Integrand size = 24, antiderivative size = 214 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2}{5 e^6 (d+e x)^{5/2}}-\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 \sqrt {d+e x}}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6} \] Output:
2/5*(-A*e+B*d)*(a*e^2+c*d^2)^2/e^6/(e*x+d)^(5/2)-2/3*(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)^(3/2)+4*c*(-A*a*e^3-3*A*c*d^2*e+3*B*a* d*e^2+5*B*c*d^3)/e^6/(e*x+d)^(1/2)+4*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)*(e*x +d)^(1/2)/e^6-2/3*c^2*(-A*e+5*B*d)*(e*x+d)^(3/2)/e^6+2/5*B*c^2*(e*x+d)^(5/ 2)/e^6
Time = 0.23 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (3 a^2 A e^5+a^2 B e^4 (2 d+5 e x)+2 a A c e^3 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a B c e^2 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+A c^2 e \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-B c^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )}{15 e^6 (d+e x)^{5/2}} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(7/2),x]
Output:
(-2*(3*a^2*A*e^5 + a^2*B*e^4*(2*d + 5*e*x) + 2*a*A*c*e^3*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*B*c*e^2*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x ^3) + A*c^2*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5* e^4*x^4) - B*c^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5)))/(15*e^6*(d + e*x)^(5/2))
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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {2 c \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5 \sqrt {d+e x}}+\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5 (d+e x)^{5/2}}+\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 (d+e x)^{7/2}}+\frac {2 c \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5 (d+e x)^{3/2}}+\frac {c^2 \sqrt {d+e x} (A e-5 B d)}{e^5}+\frac {B c^2 (d+e x)^{3/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c \sqrt {d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}-\frac {2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac {4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt {d+e x}}-\frac {2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac {2 B c^2 (d+e x)^{5/2}}{5 e^6}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(7/2),x]
Output:
(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(5*e^6*(d + e*x)^(5/2)) - (2*(c*d^2 + a* e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^(3/2)) + (4*c*(5* B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*Sqrt[d + e*x]) + (4*c *(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/e^6 - (2*c^2*(5*B*d - A* e)*(d + e*x)^(3/2))/(3*e^6) + (2*B*c^2*(d + e*x)^(5/2))/(5*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 = 176, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (6 B \,x^{5}+10 A \,x^{4}\right ) c^{2}-60 a \,x^{2} \left (-B x +A \right ) c -6 a^{2} \left (\frac {5 B x}{3}+A \right )\right ) e^{5}-80 d \left (x^{3} \left (\frac {B x}{4}+A \right ) c^{2}+a x \left (-\frac {9 B x}{2}+A \right ) c +\frac {a^{2} B}{20}\right ) e^{4}-32 c \,d^{2} \left (\left (-5 B \,x^{3}+15 A \,x^{2}\right ) c +a \left (-15 B x +A \right )\right ) e^{3}-640 c \,d^{3} \left (x \left (-\frac {3 B x}{2}+A \right ) c -\frac {3 B a}{10}\right ) e^{2}-256 c^{2} d^{4} \left (-5 B x +A \right ) e +512 B \,c^{2} d^{5}}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(176\) |
| gosper | \(-\frac {2 \left (-3 B \,x^{5} c^{2} e^{5}-5 A \,x^{4} c^{2} e^{5}+10 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} c^{2} d \,e^{4}-30 B \,x^{3} a c \,e^{5}-80 B \,x^{3} c^{2} d^{2} e^{3}+30 A \,x^{2} a c \,e^{5}+240 A \,x^{2} c^{2} d^{2} e^{3}-180 B \,x^{2} a c d \,e^{4}-480 B \,x^{2} c^{2} d^{3} e^{2}+40 A x a c d \,e^{4}+320 A x \,c^{2} d^{3} e^{2}+5 B x \,a^{2} e^{5}-240 B x a c \,d^{2} e^{3}-640 B x \,c^{2} d^{4} e +3 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}+128 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}-96 B a c \,d^{3} e^{2}-256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(259\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}-8 A \,c^{2} d e \sqrt {e x +d}+4 B a c \,e^{2} \sqrt {e x +d}+20 B \,c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(259\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A \,c^{2} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {10 B \,c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}-8 A \,c^{2} d e \sqrt {e x +d}+4 B a c \,e^{2} \sqrt {e x +d}+20 B \,c^{2} d^{2} \sqrt {e x +d}-\frac {2 \left (A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {4 c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{6}}\) | \(259\) |
| trager | \(-\frac {2 \left (-3 B \,x^{5} c^{2} e^{5}-5 A \,x^{4} c^{2} e^{5}+10 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} c^{2} d \,e^{4}-30 B \,x^{3} a c \,e^{5}-80 B \,x^{3} c^{2} d^{2} e^{3}+30 A \,x^{2} a c \,e^{5}+240 A \,x^{2} c^{2} d^{2} e^{3}-180 B \,x^{2} a c d \,e^{4}-480 B \,x^{2} c^{2} d^{3} e^{2}+40 A x a c d \,e^{4}+320 A x \,c^{2} d^{3} e^{2}+5 B x \,a^{2} e^{5}-240 B x a c \,d^{2} e^{3}-640 B x \,c^{2} d^{4} e +3 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}+128 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}-96 B a c \,d^{3} e^{2}-256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(259\) |
| orering | \(-\frac {2 \left (-3 B \,x^{5} c^{2} e^{5}-5 A \,x^{4} c^{2} e^{5}+10 B \,x^{4} c^{2} d \,e^{4}+40 A \,x^{3} c^{2} d \,e^{4}-30 B \,x^{3} a c \,e^{5}-80 B \,x^{3} c^{2} d^{2} e^{3}+30 A \,x^{2} a c \,e^{5}+240 A \,x^{2} c^{2} d^{2} e^{3}-180 B \,x^{2} a c d \,e^{4}-480 B \,x^{2} c^{2} d^{3} e^{2}+40 A x a c d \,e^{4}+320 A x \,c^{2} d^{3} e^{2}+5 B x \,a^{2} e^{5}-240 B x a c \,d^{2} e^{3}-640 B x \,c^{2} d^{4} e +3 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}+128 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}-96 B a c \,d^{3} e^{2}-256 B \,c^{2} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(259\) |
| risch | \(-\frac {2 c \left (-3 e^{2} B c \,x^{2}-5 A c \,e^{2} x +19 B c d e x +55 A c d e -30 B a \,e^{2}-128 B c \,d^{2}\right ) \sqrt {e x +d}}{15 e^{6}}-\frac {2 \left (30 A \,x^{2} a c \,e^{5}+90 A \,x^{2} c^{2} d^{2} e^{3}-90 B \,x^{2} a c d \,e^{4}-150 B \,x^{2} c^{2} d^{3} e^{2}+40 A x a c d \,e^{4}+160 A x \,c^{2} d^{3} e^{2}+5 B x \,a^{2} e^{5}-150 B x a c \,d^{2} e^{3}-275 B x \,c^{2} d^{4} e +3 A \,a^{2} e^{5}+16 A a c \,d^{2} e^{3}+73 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}-66 B a c \,d^{3} e^{2}-128 B \,c^{2} d^{5}\right )}{15 e^{6} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\) | \(261\) |
Input:
int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/15*(((6*B*x^5+10*A*x^4)*c^2-60*a*x^2*(-B*x+A)*c-6*a^2*(5/3*B*x+A))*e^5-8 0*d*(x^3*(1/4*B*x+A)*c^2+a*x*(-9/2*B*x+A)*c+1/20*a^2*B)*e^4-32*c*d^2*((-5* B*x^3+15*A*x^2)*c+a*(-15*B*x+A))*e^3-640*c*d^3*(x*(-3/2*B*x+A)*c-3/10*B*a) *e^2-256*c^2*d^4*(-5*B*x+A)*e+512*B*c^2*d^5)/(e*x+d)^(5/2)/e^6
Time = 0.08 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 128 \, A c^{2} d^{4} e + 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 2 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 5 \, {\left (2 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 10 \, {\left (8 \, B c^{2} d^{2} e^{3} - 4 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 30 \, {\left (16 \, B c^{2} d^{3} e^{2} - 8 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 5 \, {\left (128 \, B c^{2} d^{4} e - 64 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} - 8 \, A a c d e^{4} - B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(7/2),x, algorithm="fricas")
Output:
2/15*(3*B*c^2*e^5*x^5 + 256*B*c^2*d^5 - 128*A*c^2*d^4*e + 96*B*a*c*d^3*e^2 - 16*A*a*c*d^2*e^3 - 2*B*a^2*d*e^4 - 3*A*a^2*e^5 - 5*(2*B*c^2*d*e^4 - A*c ^2*e^5)*x^4 + 10*(8*B*c^2*d^2*e^3 - 4*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 30* (16*B*c^2*d^3*e^2 - 8*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 - A*a*c*e^5)*x^2 + 5*( 128*B*c^2*d^4*e - 64*A*c^2*d^3*e^2 + 48*B*a*c*d^2*e^3 - 8*A*a*c*d*e^4 - B* a^2*e^5)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 1426 vs. \(2 (226) = 452\).
Time = 0.78 (sec) , antiderivative size = 1426, normalized size of antiderivative = 6.66 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(7/2),x)
Output:
Piecewise((-6*A*a**2*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 32*A*a*c*d**2*e**3/(15*d**2*e**6*s qrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 8 0*A*a*c*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 60*A*a*c*e**5*x**2/(15*d**2*e**6*sqrt(d + e *x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 256*A*c**2 *d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8* x**2*sqrt(d + e*x)) - 640*A*c**2*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 480*A*c**2*d**2 *e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e* *8*x**2*sqrt(d + e*x)) - 80*A*c**2*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 10*A*c**2*e** 5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x **2*sqrt(d + e*x)) - 4*B*a**2*d*e**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e* *7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 10*B*a**2*e**5*x/(15*d* *2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 192*B*a*c*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt (d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 480*B*a*c*d**2*e**3*x/(15*d**2*e **6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x) ) + 360*B*a*c*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqr...
Time = 0.06 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} - 5 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 30 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, B c^{2} d^{5} - 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} + 30 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(7/2),x, algorithm="maxima")
Output:
2/15*((3*(e*x + d)^(5/2)*B*c^2 - 5*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(3/2) + 30*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*sqrt(e*x + d))/e^5 + (3*B*c^2* d^5 - 3*A*c^2*d^4*e + 6*B*a*c*d^3*e^2 - 6*A*a*c*d^2*e^3 + 3*B*a^2*d*e^4 - 3*A*a^2*e^5 + 30*(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 - 5*(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)^(5/2)*e^5))/e
Time = 0.14 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.48 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (150 \, {\left (e x + d\right )}^{2} B c^{2} d^{3} - 25 \, {\left (e x + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 90 \, {\left (e x + d\right )}^{2} A c^{2} d^{2} e + 20 \, {\left (e x + d\right )} A c^{2} d^{3} e - 3 \, A c^{2} d^{4} e + 90 \, {\left (e x + d\right )}^{2} B a c d e^{2} - 30 \, {\left (e x + d\right )} B a c d^{2} e^{2} + 6 \, B a c d^{3} e^{2} - 30 \, {\left (e x + d\right )}^{2} A a c e^{3} + 20 \, {\left (e x + d\right )} A a c d e^{3} - 6 \, A a c d^{2} e^{3} - 5 \, {\left (e x + d\right )} B a^{2} e^{4} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{2} e^{24} - 25 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{2} d e^{24} + 150 \, \sqrt {e x + d} B c^{2} d^{2} e^{24} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{2} e^{25} - 60 \, \sqrt {e x + d} A c^{2} d e^{25} + 30 \, \sqrt {e x + d} B a c e^{26}\right )}}{15 \, e^{30}} \] Input:
integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^(7/2),x, algorithm="giac")
Output:
2/15*(150*(e*x + d)^2*B*c^2*d^3 - 25*(e*x + d)*B*c^2*d^4 + 3*B*c^2*d^5 - 9 0*(e*x + d)^2*A*c^2*d^2*e + 20*(e*x + d)*A*c^2*d^3*e - 3*A*c^2*d^4*e + 90* (e*x + d)^2*B*a*c*d*e^2 - 30*(e*x + d)*B*a*c*d^2*e^2 + 6*B*a*c*d^3*e^2 - 3 0*(e*x + d)^2*A*a*c*e^3 + 20*(e*x + d)*A*a*c*d*e^3 - 6*A*a*c*d^2*e^3 - 5*( e*x + d)*B*a^2*e^4 + 3*B*a^2*d*e^4 - 3*A*a^2*e^5)/((e*x + d)^(5/2)*e^6) + 2/15*(3*(e*x + d)^(5/2)*B*c^2*e^24 - 25*(e*x + d)^(3/2)*B*c^2*d*e^24 + 150 *sqrt(e*x + d)*B*c^2*d^2*e^24 + 5*(e*x + d)^(3/2)*A*c^2*e^25 - 60*sqrt(e*x + d)*A*c^2*d*e^25 + 30*sqrt(e*x + d)*B*a*c*e^26)/e^30
Time = 0.06 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.17 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{e^6}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^2\,e^4}{3}+4\,B\,a\,c\,d^2\,e^2-\frac {8\,A\,a\,c\,d\,e^3}{3}+\frac {10\,B\,c^2\,d^4}{3}-\frac {8\,A\,c^2\,d^3\,e}{3}\right )-{\left (d+e\,x\right )}^2\,\left (20\,B\,c^2\,d^3-12\,A\,c^2\,d^2\,e+12\,B\,a\,c\,d\,e^2-4\,A\,a\,c\,e^3\right )+\frac {2\,A\,a^2\,e^5}{5}-\frac {2\,B\,c^2\,d^5}{5}-\frac {2\,B\,a^2\,d\,e^4}{5}+\frac {2\,A\,c^2\,d^4\,e}{5}+\frac {4\,A\,a\,c\,d^2\,e^3}{5}-\frac {4\,B\,a\,c\,d^3\,e^2}{5}}{e^6\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \] Input:
int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^(7/2),x)
Output:
((d + e*x)^(1/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/e^6 - ((d + e *x)*((2*B*a^2*e^4)/3 + (10*B*c^2*d^4)/3 - (8*A*c^2*d^3*e)/3 - (8*A*a*c*d*e ^3)/3 + 4*B*a*c*d^2*e^2) - (d + e*x)^2*(20*B*c^2*d^3 - 4*A*a*c*e^3 - 12*A* c^2*d^2*e + 12*B*a*c*d*e^2) + (2*A*a^2*e^5)/5 - (2*B*c^2*d^5)/5 - (2*B*a^2 *d*e^4)/5 + (2*A*c^2*d^4*e)/5 + (4*A*a*c*d^2*e^3)/5 - (4*B*a*c*d^3*e^2)/5) /(e^6*(d + e*x)^(5/2)) + (2*B*c^2*(d + e*x)^(5/2))/(5*e^6) + (2*c^2*(A*e - 5*B*d)*(d + e*x)^(3/2))/(3*e^6)
Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {\frac {2}{5} b \,c^{2} e^{5} x^{5}+\frac {2}{3} a \,c^{2} e^{5} x^{4}-\frac {4}{3} b \,c^{2} d \,e^{4} x^{4}+4 a b c \,e^{5} x^{3}-\frac {16}{3} a \,c^{2} d \,e^{4} x^{3}+\frac {32}{3} b \,c^{2} d^{2} e^{3} x^{3}-4 a^{2} c \,e^{5} x^{2}+24 a b c d \,e^{4} x^{2}-32 a \,c^{2} d^{2} e^{3} x^{2}+64 b \,c^{2} d^{3} e^{2} x^{2}-\frac {2}{3} a^{2} b \,e^{5} x -\frac {16}{3} a^{2} c d \,e^{4} x +32 a b c \,d^{2} e^{3} x -\frac {128}{3} a \,c^{2} d^{3} e^{2} x +\frac {256}{3} b \,c^{2} d^{4} e x -\frac {2}{5} a^{3} e^{5}-\frac {4}{15} a^{2} b d \,e^{4}-\frac {32}{15} a^{2} c \,d^{2} e^{3}+\frac {64}{5} a b c \,d^{3} e^{2}-\frac {256}{15} a \,c^{2} d^{4} e +\frac {512}{15} b \,c^{2} d^{5}}{\sqrt {e x +d}\, e^{6} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(7/2),x)
Output:
(2*( - 3*a**3*e**5 - 2*a**2*b*d*e**4 - 5*a**2*b*e**5*x - 16*a**2*c*d**2*e* *3 - 40*a**2*c*d*e**4*x - 30*a**2*c*e**5*x**2 + 96*a*b*c*d**3*e**2 + 240*a *b*c*d**2*e**3*x + 180*a*b*c*d*e**4*x**2 + 30*a*b*c*e**5*x**3 - 128*a*c**2 *d**4*e - 320*a*c**2*d**3*e**2*x - 240*a*c**2*d**2*e**3*x**2 - 40*a*c**2*d *e**4*x**3 + 5*a*c**2*e**5*x**4 + 256*b*c**2*d**5 + 640*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 - 10*b*c**2*d*e**4*x* *4 + 3*b*c**2*e**5*x**5))/(15*sqrt(d + e*x)*e**6*(d**2 + 2*d*e*x + e**2*x* *2))