* f^5 - 45*a^2*b*c^3*e*f^5 - 45*a^3*c^2*d*e*f^5 + 15*a^3*c^3*f^6)/((d^4*e...
Time = 2.15 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^3}{(c+d x) (e+f x)^{9/2}} \, dx=\frac {2\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\left (c^4\,f^4-4\,c^3\,d\,e\,f^3+6\,c^2\,d^2\,e^2\,f^2-4\,c\,d^3\,e^3\,f+d^4\,e^4\right )}{{\left (c\,f-d\,e\right )}^{9/2}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{{\left (c\,f-d\,e\right )}^{9/2}}-\frac {\frac {2\,\left (a^3\,f^3-3\,a^2\,b\,e\,f^2+3\,a\,b^2\,e^2\,f-b^3\,e^3\right )}{7\,\left (c\,f-d\,e\right )}-\frac {2\,{\left (e+f\,x\right )}^3\,\left (a^3\,d^3\,f^3-3\,a^2\,b\,c\,d^2\,f^3+3\,a\,b^2\,c^2\,d\,f^3-b^3\,c^3\,f^3\right )}{{\left (c\,f-d\,e\right )}^4}+\frac {2\,{\left (e+f\,x\right )}^2\,\left (a^3\,d^2\,f^3-3\,a^2\,b\,c\,d\,f^3+3\,a\,b^2\,c^2\,f^3-3\,b^3\,c^2\,e\,f^2+3\,b^3\,c\,d\,e^2\,f-b^3\,d^2\,e^3\right )}{3\,{\left (c\,f-d\,e\right )}^3}-\frac {2\,\left (e+f\,x\right )\,\left (d\,a^3\,f^3-3\,c\,a^2\,b\,f^3-3\,d\,a\,b^2\,e^2\,f+6\,c\,a\,b^2\,e\,f^2+2\,d\,b^3\,e^3-3\,c\,b^3\,e^2\,f\right )}{5\,{\left (c\,f-d\,e\right )}^2}}{f^3\,{\left (e+f\,x\right )}^{7/2}} \]
(2*d^(1/2)*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^3*(c^4*f^4 + d^4*e^4 + 6*c^2*d^2*e^2*f^2 - 4*c*d^3*e^3*f - 4*c^3*d*e*f^3))/((c*f - d*e)^(9/2)*( a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))*(a*d - b*c)^3)/(c*f - d*e)^(9/2) - ((2*(a^3*f^3 - b^3*e^3 + 3*a*b^2*e^2*f - 3*a^2*b*e*f^2))/(7* (c*f - d*e)) - (2*(e + f*x)^3*(a^3*d^3*f^3 - b^3*c^3*f^3 + 3*a*b^2*c^2*d*f ^3 - 3*a^2*b*c*d^2*f^3))/(c*f - d*e)^4 + (2*(e + f*x)^2*(a^3*d^2*f^3 - b^3 *d^2*e^3 + 3*a*b^2*c^2*f^3 - 3*b^3*c^2*e*f^2 - 3*a^2*b*c*d*f^3 + 3*b^3*c*d *e^2*f))/(3*(c*f - d*e)^3) - (2*(e + f*x)*(a^3*d*f^3 + 2*b^3*d*e^3 - 3*a^2 *b*c*f^3 - 3*b^3*c*e^2*f + 6*a*b^2*c*e*f^2 - 3*a*b^2*d*e^2*f))/(5*(c*f - d *e)^2))/(f^3*(e + f*x)^(7/2))