Integrand size = 24, antiderivative size = 214 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2}{3 e^6 (d+e x)^{3/2}}-\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^6 \sqrt {d+e x}}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) \sqrt {d+e x}}{e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{5/2}}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6} \] Output:
2/3*(-A*e+B*d)*(a*e^2+c*d^2)^2/e^6/(e*x+d)^(3/2)-2*(a*e^2+c*d^2)*(-4*A*c*d *e+B*a*e^2+5*B*c*d^2)/e^6/(e*x+d)^(1/2)-4*c*(-A*a*e^3-3*A*c*d^2*e+3*B*a*d* e^2+5*B*c*d^3)*(e*x+d)^(1/2)/e^6+4/3*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)*(e*x +d)^(3/2)/e^6-2/5*c^2*(-A*e+5*B*d)*(e*x+d)^(5/2)/e^6+2/7*B*c^2*(e*x+d)^(7/ 2)/e^6
Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {14 A e \left (-5 a^2 e^4+10 a c e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-10 B \left (7 a^2 e^4 (2 d+3 e x)+14 a c e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )+c^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{105 e^6 (d+e x)^{3/2}} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(5/2),x]
Output:
(14*A*e*(-5*a^2*e^4 + 10*a*c*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + c^2*(128 *d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) - 10*B*(7* a^2*e^4*(2*d + 3*e*x) + 14*a*c*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^ 3*x^3) + c^2*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6* d*e^4*x^4 - 3*e^5*x^5)))/(105*e^6*(d + e*x)^(3/2))
Time = 0.34 (sec) , antiderivative size = 214, 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 = {652, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {2 c \sqrt {d+e x} \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5}+\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5 (d+e x)^{3/2}}+\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 (d+e x)^{5/2}}+\frac {2 c \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5 \sqrt {d+e x}}+\frac {c^2 (d+e x)^{3/2} (A e-5 B d)}{e^5}+\frac {B c^2 (d+e x)^{5/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c (d+e x)^{3/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^6 \sqrt {d+e x}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6 (d+e x)^{3/2}}-\frac {4 c \sqrt {d+e x} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6}-\frac {2 c^2 (d+e x)^{5/2} (5 B d-A e)}{5 e^6}+\frac {2 B c^2 (d+e x)^{7/2}}{7 e^6}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(5/2),x]
Output:
(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(3*e^6*(d + e*x)^(3/2)) - (2*(c*d^2 + a* e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(e^6*Sqrt[d + e*x]) - (4*c*(5*B*c* d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*Sqrt[d + e*x])/e^6 + (4*c*(5*B* c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/(3*e^6) - (2*c^2*(5*B*d - A* e)*(d + e*x)^(5/2))/(5*e^6) + (2*B*c^2*(d + e*x)^(7/2))/(7*e^6)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 1.00 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (30 B \,x^{5}+42 A \,x^{4}\right ) c^{2}+420 x^{2} a \left (\frac {B x}{3}+A \right ) c -70 a^{2} \left (3 B x +A \right )\right ) e^{5}+1680 d \left (\left (-\frac {1}{28} B \,x^{4}-\frac {1}{15} A \,x^{3}\right ) c^{2}+a x \left (-\frac {B x}{2}+A \right ) c -\frac {a^{2} B}{12}\right ) e^{4}+1120 c \,d^{2} \left (\left (\frac {1}{7} B \,x^{3}+\frac {3}{5} A \,x^{2}\right ) c +a \left (-3 B x +A \right )\right ) e^{3}+2688 c \,d^{3} \left (x \left (-\frac {5 B x}{14}+A \right ) c -\frac {5 B a}{6}\right ) e^{2}+1792 c^{2} d^{4} \left (-\frac {15 B x}{7}+A \right ) e -2560 B \,c^{2} d^{5}}{105 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(180\) |
| risch | \(\frac {2 c \left (15 B c \,x^{3} e^{3}+21 A \,x^{2} c \,e^{3}-60 B \,x^{2} c d \,e^{2}-98 A x c d \,e^{2}+70 B x a \,e^{3}+185 B x c \,d^{2} e +210 A a \,e^{3}+511 A c \,d^{2} e -560 B a d \,e^{2}-790 B c \,d^{3}\right ) \sqrt {e x +d}}{105 e^{6}}-\frac {2 \left (-12 A x c d \,e^{2}+3 B x a \,e^{3}+15 B x c \,d^{2} e +A a \,e^{3}-11 A c \,d^{2} e +2 B a d \,e^{2}+14 B c \,d^{3}\right ) \left (a \,e^{2}+c \,d^{2}\right )}{3 e^{6} \left (e x +d \right )^{\frac {3}{2}}}\) | \(182\) |
| gosper | \(-\frac {2 \left (-15 B \,x^{5} c^{2} e^{5}-21 A \,x^{4} c^{2} e^{5}+30 B \,x^{4} c^{2} d \,e^{4}+56 A \,x^{3} c^{2} d \,e^{4}-70 B \,x^{3} a c \,e^{5}-80 B \,x^{3} c^{2} d^{2} e^{3}-210 A \,x^{2} a c \,e^{5}-336 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+480 B \,x^{2} c^{2} d^{3} e^{2}-840 A x a c d \,e^{4}-1344 A x \,c^{2} d^{3} e^{2}+105 B x \,a^{2} e^{5}+1680 B x a c \,d^{2} e^{3}+1920 B x \,c^{2} d^{4} e +35 A \,a^{2} e^{5}-560 A a c \,d^{2} e^{3}-896 A \,c^{2} d^{4} e +70 B \,a^{2} d \,e^{4}+1120 B a c \,d^{3} e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(259\) |
| trager | \(-\frac {2 \left (-15 B \,x^{5} c^{2} e^{5}-21 A \,x^{4} c^{2} e^{5}+30 B \,x^{4} c^{2} d \,e^{4}+56 A \,x^{3} c^{2} d \,e^{4}-70 B \,x^{3} a c \,e^{5}-80 B \,x^{3} c^{2} d^{2} e^{3}-210 A \,x^{2} a c \,e^{5}-336 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+480 B \,x^{2} c^{2} d^{3} e^{2}-840 A x a c d \,e^{4}-1344 A x \,c^{2} d^{3} e^{2}+105 B x \,a^{2} e^{5}+1680 B x a c \,d^{2} e^{3}+1920 B x \,c^{2} d^{4} e +35 A \,a^{2} e^{5}-560 A a c \,d^{2} e^{3}-896 A \,c^{2} d^{4} e +70 B \,a^{2} d \,e^{4}+1120 B a c \,d^{3} e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(259\) |
| orering | \(-\frac {2 \left (-15 B \,x^{5} c^{2} e^{5}-21 A \,x^{4} c^{2} e^{5}+30 B \,x^{4} c^{2} d \,e^{4}+56 A \,x^{3} c^{2} d \,e^{4}-70 B \,x^{3} a c \,e^{5}-80 B \,x^{3} c^{2} d^{2} e^{3}-210 A \,x^{2} a c \,e^{5}-336 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+480 B \,x^{2} c^{2} d^{3} e^{2}-840 A x a c d \,e^{4}-1344 A x \,c^{2} d^{3} e^{2}+105 B x \,a^{2} e^{5}+1680 B x a c \,d^{2} e^{3}+1920 B x \,c^{2} d^{4} e +35 A \,a^{2} e^{5}-560 A a c \,d^{2} e^{3}-896 A \,c^{2} d^{4} e +70 B \,a^{2} d \,e^{4}+1120 B a c \,d^{3} e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(259\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-2 B \,c^{2} d \left (e x +d \right )^{\frac {5}{2}}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 B a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 A a c \,e^{3} \sqrt {e x +d}+12 A \,c^{2} d^{2} e \sqrt {e x +d}-12 B a c d \,e^{2} \sqrt {e x +d}-20 B \,c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(283\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}-2 B \,c^{2} d \left (e x +d \right )^{\frac {5}{2}}-\frac {8 A \,c^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 B a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 B \,c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+4 A a c \,e^{3} \sqrt {e x +d}+12 A \,c^{2} d^{2} e \sqrt {e x +d}-12 B a c d \,e^{2} \sqrt {e x +d}-20 B \,c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(283\) |
Input:
int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/105*(((30*B*x^5+42*A*x^4)*c^2+420*x^2*a*(1/3*B*x+A)*c-70*a^2*(3*B*x+A))* e^5+1680*d*((-1/28*B*x^4-1/15*A*x^3)*c^2+a*x*(-1/2*B*x+A)*c-1/12*a^2*B)*e^ 4+1120*c*d^2*((1/7*B*x^3+3/5*A*x^2)*c+a*(-3*B*x+A))*e^3+2688*c*d^3*(x*(-5/ 14*B*x+A)*c-5/6*B*a)*e^2+1792*c^2*d^4*(-15/7*B*x+A)*e-2560*B*c^2*d^5)/(e*x +d)^(3/2)/e^6
Time = 0.08 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.26 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (15 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 896 \, A c^{2} d^{4} e - 1120 \, B a c d^{3} e^{2} + 560 \, A a c d^{2} e^{3} - 70 \, B a^{2} d e^{4} - 35 \, A a^{2} e^{5} - 3 \, {\left (10 \, B c^{2} d e^{4} - 7 \, A c^{2} e^{5}\right )} x^{4} + 2 \, {\left (40 \, B c^{2} d^{2} e^{3} - 28 \, A c^{2} d e^{4} + 35 \, B a c e^{5}\right )} x^{3} - 6 \, {\left (80 \, B c^{2} d^{3} e^{2} - 56 \, A c^{2} d^{2} e^{3} + 70 \, B a c d e^{4} - 35 \, A a c e^{5}\right )} x^{2} - 3 \, {\left (640 \, B c^{2} d^{4} e - 448 \, A c^{2} d^{3} e^{2} + 560 \, B a c d^{2} e^{3} - 280 \, A a c d e^{4} + 35 \, B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
2/105*(15*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 896*A*c^2*d^4*e - 1120*B*a*c*d^ 3*e^2 + 560*A*a*c*d^2*e^3 - 70*B*a^2*d*e^4 - 35*A*a^2*e^5 - 3*(10*B*c^2*d* e^4 - 7*A*c^2*e^5)*x^4 + 2*(40*B*c^2*d^2*e^3 - 28*A*c^2*d*e^4 + 35*B*a*c*e ^5)*x^3 - 6*(80*B*c^2*d^3*e^2 - 56*A*c^2*d^2*e^3 + 70*B*a*c*d*e^4 - 35*A*a *c*e^5)*x^2 - 3*(640*B*c^2*d^4*e - 448*A*c^2*d^3*e^2 + 560*B*a*c*d^2*e^3 - 280*A*a*c*d*e^4 + 35*B*a^2*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d ^2*e^6)
Time = 6.76 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (A c^{2} e - 5 B c^{2} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 4 A c^{2} d e + 2 B a c e^{2} + 10 B c^{2} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (2 A a c e^{3} + 6 A c^{2} d^{2} e - 6 B a c d e^{2} - 10 B c^{2} d^{3}\right )}{e^{5}} - \frac {\left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{5} \sqrt {d + e x}} + \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + \frac {2 A a c x^{3}}{3} + \frac {A c^{2} x^{5}}{5} + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(5/2),x)
Output:
Piecewise((2*(B*c**2*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(5/2)*(A*c**2* e - 5*B*c**2*d)/(5*e**5) + (d + e*x)**(3/2)*(-4*A*c**2*d*e + 2*B*a*c*e**2 + 10*B*c**2*d**2)/(3*e**5) + sqrt(d + e*x)*(2*A*a*c*e**3 + 6*A*c**2*d**2*e - 6*B*a*c*d*e**2 - 10*B*c**2*d**3)/e**5 - (a*e**2 + c*d**2)*(-4*A*c*d*e + B*a*e**2 + 5*B*c*d**2)/(e**5*sqrt(d + e*x)) + (-A*e + B*d)*(a*e**2 + c*d* *2)**2/(3*e**5*(d + e*x)**(3/2)))/e, Ne(e, 0)), ((A*a**2*x + 2*A*a*c*x**3/ 3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6)/d**(5/2) , True))
Time = 0.04 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} - 21 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 70 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 210 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} \sqrt {e x + d}}{e^{5}} + \frac {35 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5} - 3 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{105 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
2/105*((15*(e*x + d)^(7/2)*B*c^2 - 21*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(5/2 ) + 70*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*(e*x + d)^(3/2) - 210*(5*B* c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*sqrt(e*x + d))/e^5 + 35*(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d* e^4 - A*a^2*e^5 - 3*(5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a *c*d*e^3 + B*a^2*e^4)*(e*x + d))/((e*x + d)^(3/2)*e^5))/e
Time = 0.14 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (e x + d\right )} B c^{2} d^{4} - B c^{2} d^{5} - 12 \, {\left (e x + d\right )} A c^{2} d^{3} e + A c^{2} d^{4} e + 18 \, {\left (e x + d\right )} B a c d^{2} e^{2} - 2 \, B a c d^{3} e^{2} - 12 \, {\left (e x + d\right )} A a c d e^{3} + 2 \, A a c d^{2} e^{3} + 3 \, {\left (e x + d\right )} B a^{2} e^{4} - B a^{2} d e^{4} + A a^{2} e^{5}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{6}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{2} e^{36} - 105 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} d e^{36} + 350 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{2} d^{2} e^{36} - 1050 \, \sqrt {e x + d} B c^{2} d^{3} e^{36} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{2} e^{37} - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} d e^{37} + 630 \, \sqrt {e x + d} A c^{2} d^{2} e^{37} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} B a c e^{38} - 630 \, \sqrt {e x + d} B a c d e^{38} + 210 \, \sqrt {e x + d} A a c e^{39}\right )}}{105 \, e^{42}} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(5/2),x, algorithm="giac")
Output:
-2/3*(15*(e*x + d)*B*c^2*d^4 - B*c^2*d^5 - 12*(e*x + d)*A*c^2*d^3*e + A*c^ 2*d^4*e + 18*(e*x + d)*B*a*c*d^2*e^2 - 2*B*a*c*d^3*e^2 - 12*(e*x + d)*A*a* c*d*e^3 + 2*A*a*c*d^2*e^3 + 3*(e*x + d)*B*a^2*e^4 - B*a^2*d*e^4 + A*a^2*e^ 5)/((e*x + d)^(3/2)*e^6) + 2/105*(15*(e*x + d)^(7/2)*B*c^2*e^36 - 105*(e*x + d)^(5/2)*B*c^2*d*e^36 + 350*(e*x + d)^(3/2)*B*c^2*d^2*e^36 - 1050*sqrt( e*x + d)*B*c^2*d^3*e^36 + 21*(e*x + d)^(5/2)*A*c^2*e^37 - 140*(e*x + d)^(3 /2)*A*c^2*d*e^37 + 630*sqrt(e*x + d)*A*c^2*d^2*e^37 + 70*(e*x + d)^(3/2)*B *a*c*e^38 - 630*sqrt(e*x + d)*B*a*c*d*e^38 + 210*sqrt(e*x + d)*A*a*c*e^39) /e^42
Time = 6.44 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.16 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {{\left (d+e\,x\right )}^{3/2}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{3\,e^6}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^2\,e^4+12\,B\,a\,c\,d^2\,e^2-8\,A\,a\,c\,d\,e^3+10\,B\,c^2\,d^4-8\,A\,c^2\,d^3\,e\right )+\frac {2\,A\,a^2\,e^5}{3}-\frac {2\,B\,c^2\,d^5}{3}-\frac {2\,B\,a^2\,d\,e^4}{3}+\frac {2\,A\,c^2\,d^4\,e}{3}+\frac {4\,A\,a\,c\,d^2\,e^3}{3}-\frac {4\,B\,a\,c\,d^3\,e^2}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {4\,c\,\sqrt {d+e\,x}\,\left (-5\,B\,c\,d^3+3\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6} \] Input:
int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^(5/2),x)
Output:
((d + e*x)^(3/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/(3*e^6) - ((d + e*x)*(2*B*a^2*e^4 + 10*B*c^2*d^4 - 8*A*c^2*d^3*e - 8*A*a*c*d*e^3 + 12*B *a*c*d^2*e^2) + (2*A*a^2*e^5)/3 - (2*B*c^2*d^5)/3 - (2*B*a^2*d*e^4)/3 + (2 *A*c^2*d^4*e)/3 + (4*A*a*c*d^2*e^3)/3 - (4*B*a*c*d^3*e^2)/3)/(e^6*(d + e*x )^(3/2)) + (4*c*(d + e*x)^(1/2)*(A*a*e^3 - 5*B*c*d^3 - 3*B*a*d*e^2 + 3*A*c *d^2*e))/e^6 + (2*B*c^2*(d + e*x)^(7/2))/(7*e^6) + (2*c^2*(A*e - 5*B*d)*(d + e*x)^(5/2))/(5*e^6)
Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {\frac {2}{7} b \,c^{2} e^{5} x^{5}+\frac {2}{5} a \,c^{2} e^{5} x^{4}-\frac {4}{7} b \,c^{2} d \,e^{4} x^{4}+\frac {4}{3} a b c \,e^{5} x^{3}-\frac {16}{15} a \,c^{2} d \,e^{4} x^{3}+\frac {32}{21} b \,c^{2} d^{2} e^{3} x^{3}+4 a^{2} c \,e^{5} x^{2}-8 a b c d \,e^{4} x^{2}+\frac {32}{5} a \,c^{2} d^{2} e^{3} x^{2}-\frac {64}{7} b \,c^{2} d^{3} e^{2} x^{2}-2 a^{2} b \,e^{5} x +16 a^{2} c d \,e^{4} x -32 a b c \,d^{2} e^{3} x +\frac {128}{5} a \,c^{2} d^{3} e^{2} x -\frac {256}{7} b \,c^{2} d^{4} e x -\frac {2}{3} a^{3} e^{5}-\frac {4}{3} a^{2} b d \,e^{4}+\frac {32}{3} a^{2} c \,d^{2} e^{3}-\frac {64}{3} a b c \,d^{3} e^{2}+\frac {256}{15} a \,c^{2} d^{4} e -\frac {512}{21} b \,c^{2} d^{5}}{\sqrt {e x +d}\, e^{6} \left (e x +d \right )} \] Input:
int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(5/2),x)
Output:
(2*( - 35*a**3*e**5 - 70*a**2*b*d*e**4 - 105*a**2*b*e**5*x + 560*a**2*c*d* *2*e**3 + 840*a**2*c*d*e**4*x + 210*a**2*c*e**5*x**2 - 1120*a*b*c*d**3*e** 2 - 1680*a*b*c*d**2*e**3*x - 420*a*b*c*d*e**4*x**2 + 70*a*b*c*e**5*x**3 + 896*a*c**2*d**4*e + 1344*a*c**2*d**3*e**2*x + 336*a*c**2*d**2*e**3*x**2 - 56*a*c**2*d*e**4*x**3 + 21*a*c**2*e**5*x**4 - 1280*b*c**2*d**5 - 1920*b*c* *2*d**4*e*x - 480*b*c**2*d**3*e**2*x**2 + 80*b*c**2*d**2*e**3*x**3 - 30*b* c**2*d*e**4*x**4 + 15*b*c**2*e**5*x**5))/(105*sqrt(d + e*x)*e**6*(d + e*x) )