Integrand size = 26, antiderivative size = 363 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {(d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac {d^{3/2} \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5}+\frac {(c d-b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 c^{5/2}} \]
-1/2*(e*x+d)^(5/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c* x^2+b*x)^2-1/4*d^(3/2)*(48*A*c^2*d^2+7*b^2*e*(5*A*e+4*B*d)-12*b*c*d*(7*A*e +2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5+1/4*(-b*e+c*d)^(3/2)*(48*A*c^3 *d^2-3*b^3*B*e^2-12*b*c^2*d*(A*e+2*B*d)-b^2*c*e*(A*e+8*B*d))*arctanh(c^(1/ 2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/c^(5/2)+1/4*(b*c*d^2*(12*A*c^2*d+2* b^2*B*e-b*c*(11*A*e+6*B*d))+(24*A*c^4*d^3-3*b^4*B*e^3-A*b^3*c*e^3-12*b*c^3 *d^2*(3*A*e+B*d)+b^2*c^2*d*e*(14*A*e+11*B*d))*x)*(e*x+d)^(1/2)/b^4/c^2/(c* x^2+b*x)
Time = 2.01 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b \sqrt {d+e x} \left (b B x \left (3 b^4 e^3 x+12 c^4 d^3 x^2+b c^3 d^2 x (18 d-11 e x)+b^3 c e^2 x (4 d+5 e x)+b^2 c^2 d \left (4 d^2-17 d e x-2 e^2 x^2\right )\right )+A c \left (b^4 e^3 x^2-24 c^4 d^3 x^3-36 b c^3 d^2 x^2 (d-e x)+b^2 c^2 d x \left (-8 d^2+55 d e x-10 e^2 x^2\right )+b^3 c \left (2 d^3+13 d^2 e x-16 d e^2 x^2-e^3 x^3\right )\right )\right )}{c^2 x^2 (b+c x)^2}+\frac {(-c d+b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{c^{5/2}}-d^{3/2} \left (-48 A c^2 d^2-7 b^2 e (4 B d+5 A e)+12 b c d (2 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5} \]
-1/4*((b*Sqrt[d + e*x]*(b*B*x*(3*b^4*e^3*x + 12*c^4*d^3*x^2 + b*c^3*d^2*x* (18*d - 11*e*x) + b^3*c*e^2*x*(4*d + 5*e*x) + b^2*c^2*d*(4*d^2 - 17*d*e*x - 2*e^2*x^2)) + A*c*(b^4*e^3*x^2 - 24*c^4*d^3*x^3 - 36*b*c^3*d^2*x^2*(d - e*x) + b^2*c^2*d*x*(-8*d^2 + 55*d*e*x - 10*e^2*x^2) + b^3*c*(2*d^3 + 13*d^ 2*e*x - 16*d*e^2*x^2 - e^3*x^3))))/(c^2*x^2*(b + c*x)^2) + ((-(c*d) + b*e) ^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8 *B*d + A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/c^(5/2) - d^(3/2)*(-48*A*c^2*d^2 - 7*b^2*e*(4*B*d + 5*A*e) + 12*b*c*d*(2*B*d + 7*A* e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^5
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{2 \left (c x^2+b x\right )^2}dx}{2 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^{3/2} \left (d \left (2 B e b^2-c (6 B d+11 A e) b+12 A c^2 d\right )+e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {(d+e x)^{3/2} \left (d \left (-2 B e b^2+6 B c d b+11 A c e b-12 A c^2 d\right )-e \left (-3 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{\left (c x^2+b x\right )^2}dx}{4 b^2 c}-\frac {(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}\) |
3.13.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Time = 0.72 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(2 e^{4} \left (\frac {\left (b e -c d \right )^{2} \left (\frac {\frac {b e \left (A b c e +12 A \,c^{2} d -5 b^{2} B e -8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {b e \left (A \,b^{2} c \,e^{2}-13 A b \,c^{2} d e +12 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+5 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c^{2}}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (A \,b^{2} c \,e^{2}+12 A b \,c^{2} d e -48 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+8 B \,b^{2} c d e +24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 c^{2} \sqrt {\left (b e -c d \right ) c}}\right )}{e^{4} b^{5}}-\frac {d^{2} \left (\frac {\left (\frac {13}{8} A \,b^{2} e^{2}-\frac {3}{2} A b c d e +\frac {1}{2} B \,b^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 A \,b^{2} e^{2}-84 A b c d e +48 A \,c^{2} d^{2}+28 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{4}}\right )\) | \(399\) |
| default | \(2 e^{4} \left (\frac {\left (b e -c d \right )^{2} \left (\frac {\frac {b e \left (A b c e +12 A \,c^{2} d -5 b^{2} B e -8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}-\frac {b e \left (A \,b^{2} c \,e^{2}-13 A b \,c^{2} d e +12 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+5 B \,b^{2} c d e -8 B b \,c^{2} d^{2}\right ) \sqrt {e x +d}}{8 c^{2}}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {\left (A \,b^{2} c \,e^{2}+12 A b \,c^{2} d e -48 A \,c^{3} d^{2}+3 b^{3} B \,e^{2}+8 B \,b^{2} c d e +24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 c^{2} \sqrt {\left (b e -c d \right ) c}}\right )}{e^{4} b^{5}}-\frac {d^{2} \left (\frac {\left (\frac {13}{8} A \,b^{2} e^{2}-\frac {3}{2} A b c d e +\frac {1}{2} B \,b^{2} d e \right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} A \,b^{2} d \,e^{2}+\frac {3}{2} A b c \,d^{2} e -\frac {1}{2} B \,b^{2} d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {\left (35 A \,b^{2} e^{2}-84 A b c d e +48 A \,c^{2} d^{2}+28 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{b^{5} e^{4}}\right )\) | \(399\) |
| pseudoelliptic | \(-\frac {12 \left (\left (-\frac {b^{4} e^{4} \left (A c +3 B b \right ) \sqrt {d}}{48}+c \left (-\frac {b^{4} B \,e^{3}}{24}-\frac {5 c \left (A e +\frac {11 B d}{10}\right ) e^{2} b^{3}}{24}+\frac {71 c^{2} \left (A e +\frac {40 B d}{71}\right ) d e \,b^{2}}{48}-\frac {9 c^{3} \left (A e +\frac {2 B d}{9}\right ) d^{2} b}{4}+A \,c^{4} d^{3}\right ) d^{\frac {3}{2}}\right ) x^{2} \left (c x +b \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )+\frac {\sqrt {\left (b e -c d \right ) c}\, \left (\frac {35 c^{2} x^{2} \left (c x +b \right )^{2} d^{2} \left (\frac {48 A \,c^{2} d^{2}}{35}-\frac {12 d \left (A e +\frac {2 B d}{7}\right ) b c}{5}+b^{2} e \left (A e +\frac {4 B d}{5}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\sqrt {e x +d}\, \left (\frac {x^{2} \left (\left (-A \,c^{2}+5 B b c \right ) x +A b c +3 B \,b^{2}\right ) e^{3} b^{3} \sqrt {d}}{2}+c \,d^{\frac {3}{2}} \left (-5 c \left (\frac {b^{3} B \,e^{2}}{5}+c e \left (A e +\frac {11 B d}{10}\right ) b^{2}-\frac {18 c^{2} d \left (A e +\frac {B d}{3}\right ) b}{5}+\frac {12 A \,c^{3} d^{2}}{5}\right ) x^{3}-8 \left (-\frac {b^{3} B \,e^{2}}{4}+c e \left (A e +\frac {17 B d}{16}\right ) b^{2}-\frac {55 c^{2} d \left (A e +\frac {18 B d}{55}\right ) b}{16}+\frac {9 A \,c^{3} d^{2}}{4}\right ) b \,x^{2}+\left (\left (\frac {13}{2} A d e +2 B \,d^{2}\right ) c \,b^{3}-4 A \,b^{2} c^{2} d^{2}\right ) x +A \,b^{3} c \,d^{2}\right )\right ) b \right )}{24}\right )}{\sqrt {\left (b e -c d \right ) c}\, \sqrt {d}\, \left (c x +b \right )^{2} b^{5} c^{2} x^{2}}\) | \(448\) |
| risch | \(-\frac {d^{2} \sqrt {e x +d}\, \left (13 A b e x -12 A c d x +4 B b d x +2 A b d \right )}{4 b^{4} x^{2}}-\frac {e \left (\frac {\frac {8 \left (-\frac {b e \left (A \,b^{3} c \,e^{3}+10 A \,b^{2} c^{2} d \,e^{2}-23 A b \,c^{3} d^{2} e +12 A \,c^{4} d^{3}-5 b^{4} B \,e^{3}+2 b^{3} B c d \,e^{2}+11 b^{2} B \,c^{2} d^{2} e -8 B b \,c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 c}+\frac {b e \left (A \,b^{4} c \,e^{4}-15 A \,b^{3} c^{2} d \,e^{3}+39 A \,b^{2} c^{3} d^{2} e^{2}-37 A b \,c^{4} d^{3} e +12 d^{4} A \,c^{5}+3 b^{5} B \,e^{4}-B \,b^{4} c d \,e^{3}-15 B \,b^{3} c^{2} d^{2} e^{2}+21 B \,b^{2} c^{3} d^{3} e -8 B b \,c^{4} d^{4}\right ) \sqrt {e x +d}}{8 c^{2}}\right )}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}-\frac {\left (A \,b^{4} c \,e^{4}+10 A \,b^{3} c^{2} d \,e^{3}-71 A \,b^{2} c^{3} d^{2} e^{2}+108 A b \,c^{4} d^{3} e -48 d^{4} A \,c^{5}+3 b^{5} B \,e^{4}+2 B \,b^{4} c d \,e^{3}+11 B \,b^{3} c^{2} d^{2} e^{2}-40 B \,b^{2} c^{3} d^{3} e +24 B b \,c^{4} d^{4}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{c^{2} \sqrt {\left (b e -c d \right ) c}}}{b e}+\frac {d^{\frac {3}{2}} \left (35 A \,b^{2} e^{2}-84 A b c d e +48 A \,c^{2} d^{2}+28 B \,b^{2} d e -24 c \,d^{2} B b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e}\right )}{4 b^{4}}\) | \(528\) |
2*e^4*((b*e-c*d)^2/e^4/b^5*((1/8*b*e*(A*b*c*e+12*A*c^2*d-5*B*b^2*e-8*B*b*c *d)/c*(e*x+d)^(3/2)-1/8*b/c^2*e*(A*b^2*c*e^2-13*A*b*c^2*d*e+12*A*c^3*d^2+3 *B*b^3*e^2+5*B*b^2*c*d*e-8*B*b*c^2*d^2)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d) ^2+1/8*(A*b^2*c*e^2+12*A*b*c^2*d*e-48*A*c^3*d^2+3*B*b^3*e^2+8*B*b^2*c*d*e+ 24*B*b*c^2*d^2)/c^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)* c)^(1/2)))-d^2/b^5/e^4*(((13/8*A*b^2*e^2-3/2*A*b*c*d*e+1/2*B*b^2*d*e)*(e*x +d)^(3/2)+(-11/8*A*b^2*d*e^2+3/2*A*b*c*d^2*e-1/2*B*b^2*d^2*e)*(e*x+d)^(1/2 ))/e^2/x^2+1/8*(35*A*b^2*e^2-84*A*b*c*d*e+48*A*c^2*d^2+28*B*b^2*d*e-24*B*b *c*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 836 vs. \(2 (335) = 670\).
Time = 24.00 (sec) , antiderivative size = 3378, normalized size of antiderivative = 9.31 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt((c* d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/ c))/(c*x + b)) - ((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B *b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2* d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^ 2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^ 4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4 )*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3) *x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d ^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (1 3*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^ 5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2), -1/8*(2*((24*(B*b*c^5 - 2*A*c^6) *d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d *e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1025 vs. \(2 (335) = 670\).
Time = 0.31 (sec) , antiderivative size = 1025, normalized size of antiderivative = 2.82 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/4*(24*B*b*c*d^4 - 48*A*c^2*d^4 - 28*B*b^2*d^3*e + 84*A*b*c*d^3*e - 35*A *b^2*d^2*e^2)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b* c^4*d^4 - 48*A*c^5*d^4 - 40*B*b^2*c^3*d^3*e + 108*A*b*c^4*d^3*e + 11*B*b^3 *c^2*d^2*e^2 - 71*A*b^2*c^3*d^2*e^2 + 2*B*b^4*c*d*e^3 + 10*A*b^3*c^2*d*e^3 + 3*B*b^5*e^4 + A*b^4*c*e^4)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e)) /(sqrt(-c^2*d + b*c*e)*b^5*c^2) - 1/4*(12*(e*x + d)^(7/2)*B*b*c^4*d^3*e - 24*(e*x + d)^(7/2)*A*c^5*d^3*e - 36*(e*x + d)^(5/2)*B*b*c^4*d^4*e + 72*(e* x + d)^(5/2)*A*c^5*d^4*e + 36*(e*x + d)^(3/2)*B*b*c^4*d^5*e - 72*(e*x + d) ^(3/2)*A*c^5*d^5*e - 12*sqrt(e*x + d)*B*b*c^4*d^6*e + 24*sqrt(e*x + d)*A*c ^5*d^6*e - 11*(e*x + d)^(7/2)*B*b^2*c^3*d^2*e^2 + 36*(e*x + d)^(7/2)*A*b*c ^4*d^2*e^2 + 51*(e*x + d)^(5/2)*B*b^2*c^3*d^3*e^2 - 144*(e*x + d)^(5/2)*A* b*c^4*d^3*e^2 - 69*(e*x + d)^(3/2)*B*b^2*c^3*d^4*e^2 + 180*(e*x + d)^(3/2) *A*b*c^4*d^4*e^2 + 29*sqrt(e*x + d)*B*b^2*c^3*d^5*e^2 - 72*sqrt(e*x + d)*A *b*c^4*d^5*e^2 - 2*(e*x + d)^(7/2)*B*b^3*c^2*d*e^3 - 10*(e*x + d)^(7/2)*A* b^2*c^3*d*e^3 - 11*(e*x + d)^(5/2)*B*b^3*c^2*d^2*e^3 + 85*(e*x + d)^(5/2)* A*b^2*c^3*d^2*e^3 + 32*(e*x + d)^(3/2)*B*b^3*c^2*d^3*e^3 - 148*(e*x + d)^( 3/2)*A*b^2*c^3*d^3*e^3 - 19*sqrt(e*x + d)*B*b^3*c^2*d^4*e^3 + 73*sqrt(e*x + d)*A*b^2*c^3*d^4*e^3 + 5*(e*x + d)^(7/2)*B*b^4*c*e^4 - (e*x + d)^(7/2)*A *b^3*c^2*e^4 - 11*(e*x + d)^(5/2)*B*b^4*c*d*e^4 - 13*(e*x + d)^(5/2)*A*b^3 *c^2*d*e^4 + 7*(e*x + d)^(3/2)*B*b^4*c*d^2*e^4 + 42*(e*x + d)^(3/2)*A*b...
Time = 15.77 (sec) , antiderivative size = 11072, normalized size of antiderivative = 30.50 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((((64*A*b^13*c^4*d*e^6 + 192*B*b^14*c^3*d*e^6 - 1536*A*b^10*c^7*d^4* e^3 + 3072*A*b^11*c^6*d^3*e^4 - 1600*A*b^12*c^5*d^2*e^5 + 768*B*b^11*c^6*d ^4*e^3 - 1088*B*b^12*c^5*d^3*e^4 + 128*B*b^13*c^4*d^2*e^5)/(64*b^12*c^3) - ((64*b^11*c^5*e^3 - 128*b^10*c^6*d*e^2)*(d + e*x)^(1/2)*(-(9*B^2*b^9*e^7 - 2304*A^2*c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2 *c^7*d^5*e^2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^ 2*b^5*c^4*d^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91 *B^2*b^7*c^2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2* b^6*c^3*d*e^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c* e^7 - 6720*A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5 *e^2 - 1960*A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^ 3*d^2*e^5)/(64*b^10*c^5))^(1/2))/(8*b^8*c^3))*(-(9*B^2*b^9*e^7 - 2304*A^2* c^9*d^7 + A^2*b^7*c^2*e^7 - 576*B^2*b^2*c^7*d^7 - 10416*A^2*b^2*c^7*d^5*e^ 2 + 5880*A^2*b^3*c^6*d^4*e^3 - 1225*A^2*b^4*c^5*d^3*e^4 - 21*A^2*b^5*c^4*d ^2*e^5 - 784*B^2*b^4*c^5*d^5*e^2 - 105*B^2*b^6*c^3*d^3*e^4 + 91*B^2*b^7*c^ 2*d^2*e^5 + 8064*A^2*b*c^8*d^6*e + 21*B^2*b^8*c*d*e^6 + 21*A^2*b^6*c^3*d*e ^6 + 1344*B^2*b^3*c^6*d^6*e + 2304*A*B*b*c^8*d^7 + 6*A*B*b^8*c*e^7 - 6720* A*B*b^2*c^7*d^6*e + 70*A*B*b^7*c^2*d*e^6 + 6384*A*B*b^3*c^6*d^5*e^2 - 1960 *A*B*b^4*c^5*d^4*e^3 + 210*A*B*b^5*c^4*d^3*e^4 - 294*A*B*b^6*c^3*d^2*e^5)/ (64*b^10*c^5))^(1/2) - ((d + e*x)^(1/2)*(9*B^2*b^10*e^10 + A^2*b^8*c^2*...