Integrand size = 24, antiderivative size = 214 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2}{7 e^6 (d+e x)^{7/2}}-\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{e^6 \sqrt {d+e x}}-\frac {2 c^2 (5 B d-A e) \sqrt {d+e x}}{e^6}+\frac {2 B c^2 (d+e x)^{3/2}}{3 e^6} \] Output:
2/7*(-A*e+B*d)*(a*e^2+c*d^2)^2/e^6/(e*x+d)^(7/2)-2/5*(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)^(5/2)+4/3*c*(-A*a*e^3-3*A*c*d^2*e+3*B* a*d*e^2+5*B*c*d^3)/e^6/(e*x+d)^(3/2)-4*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)/e^ 6/(e*x+d)^(1/2)-2*c^2*(-A*e+5*B*d)*(e*x+d)^(1/2)/e^6+2/3*B*c^2*(e*x+d)^(3/ 2)/e^6
Time = 0.40 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=-\frac {2 \left (A e \left (15 a^2 e^4+2 a c e^2 \left (8 d^2+28 d e x+35 e^2 x^2\right )-3 c^2 \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (3 a^2 e^4 (2 d+7 e x)+6 a c e^2 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )+5 c^2 \left (256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right )\right )\right )}{105 e^6 (d+e x)^{7/2}} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(9/2),x]
Output:
(-2*(A*e*(15*a^2*e^4 + 2*a*c*e^2*(8*d^2 + 28*d*e*x + 35*e^2*x^2) - 3*c^2*( 128*d^4 + 448*d^3*e*x + 560*d^2*e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4)) + B *(3*a^2*e^4*(2*d + 7*e*x) + 6*a*c*e^2*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) + 5*c^2*(256*d^5 + 896*d^4*e*x + 1120*d^3*e^2*x^2 + 560*d^2* e^3*x^3 + 70*d*e^4*x^4 - 7*e^5*x^5))))/(105*e^6*(d + e*x)^(7/2))
Time = 0.33 (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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {2 c \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5 (d+e x)^{3/2}}+\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5 (d+e x)^{7/2}}+\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 (d+e x)^{9/2}}+\frac {2 c \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5 (d+e x)^{5/2}}+\frac {c^2 (A e-5 B d)}{e^5 \sqrt {d+e x}}+\frac {B c^2 \sqrt {d+e x}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^{5/2}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{7 e^6 (d+e x)^{7/2}}+\frac {4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6 (d+e x)^{3/2}}-\frac {2 c^2 \sqrt {d+e x} (5 B d-A e)}{e^6}+\frac {2 B c^2 (d+e x)^{3/2}}{3 e^6}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(9/2),x]
Output:
(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(7*e^6*(d + e*x)^(7/2)) - (2*(c*d^2 + a* e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(5*e^6*(d + e*x)^(5/2)) + (4*c*(5* B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(3*e^6*(d + e*x)^(3/2)) - (4*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^6*Sqrt[d + e*x]) - (2*c^2*(5*B* d - A*e)*Sqrt[d + e*x])/e^6 + (2*B*c^2*(d + e*x)^(3/2))/(3*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 = 1.02 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (70 B \,x^{5}+210 A \,x^{4}\right ) c^{2}-140 a \,x^{2} \left (3 B x +A \right ) c -30 a^{2} \left (\frac {7 B x}{5}+A \right )\right ) e^{5}-112 d \left (\left (\frac {25}{4} B \,x^{4}-15 A \,x^{3}\right ) c^{2}+a x \left (\frac {15 B x}{2}+A \right ) c +\frac {3 a^{2} B}{28}\right ) e^{4}-32 c \left (\left (175 B \,x^{3}-105 A \,x^{2}\right ) c +a \left (21 B x +A \right )\right ) d^{2} e^{3}+2688 c \left (x \left (-\frac {25 B x}{6}+A \right ) c -\frac {B a}{14}\right ) d^{3} e^{2}+768 c^{2} d^{4} \left (-\frac {35 B x}{3}+A \right ) e -2560 B \,c^{2} d^{5}}{105 \left (e x +d \right )^{\frac {7}{2}} e^{6}}\) | \(180\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 c^{2} e A \sqrt {e x +d}-10 B \,c^{2} d \sqrt {e x +d}-\frac {2 \left (-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {4 c \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}-\frac {4 c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(243\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+2 c^{2} e A \sqrt {e x +d}-10 B \,c^{2} d \sqrt {e x +d}-\frac {2 \left (-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}+\frac {4 c \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 \left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right )}{7 \left (e x +d \right )^{\frac {7}{2}}}-\frac {4 c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(243\) |
| gosper | \(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-105 A \,x^{4} c^{2} e^{5}+350 B \,x^{4} c^{2} d \,e^{4}-840 A \,x^{3} c^{2} d \,e^{4}+210 B \,x^{3} a c \,e^{5}+2800 B \,x^{3} c^{2} d^{2} e^{3}+70 A \,x^{2} a c \,e^{5}-1680 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+5600 B \,x^{2} c^{2} d^{3} e^{2}+56 A x a c d \,e^{4}-1344 A x \,c^{2} d^{3} e^{2}+21 B x \,a^{2} e^{5}+336 B x a c \,d^{2} e^{3}+4480 B x \,c^{2} d^{4} e +15 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}-384 A \,c^{2} d^{4} e +6 B \,a^{2} d \,e^{4}+96 B a c \,d^{3} e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} e^{6}}\) | \(259\) |
| trager | \(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-105 A \,x^{4} c^{2} e^{5}+350 B \,x^{4} c^{2} d \,e^{4}-840 A \,x^{3} c^{2} d \,e^{4}+210 B \,x^{3} a c \,e^{5}+2800 B \,x^{3} c^{2} d^{2} e^{3}+70 A \,x^{2} a c \,e^{5}-1680 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+5600 B \,x^{2} c^{2} d^{3} e^{2}+56 A x a c d \,e^{4}-1344 A x \,c^{2} d^{3} e^{2}+21 B x \,a^{2} e^{5}+336 B x a c \,d^{2} e^{3}+4480 B x \,c^{2} d^{4} e +15 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}-384 A \,c^{2} d^{4} e +6 B \,a^{2} d \,e^{4}+96 B a c \,d^{3} e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} e^{6}}\) | \(259\) |
| orering | \(-\frac {2 \left (-35 B \,x^{5} c^{2} e^{5}-105 A \,x^{4} c^{2} e^{5}+350 B \,x^{4} c^{2} d \,e^{4}-840 A \,x^{3} c^{2} d \,e^{4}+210 B \,x^{3} a c \,e^{5}+2800 B \,x^{3} c^{2} d^{2} e^{3}+70 A \,x^{2} a c \,e^{5}-1680 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+5600 B \,x^{2} c^{2} d^{3} e^{2}+56 A x a c d \,e^{4}-1344 A x \,c^{2} d^{3} e^{2}+21 B x \,a^{2} e^{5}+336 B x a c \,d^{2} e^{3}+4480 B x \,c^{2} d^{4} e +15 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}-384 A \,c^{2} d^{4} e +6 B \,a^{2} d \,e^{4}+96 B a c \,d^{3} e^{2}+1280 B \,c^{2} d^{5}\right )}{105 \left (e x +d \right )^{\frac {7}{2}} e^{6}}\) | \(259\) |
| risch | \(\frac {2 c^{2} \left (B e x +3 A e -14 B d \right ) \sqrt {e x +d}}{3 e^{6}}-\frac {2 \left (-420 A \,x^{3} c^{2} d \,e^{4}+210 B \,x^{3} a c \,e^{5}+1050 B \,x^{3} c^{2} d^{2} e^{3}+70 A \,x^{2} a c \,e^{5}-1050 A \,x^{2} c^{2} d^{2} e^{3}+420 B \,x^{2} a c d \,e^{4}+2800 B \,x^{2} c^{2} d^{3} e^{2}+56 A x a c d \,e^{4}-924 A x \,c^{2} d^{3} e^{2}+21 B x \,a^{2} e^{5}+336 B x a c \,d^{2} e^{3}+2555 B x \,c^{2} d^{4} e +15 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}-279 A \,c^{2} d^{4} e +6 B \,a^{2} d \,e^{4}+96 B a c \,d^{3} e^{2}+790 B \,c^{2} d^{5}\right )}{105 e^{6} \sqrt {e x +d}\, \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )}\) | \(280\) |
Input:
int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/105*(((70*B*x^5+210*A*x^4)*c^2-140*a*x^2*(3*B*x+A)*c-30*a^2*(7/5*B*x+A)) *e^5-112*d*((25/4*B*x^4-15*A*x^3)*c^2+a*x*(15/2*B*x+A)*c+3/28*a^2*B)*e^4-3 2*c*((175*B*x^3-105*A*x^2)*c+a*(21*B*x+A))*d^2*e^3+2688*c*(x*(-25/6*B*x+A) *c-1/14*B*a)*d^3*e^2+768*c^2*d^4*(-35/3*B*x+A)*e-2560*B*c^2*d^5)/(e*x+d)^( 7/2)/e^6
Time = 0.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (35 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 384 \, A c^{2} d^{4} e - 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 6 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 35 \, {\left (10 \, B c^{2} d e^{4} - 3 \, A c^{2} e^{5}\right )} x^{4} - 70 \, {\left (40 \, B c^{2} d^{2} e^{3} - 12 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} - 70 \, {\left (80 \, B c^{2} d^{3} e^{2} - 24 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} + A a c e^{5}\right )} x^{2} - 7 \, {\left (640 \, B c^{2} d^{4} e - 192 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} + 8 \, A a c d e^{4} + 3 \, B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x, algorithm="fricas")
Output:
2/105*(35*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 384*A*c^2*d^4*e - 96*B*a*c*d^3* e^2 - 16*A*a*c*d^2*e^3 - 6*B*a^2*d*e^4 - 15*A*a^2*e^5 - 35*(10*B*c^2*d*e^4 - 3*A*c^2*e^5)*x^4 - 70*(40*B*c^2*d^2*e^3 - 12*A*c^2*d*e^4 + 3*B*a*c*e^5) *x^3 - 70*(80*B*c^2*d^3*e^2 - 24*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + A*a*c*e^5 )*x^2 - 7*(640*B*c^2*d^4*e - 192*A*c^2*d^3*e^2 + 48*B*a*c*d^2*e^3 + 8*A*a* c*d*e^4 + 3*B*a^2*e^5)*x)*sqrt(e*x + d)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^ 8*x^2 + 4*d^3*e^7*x + d^4*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 1855 vs. \(2 (226) = 452\).
Time = 0.91 (sec) , antiderivative size = 1855, normalized size of antiderivative = 8.67 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(9/2),x)
Output:
Piecewise((-30*A*a**2*e**5/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x* sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x )) - 32*A*a*c*d**2*e**3/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqr t(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 112*A*a*c*d*e**4*x/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 1 40*A*a*c*e**5*x**2/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 768 *A*c**2*d**4*e/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x ) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 2688*A* c**2*d**3*e**2*x/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e *x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 3360* A*c**2*d**2*e**3*x**2/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt( d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 1680*A*c**2*d*e**4*x**3/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqr t(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) + 210*A*c**2*e**5*x**4/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt (d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 12*B*a**2*d*e**4/(105*d**3*e**6*sqrt(d + e*x) + 315*d**2*e**7*x*sqrt(d + e*x) + 315*d*e**8*x**2*sqrt(d + e*x) + 105*e**9*x**3*sqrt(d + e*x)) - 4...
Time = 0.05 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B c^{2} - 3 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} \sqrt {e x + d}\right )}}{e^{5}} + \frac {15 \, B c^{2} d^{5} - 15 \, A c^{2} d^{4} e + 30 \, B a c d^{3} e^{2} - 30 \, A a c d^{2} e^{3} + 15 \, B a^{2} d e^{4} - 15 \, A a^{2} e^{5} - 210 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{3} + 70 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{2} - 21 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {7}{2}} e^{5}}\right )}}{105 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x, algorithm="maxima")
Output:
2/105*(35*((e*x + d)^(3/2)*B*c^2 - 3*(5*B*c^2*d - A*c^2*e)*sqrt(e*x + d))/ e^5 + (15*B*c^2*d^5 - 15*A*c^2*d^4*e + 30*B*a*c*d^3*e^2 - 30*A*a*c*d^2*e^3 + 15*B*a^2*d*e^4 - 15*A*a^2*e^5 - 210*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c* e^2)*(e*x + d)^3 + 70*(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)*(e*x + d)^2 - 21*(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)^(7/2)*e^5))/e
Time = 0.14 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=-\frac {2 \, {\left (1050 \, {\left (e x + d\right )}^{3} B c^{2} d^{2} - 350 \, {\left (e x + d\right )}^{2} B c^{2} d^{3} + 105 \, {\left (e x + d\right )} B c^{2} d^{4} - 15 \, B c^{2} d^{5} - 420 \, {\left (e x + d\right )}^{3} A c^{2} d e + 210 \, {\left (e x + d\right )}^{2} A c^{2} d^{2} e - 84 \, {\left (e x + d\right )} A c^{2} d^{3} e + 15 \, A c^{2} d^{4} e + 210 \, {\left (e x + d\right )}^{3} B a c e^{2} - 210 \, {\left (e x + d\right )}^{2} B a c d e^{2} + 126 \, {\left (e x + d\right )} B a c d^{2} e^{2} - 30 \, B a c d^{3} e^{2} + 70 \, {\left (e x + d\right )}^{2} A a c e^{3} - 84 \, {\left (e x + d\right )} A a c d e^{3} + 30 \, A a c d^{2} e^{3} + 21 \, {\left (e x + d\right )} B a^{2} e^{4} - 15 \, B a^{2} d e^{4} + 15 \, A a^{2} e^{5}\right )}}{105 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{6}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B c^{2} e^{12} - 15 \, \sqrt {e x + d} B c^{2} d e^{12} + 3 \, \sqrt {e x + d} A c^{2} e^{13}\right )}}{3 \, e^{18}} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x, algorithm="giac")
Output:
-2/105*(1050*(e*x + d)^3*B*c^2*d^2 - 350*(e*x + d)^2*B*c^2*d^3 + 105*(e*x + d)*B*c^2*d^4 - 15*B*c^2*d^5 - 420*(e*x + d)^3*A*c^2*d*e + 210*(e*x + d)^ 2*A*c^2*d^2*e - 84*(e*x + d)*A*c^2*d^3*e + 15*A*c^2*d^4*e + 210*(e*x + d)^ 3*B*a*c*e^2 - 210*(e*x + d)^2*B*a*c*d*e^2 + 126*(e*x + d)*B*a*c*d^2*e^2 - 30*B*a*c*d^3*e^2 + 70*(e*x + d)^2*A*a*c*e^3 - 84*(e*x + d)*A*a*c*d*e^3 + 3 0*A*a*c*d^2*e^3 + 21*(e*x + d)*B*a^2*e^4 - 15*B*a^2*d*e^4 + 15*A*a^2*e^5)/ ((e*x + d)^(7/2)*e^6) + 2/3*((e*x + d)^(3/2)*B*c^2*e^12 - 15*sqrt(e*x + d) *B*c^2*d*e^12 + 3*sqrt(e*x + d)*A*c^2*e^13)/e^18
Time = 6.41 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.21 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=-\frac {2\,\left (6\,B\,a^2\,d\,e^4+21\,B\,a^2\,e^5\,x+15\,A\,a^2\,e^5+96\,B\,a\,c\,d^3\,e^2+336\,B\,a\,c\,d^2\,e^3\,x+16\,A\,a\,c\,d^2\,e^3+420\,B\,a\,c\,d\,e^4\,x^2+56\,A\,a\,c\,d\,e^4\,x+210\,B\,a\,c\,e^5\,x^3+70\,A\,a\,c\,e^5\,x^2+1280\,B\,c^2\,d^5+4480\,B\,c^2\,d^4\,e\,x-384\,A\,c^2\,d^4\,e+5600\,B\,c^2\,d^3\,e^2\,x^2-1344\,A\,c^2\,d^3\,e^2\,x+2800\,B\,c^2\,d^2\,e^3\,x^3-1680\,A\,c^2\,d^2\,e^3\,x^2+350\,B\,c^2\,d\,e^4\,x^4-840\,A\,c^2\,d\,e^4\,x^3-35\,B\,c^2\,e^5\,x^5-105\,A\,c^2\,e^5\,x^4\right )}{105\,e^6\,{\left (d+e\,x\right )}^{7/2}} \] Input:
int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^(9/2),x)
Output:
-(2*(15*A*a^2*e^5 + 1280*B*c^2*d^5 + 6*B*a^2*d*e^4 - 384*A*c^2*d^4*e + 21* B*a^2*e^5*x - 105*A*c^2*e^5*x^4 - 35*B*c^2*e^5*x^5 + 70*A*a*c*e^5*x^2 + 21 0*B*a*c*e^5*x^3 + 4480*B*c^2*d^4*e*x - 1344*A*c^2*d^3*e^2*x - 840*A*c^2*d* e^4*x^3 + 350*B*c^2*d*e^4*x^4 - 1680*A*c^2*d^2*e^3*x^2 + 5600*B*c^2*d^3*e^ 2*x^2 + 2800*B*c^2*d^2*e^3*x^3 + 16*A*a*c*d^2*e^3 + 96*B*a*c*d^3*e^2 + 56* A*a*c*d*e^4*x + 336*B*a*c*d^2*e^3*x + 420*B*a*c*d*e^4*x^2))/(105*e^6*(d + e*x)^(7/2))
Time = 0.22 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{9/2}} \, dx=\frac {\frac {2}{3} b \,c^{2} e^{5} x^{5}+2 a \,c^{2} e^{5} x^{4}-\frac {20}{3} b \,c^{2} d \,e^{4} x^{4}-4 a b c \,e^{5} x^{3}+16 a \,c^{2} d \,e^{4} x^{3}-\frac {160}{3} b \,c^{2} d^{2} e^{3} x^{3}-\frac {4}{3} a^{2} c \,e^{5} x^{2}-8 a b c d \,e^{4} x^{2}+32 a \,c^{2} d^{2} e^{3} x^{2}-\frac {320}{3} b \,c^{2} d^{3} e^{2} x^{2}-\frac {2}{5} a^{2} b \,e^{5} x -\frac {16}{15} a^{2} c d \,e^{4} x -\frac {32}{5} a b c \,d^{2} e^{3} x +\frac {128}{5} a \,c^{2} d^{3} e^{2} x -\frac {256}{3} b \,c^{2} d^{4} e x -\frac {2}{7} a^{3} e^{5}-\frac {4}{35} a^{2} b d \,e^{4}-\frac {32}{105} a^{2} c \,d^{2} e^{3}-\frac {64}{35} a b c \,d^{3} e^{2}+\frac {256}{35} a \,c^{2} d^{4} e -\frac {512}{21} b \,c^{2} d^{5}}{\sqrt {e x +d}\, e^{6} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(9/2),x)
Output:
(2*( - 15*a**3*e**5 - 6*a**2*b*d*e**4 - 21*a**2*b*e**5*x - 16*a**2*c*d**2* e**3 - 56*a**2*c*d*e**4*x - 70*a**2*c*e**5*x**2 - 96*a*b*c*d**3*e**2 - 336 *a*b*c*d**2*e**3*x - 420*a*b*c*d*e**4*x**2 - 210*a*b*c*e**5*x**3 + 384*a*c **2*d**4*e + 1344*a*c**2*d**3*e**2*x + 1680*a*c**2*d**2*e**3*x**2 + 840*a* c**2*d*e**4*x**3 + 105*a*c**2*e**5*x**4 - 1280*b*c**2*d**5 - 4480*b*c**2*d **4*e*x - 5600*b*c**2*d**3*e**2*x**2 - 2800*b*c**2*d**2*e**3*x**3 - 350*b* c**2*d*e**4*x**4 + 35*b*c**2*e**5*x**5))/(105*sqrt(d + e*x)*e**6*(d**3 + 3 *d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))