Integrand size = 34, antiderivative size = 826 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {\left (15 a^3 d^6 D-161 a^2 b d^4 \left (2 c C d-B d^2-3 c^2 D\right )-105 b^3 c^3 \left (6 c^2 C d-5 B c d^2+4 A d^3-7 c^3 D\right )-245 a b^2 c d^2 \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) \sqrt {a+b x^2}}{105 b d^8}+\frac {\left (11 a^2 d^4 (C d-2 c D)+8 b^2 c^2 \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )+18 a b d^2 \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) x \sqrt {a+b x^2}}{16 d^7}+\frac {\left (45 a^2 d^4 D-77 a b d^2 \left (2 c C d-B d^2-3 c^2 D\right )-35 b^2 c \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) x^2 \sqrt {a+b x^2}}{105 d^6}+\frac {b \left (13 a d^2 (C d-2 c D)+6 b \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) x^3 \sqrt {a+b x^2}}{24 d^5}+\frac {b \left (15 a d^2 D-7 b \left (2 c C d-B d^2-3 c^2 D\right )\right ) x^4 \sqrt {a+b x^2}}{35 d^4}+\frac {b^2 (C d-2 c D) x^5 \sqrt {a+b x^2}}{6 d^3}+\frac {b^2 D x^6 \sqrt {a+b x^2}}{7 d^2}-\frac {\left (b c^2+a d^2\right )^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{d^8 (c+d x)}+\frac {\left (5 a^3 d^6 (C d-2 c D)+16 b^3 c^4 \left (7 c^2 C d-6 B c d^2+5 A d^3-8 c^3 D\right )+40 a b^2 c^2 d^2 \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )+30 a^2 b d^4 \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{16 \sqrt {b} d^9}+\frac {\left (b c^2+a d^2\right )^{3/2} \left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (7 c^2 C d-6 B c d^2+5 A d^3-8 c^3 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^9} \] Output:
1/105*(15*a^3*d^6*D-161*a^2*b*d^4*(-B*d^2+2*C*c*d-3*D*c^2)-105*b^3*c^3*(4* A*d^3-5*B*c*d^2+6*C*c^2*d-7*D*c^3)-245*a*b^2*c*d^2*(2*A*d^3-3*B*c*d^2+4*C* c^2*d-5*D*c^3))*(b*x^2+a)^(1/2)/b/d^8+1/16*(11*a^2*d^4*(C*d-2*D*c)+8*b^2*c ^2*(3*A*d^3-4*B*c*d^2+5*C*c^2*d-6*D*c^3)+18*a*b*d^2*(A*d^3-2*B*c*d^2+3*C*c ^2*d-4*D*c^3))*x*(b*x^2+a)^(1/2)/d^7+1/105*(45*a^2*d^4*D-77*a*b*d^2*(-B*d^ 2+2*C*c*d-3*D*c^2)-35*b^2*c*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3))*x^2*(b* x^2+a)^(1/2)/d^6+1/24*b*(13*a*d^2*(C*d-2*D*c)+6*b*(A*d^3-2*B*c*d^2+3*C*c^2 *d-4*D*c^3))*x^3*(b*x^2+a)^(1/2)/d^5+1/35*b*(15*a*d^2*D-7*b*(-B*d^2+2*C*c* d-3*D*c^2))*x^4*(b*x^2+a)^(1/2)/d^4+1/6*b^2*(C*d-2*D*c)*x^5*(b*x^2+a)^(1/2 )/d^3+1/7*b^2*D*x^6*(b*x^2+a)^(1/2)/d^2-(a*d^2+b*c^2)^2*(A*d^3-B*c*d^2+C*c ^2*d-D*c^3)*(b*x^2+a)^(1/2)/d^8/(d*x+c)+1/16*(5*a^3*d^6*(C*d-2*D*c)+16*b^3 *c^4*(5*A*d^3-6*B*c*d^2+7*C*c^2*d-8*D*c^3)+40*a*b^2*c^2*d^2*(3*A*d^3-4*B*c *d^2+5*C*c^2*d-6*D*c^3)+30*a^2*b*d^4*(A*d^3-2*B*c*d^2+3*C*c^2*d-4*D*c^3))* arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^9+(a*d^2+b*c^2)^(3/2)*(a*d^2* (-B*d^2+2*C*c*d-3*D*c^2)+b*c*(5*A*d^3-6*B*c*d^2+7*C*c^2*d-8*D*c^3))*arctan h((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^9
Time = 10.05 (sec) , antiderivative size = 722, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (240 a^3 d^6 D (c+d x)+4 b^3 \left (3360 c^7 D-420 c^6 d (7 C-4 D x)+d^7 x^4 \left (105 A+84 B x+70 C x^2+60 D x^3\right )+70 c^5 d^2 (36 B-x (21 C+8 D x))-70 c^4 d^3 \left (30 A-x \left (18 B+7 C x+4 D x^2\right )\right )+7 c^2 d^5 x^2 \left (50 A+x \left (30 B+21 C x+16 D x^2\right )\right )-7 c^3 d^4 x \left (150 A+x \left (60 B+35 C x+24 D x^2\right )\right )-c d^6 x^3 \left (175 A+2 x \left (63 B+49 C x+40 D x^2\right )\right )\right )+a^2 b d^4 \left (9408 c^3 D+c^2 (-6832 C d+5418 d D x)+c d^2 (4256 B-x (3997 C+1590 D x))+d^3 (-1680 A+x (2576 B+15 x (77 C+48 D x)))\right )+2 a b^2 d^2 \left (11480 c^5 D-140 c^4 d (68 C-43 D x)+7 c^3 d^2 (1080 B-x (715 C+276 D x))+d^5 x^2 (945 A+x (616 B+5 x (91 C+72 D x)))+7 c^2 d^3 (-800 A+x (570 B+x (229 C+134 D x)))-c d^4 x (2975 A+x (1274 B+x (777 C+550 D x)))\right )\right )}{b (c+d x)}-3360 \left (-b c^2-a d^2\right )^{3/2} \left (a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-7 c^2 C d+6 B c d^2-5 A d^3+8 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {105 \left (-5 a^3 d^6 (C d-2 c D)-30 a^2 b d^4 \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )+40 a b^2 c^2 d^2 \left (-5 c^2 C d+4 B c d^2-3 A d^3+6 c^3 D\right )+16 b^3 c^4 \left (-7 c^2 C d+6 B c d^2-5 A d^3+8 c^3 D\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{1680 d^9} \] Input:
Integrate[((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^2,x]
Output:
((d*Sqrt[a + b*x^2]*(240*a^3*d^6*D*(c + d*x) + 4*b^3*(3360*c^7*D - 420*c^6 *d*(7*C - 4*D*x) + d^7*x^4*(105*A + 84*B*x + 70*C*x^2 + 60*D*x^3) + 70*c^5 *d^2*(36*B - x*(21*C + 8*D*x)) - 70*c^4*d^3*(30*A - x*(18*B + 7*C*x + 4*D* x^2)) + 7*c^2*d^5*x^2*(50*A + x*(30*B + 21*C*x + 16*D*x^2)) - 7*c^3*d^4*x* (150*A + x*(60*B + 35*C*x + 24*D*x^2)) - c*d^6*x^3*(175*A + 2*x*(63*B + 49 *C*x + 40*D*x^2))) + a^2*b*d^4*(9408*c^3*D + c^2*(-6832*C*d + 5418*d*D*x) + c*d^2*(4256*B - x*(3997*C + 1590*D*x)) + d^3*(-1680*A + x*(2576*B + 15*x *(77*C + 48*D*x)))) + 2*a*b^2*d^2*(11480*c^5*D - 140*c^4*d*(68*C - 43*D*x) + 7*c^3*d^2*(1080*B - x*(715*C + 276*D*x)) + d^5*x^2*(945*A + x*(616*B + 5*x*(91*C + 72*D*x))) + 7*c^2*d^3*(-800*A + x*(570*B + x*(229*C + 134*D*x) )) - c*d^4*x*(2975*A + x*(1274*B + x*(777*C + 550*D*x))))))/(b*(c + d*x)) - 3360*(-(b*c^2) - a*d^2)^(3/2)*(a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c* (-7*c^2*C*d + 6*B*c*d^2 - 5*A*d^3 + 8*c^3*D))*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + (105*(-5*a^3*d^6*(C*d - 2*c*D ) - 30*a^2*b*d^4*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D) + 40*a*b^2*c^2* d^2*(-5*c^2*C*d + 4*B*c*d^2 - 3*A*d^3 + 6*c^3*D) + 16*b^3*c^4*(-7*c^2*C*d + 6*B*c*d^2 - 5*A*d^3 + 8*c^3*D))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqr t[b])/(1680*d^9)
Time = 1.87 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2182, 25, 2185, 27, 682, 25, 27, 682, 27, 682, 27, 719, 224, 219, 488, 219}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{5/2} \left (\left (\frac {b c^2}{d}+a d\right ) D x^2+\left (a (C d-c D)-b \left (\frac {7 D c^3}{d^2}-\frac {7 C c^2}{d}+6 B c-6 A d\right )\right ) x+A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{c+d x}dx}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{5/2} \left (\left (\frac {b c^2}{d}+a d\right ) D x^2+\left (a (C d-c D)-b \left (\frac {7 D c^3}{d^2}-\frac {7 C c^2}{d}+6 B c-6 A d\right )\right ) x+A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{c+d x}dx}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {\int \frac {7 b \left (d \left (A b c d-a \left (-D c^2+C d c-B d^2\right )\right )+\left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{7 b d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d \left (A b c d-a \left (-D c^2+C d c-B d^2\right )\right )+\left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{c+d x}dx}{d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {b \left (b c^2+a d^2\right ) \left (a d \left (-8 D c^2+7 C d c-6 B d^2\right )-\left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{6 b d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-5 d x \left (a d^2 (C d-2 c D)+b \left (6 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{30 d^2}}{d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {b \left (b c^2+a d^2\right ) \left (a d \left (-8 D c^2+7 C d c-6 B d^2\right )-\left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{6 b d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-5 d x \left (a d^2 (C d-2 c D)+b \left (6 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{30 d^2}}{d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\left (a d^2+b c^2\right ) \int \frac {\left (a d \left (-8 D c^2+7 C d c-6 B d^2\right )-\left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{6 d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-5 d x \left (a d^2 (C d-2 c D)+b \left (6 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{30 d^2}}{d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {-\frac {\left (a d^2+b c^2\right ) \left (\frac {\int \frac {b \left (3 a d \left (a \left (-14 D c^2+11 C d c-8 B d^2\right ) d^2+2 b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-\left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 b d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-d x \left (5 a d^2 (C d-2 c D)+6 b \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{4 d^2}\right )}{6 d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-5 d x \left (a d^2 (C d-2 c D)+b \left (6 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{30 d^2}}{d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\left (a d^2+b c^2\right ) \left (\frac {\int \frac {\left (3 a d \left (a \left (-14 D c^2+11 C d c-8 B d^2\right ) d^2+2 b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-\left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (8 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-d x \left (5 a d^2 (C d-2 c D)+6 b \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{4 d^2}\right )}{6 d^2}-\frac {\left (a+b x^2\right )^{5/2} \left (6 \left (a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (5 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )-5 d x \left (a d^2 (C d-2 c D)+b \left (6 A d^3-6 B c d^2-8 c^3 D+7 c^2 C d\right )\right )\right )}{30 d^2}}{d^2}+\frac {1}{7} D \left (a+b x^2\right )^{7/2} \left (\frac {a}{b}+\frac {c^2}{d^2}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {\int \frac {3 b \left (a d \left (a^2 \left (-38 D c^2+27 C d c-16 B d^2\right ) d^4+2 a b c \left (-52 D c^3+43 C d c^2-34 B d^2 c+25 A d^3\right ) d^2+8 b^2 c^3 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {3 \int \frac {a d \left (a^2 \left (-38 D c^2+27 C d c-16 B d^2\right ) d^4+2 a b c \left (-52 D c^3+43 C d c^2-34 B d^2 c+25 A d^3\right ) d^2+8 b^2 c^3 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {3 \left (\frac {16 \left (b c^2+a d^2\right )^2 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {3 \left (\frac {16 \left (b c^2+a d^2\right )^2 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{2 d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {3 \left (\frac {16 \left (b c^2+a d^2\right )^2 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}\right )}{2 d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {3 \left (-\frac {16 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}} \left (b c^2+a d^2\right )^2}{d}-\frac {\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}\right )}{2 d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) D \left (b x^2+a\right )^{7/2}+\frac {-\frac {\left (6 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-5 d \left (a (C d-2 c D) d^2+b \left (-8 D c^3+7 C d c^2-6 B d^2 c+6 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{30 d^2}-\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (8 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{4 d^2}+\frac {\frac {\sqrt {b x^2+a} \left (48 \left (b c^2+a d^2\right ) \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )-d \left (4 a b c \left (-8 D c^2+7 C d c-6 B d^2\right ) d^2+\left (4 b c^2+3 a d^2\right ) \left (5 a (C d-2 c D) d^2+6 b \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right )\right ) x\right )}{2 d^2}+\frac {3 \left (-\frac {16 \left (a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right ) \left (b c^2+a d^2\right )^{3/2}}{d}-\frac {\left (5 a^3 (C d-2 c D) d^6+30 a^2 b \left (-4 D c^3+3 C d c^2-2 B d^2 c+A d^3\right ) d^4+40 a b^2 c^2 \left (-6 D c^3+5 C d c^2-4 B d^2 c+3 A d^3\right ) d^2+16 b^3 c^4 \left (-8 D c^3+7 C d c^2-6 B d^2 c+5 A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}\right )}{2 d^2}}{4 d^2}\right )}{6 d^2}}{d^2}}{b c^2+a d^2}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{7/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}\) |
Input:
Int[((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^2,x]
Output:
-(((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(7/2))/(d^2*(b*c^2 + a* d^2)*(c + d*x))) + (((a/b + c^2/d^2)*D*(a + b*x^2)^(7/2))/7 + (-1/30*((6*( a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(7*c^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 8*c^3*D)) - 5*d*(a*d^2*(C*d - 2*c*D) + b*(7*c^2*C*d - 6*B*c*d^2 + 6*A*d^3 - 8*c^3*D))*x)*(a + b*x^2)^(5/2))/d^2 - ((b*c^2 + a*d^2)*(((8*(a*d^2*(2*c *C*d - B*d^2 - 3*c^2*D) + b*c*(7*c^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 8*c^3*D)) - d*(5*a*d^2*(C*d - 2*c*D) + 6*b*(7*c^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 8*c^3 *D))*x)*(a + b*x^2)^(3/2))/(4*d^2) + (((48*(b*c^2 + a*d^2)*(a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(7*c^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 8*c^3*D)) - d *(4*a*b*c*d^2*(7*c*C*d - 6*B*d^2 - 8*c^2*D) + (4*b*c^2 + 3*a*d^2)*(5*a*d^2 *(C*d - 2*c*D) + 6*b*(7*c^2*C*d - 6*B*c*d^2 + 5*A*d^3 - 8*c^3*D)))*x)*Sqrt [a + b*x^2])/(2*d^2) + (3*(-(((5*a^3*d^6*(C*d - 2*c*D) + 16*b^3*c^4*(7*c^2 *C*d - 6*B*c*d^2 + 5*A*d^3 - 8*c^3*D) + 40*a*b^2*c^2*d^2*(5*c^2*C*d - 4*B* c*d^2 + 3*A*d^3 - 6*c^3*D) + 30*a^2*b*d^4*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (16*(b*c^2 + a*d^2)^(3/2)*(a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(7*c^2*C*d - 6*B* c*d^2 + 5*A*d^3 - 8*c^3*D))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqr t[a + b*x^2])])/d))/(2*d^2))/(4*d^2)))/(6*d^2))/d^2)/(b*c^2 + a*d^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(2344\) vs. \(2(782)=1564\).
Time = 1.43 (sec) , antiderivative size = 2345, normalized size of antiderivative = 2.84
Input:
int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
1/d^3*(C*d*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2* x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2)))))+1/7*D*d*( b*x^2+a)^(7/2)/b-2*D*c*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(1/4*x*(b*x^2+a)^(3/2) +3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))) )))+1/d^4*(B*d^2-2*C*c*d+3*D*c^2)*(1/5*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2 +b*c^2)/d^2)^(5/2)-b*c/d*(1/8*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d *(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+3/16*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^ 2)/b*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^ 2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/d +b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)) ))+(a*d^2+b*c^2)/d^2*(1/3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^ (3/2)-b*c/d*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d ^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln( (-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2) ^(1/2)))+(a*d^2+b*c^2)/d^2*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2 )^(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+ c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(a*d^2+b*c^2)/d^2/((a*d^2+b*c^2)/d^2)^(1/2) *ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x +c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))))+1/d^5*(A*d^3 -B*c*d^2+C*c^2*d-D*c^3)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{2}}\, dx \] Input:
integrate((b*x**2+a)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**2,x)
Output:
Integral((a + b*x**2)**(5/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**2, x)
Time = 0.24 (sec) , antiderivative size = 1565, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x, algorithm="maxi ma")
Output:
(b*x^2 + a)^(5/2)*D*c^3/(d^5*x + c*d^4) - (b*x^2 + a)^(5/2)*C*c^2/(d^4*x + c*d^3) + (b*x^2 + a)^(5/2)*B*c/(d^3*x + c*d^2) - (b*x^2 + a)^(5/2)*A/(d^2 *x + c*d) - 4*sqrt(b*x^2 + a)*D*b^2*c^5*x/d^7 + 7/2*sqrt(b*x^2 + a)*C*b^2* c^4*x/d^6 - 2*(b*x^2 + a)^(3/2)*D*b*c^3*x/d^5 - 9/2*sqrt(b*x^2 + a)*D*a*b* c^3*x/d^5 - 3*sqrt(b*x^2 + a)*B*b^2*c^3*x/d^5 + 7/4*(b*x^2 + a)^(3/2)*C*b* c^2*x/d^4 + 29/8*sqrt(b*x^2 + a)*C*a*b*c^2*x/d^4 + 5/2*sqrt(b*x^2 + a)*A*b ^2*c^2*x/d^4 - 1/3*(b*x^2 + a)^(5/2)*D*c*x/d^3 - 5/12*(b*x^2 + a)^(3/2)*D* a*c*x/d^3 - 5/8*sqrt(b*x^2 + a)*D*a^2*c*x/d^3 - 3/2*(b*x^2 + a)^(3/2)*B*b* c*x/d^3 - 11/4*sqrt(b*x^2 + a)*B*a*b*c*x/d^3 + 1/6*(b*x^2 + a)^(5/2)*C*x/d ^2 + 5/24*(b*x^2 + a)^(3/2)*C*a*x/d^2 + 5/16*sqrt(b*x^2 + a)*C*a^2*x/d^2 + 5/4*(b*x^2 + a)^(3/2)*A*b*x/d^2 + 15/8*sqrt(b*x^2 + a)*A*a*b*x/d^2 - 8*D* b^(5/2)*c^7*arcsinh(b*x/sqrt(a*b))/d^9 + 7*C*b^(5/2)*c^6*arcsinh(b*x/sqrt( a*b))/d^8 - 15*D*a*b^(3/2)*c^5*arcsinh(b*x/sqrt(a*b))/d^7 - 6*B*b^(5/2)*c^ 5*arcsinh(b*x/sqrt(a*b))/d^7 + 25/2*C*a*b^(3/2)*c^4*arcsinh(b*x/sqrt(a*b)) /d^6 + 5*A*b^(5/2)*c^4*arcsinh(b*x/sqrt(a*b))/d^6 - 15/2*D*a^2*sqrt(b)*c^3 *arcsinh(b*x/sqrt(a*b))/d^5 - 10*B*a*b^(3/2)*c^3*arcsinh(b*x/sqrt(a*b))/d^ 5 + 45/8*C*a^2*sqrt(b)*c^2*arcsinh(b*x/sqrt(a*b))/d^4 + 15/2*A*a*b^(3/2)*c ^2*arcsinh(b*x/sqrt(a*b))/d^4 - 5/8*D*a^3*c*arcsinh(b*x/sqrt(a*b))/(sqrt(b )*d^3) - 15/4*B*a^2*sqrt(b)*c*arcsinh(b*x/sqrt(a*b))/d^3 + 5/16*C*a^3*arcs inh(b*x/sqrt(a*b))/(sqrt(b)*d^2) + 15/8*A*a^2*sqrt(b)*arcsinh(b*x/sqrt(...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x, algorithm="giac ")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^2,x)
Output:
int(((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^2, x)
\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{2}}d x \] Input:
int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x)
Output:
int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x)