Integrand size = 29, antiderivative size = 678 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )}-\frac {\sqrt {2} \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)-2 c^2 \left (3 A c d^2-20 a B d e-9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 B e-3 b c (B d+A e)+2 c (3 A c d-5 a B e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^3 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
2*(e*x+d)^(3/2)*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*(A*e+B*d)+2* c*(A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)+2/3*e*(4*b^2*B*e-3* b*c*(A*e+B*d)+2*c*(3*A*c*d-5*B*a*e))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2 /(-4*a*c+b^2)-1/3*(8*b^3*B*e^2-b^2*c*e*(6*A*e+13*B*d)-2*c^2*(-9*A*a*e^2+3* A*c*d^2-20*B*a*d*e)+b*c*(6*A*c*d*e-29*B*a*e^2+3*B*c*d^2))*EllipticE(1/2*(( b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a* c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(e*x+d)^(1/2 )*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^3/(-4*a*c+b^2)^(1/2)/(c*x^2+b*x+ a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/3*(a*e^2-b*d *e+c*d^2)*(4*b^2*B*e-3*b*c*(A*e+B*d)+2*c*(3*A*c*d-5*B*a*e))*EllipticF(1/2* ((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4* a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-c*(c*x^2 +b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^( 1/2)/c^3/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 31.99 (sec) , antiderivative size = 1287, normalized size of antiderivative = 1.90 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right )^2 \left (\frac {2 B e^2}{3 c^2}+\frac {2 \left (A b c^2 d^2-2 a B c^2 d^2+2 a b B c d e-4 a A c^2 d e-a b^2 B e^2+a A b c e^2+2 a^2 B c e^2-b B c^2 d^2 x+2 A c^3 d^2 x+2 b^2 B c d e x-2 A b c^2 d e x-4 a B c^2 d e x-b^3 B e^2 x+A b^2 c e^2 x+3 a b B c e^2 x-2 a A c^2 e^2 x\right )}{c^2 \left (-b^2+4 a c\right ) \left (a+b x+c x^2\right )}\right )}{(a+x (b+c x))^{3/2}}+\frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \left (\left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)+2 c^2 \left (-3 A c d^2+20 a B d e+9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )-\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (8 b^3 B e^2-b^2 c e (13 B d+6 A e)+2 c^2 \left (-3 A c d^2+20 a B d e+9 a A e^2\right )+b c \left (3 B c d^2+6 A c d e-29 a B e^2\right )\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (8 b^4 B e^3-b^3 e^2 \left (21 B c d+6 A c e+8 B \sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c \left (-3 c d \sqrt {\left (b^2-4 a c\right ) e^2} (B d+2 A e)+a e^2 \left (84 B c d+24 A c e+29 B \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+b^2 c e \left (6 A e \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+B \left (15 c d^2-37 a e^2+13 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+2 c^2 \left (10 a^2 B e^3+3 A c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}-a e \left (10 B d \left (3 c d+2 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+3 A e \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )}{3 c^3 \left (-b^2+4 a c\right ) e (a+x (b+c x))^{3/2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]
(Sqrt[d + e*x]*(a + b*x + c*x^2)^2*((2*B*e^2)/(3*c^2) + (2*(A*b*c^2*d^2 - 2*a*B*c^2*d^2 + 2*a*b*B*c*d*e - 4*a*A*c^2*d*e - a*b^2*B*e^2 + a*A*b*c*e^2 + 2*a^2*B*c*e^2 - b*B*c^2*d^2*x + 2*A*c^3*d^2*x + 2*b^2*B*c*d*e*x - 2*A*b* c^2*d*e*x - 4*a*B*c^2*d*e*x - b^3*B*e^2*x + A*b^2*c*e^2*x + 3*a*b*B*c*e^2* x - 2*a*A*c^2*e^2*x))/(c^2*(-b^2 + 4*a*c)*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(3/2) + (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*((8*b^3*B*e^2 - b^2*c*e*(13*B*d + 6*A*e) + 2*c^2*(-3*A*c*d^2 + 20*a*B*d*e + 9*a*A*e^2) + b *c*(3*B*c*d^2 + 6*A*c*d*e - 29*a*B*e^2))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) - ((I/2)*Sqrt[1 - (2*(c*d^ 2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x)) ]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a *c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(8*b^3*B*e^ 2 - b^2*c*e*(13*B*d + 6*A*e) + 2*c^2*(-3*A*c*d^2 + 20*a*B*d*e + 9*a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d*e - 29*a*B*e^2))*EllipticE[I*ArcSinh[(Sqrt[2]* Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sq rt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sq rt[(b^2 - 4*a*c)*e^2]))] + (8*b^4*B*e^3 - b^3*e^2*(21*B*c*d + 6*A*c*e + 8* B*Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*(-3*c*d*Sqrt[(b^2 - 4*a*c)*e^2]*(B*d + 2* A*e) + a*e^2*(84*B*c*d + 24*A*c*e + 29*B*Sqrt[(b^2 - 4*a*c)*e^2])) + b^2*c *e*(6*A*e*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + B*(15*c*d^2 - 37*a*e^2 + ...
Time = 1.22 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1233, 27, 1236, 27, 1269, 1172, 321, 327}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {e \sqrt {d+e x} \left (B d b^2+3 A c d b+3 a B e b-10 a B c d-6 a A c e+\left (4 B e b^2-3 c (B d+A e) b+2 c (3 A c d-5 a B e)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {\sqrt {d+e x} \left (B d b^2+3 A c d b+3 a B e b-10 a B c d-6 a A c e+\left (4 B e b^2-3 c (B d+A e) b+2 c (3 A c d-5 a B e)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {e \left (\frac {2 \int -\frac {4 B d e b^3-\left (6 B c d^2+3 A c e d-4 a B e^2\right ) b^2-c \left (22 a B d e+3 A \left (c d^2+a e^2\right )\right ) b+2 a c \left (12 A c d e+5 B \left (3 c d^2-a e^2\right )\right )+\left (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b-2 c^2 \left (3 A c d^2-20 a B e d-9 a A e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c}-\frac {\int \frac {4 B d e b^3-\left (6 B c d^2+3 A c e d-4 a B e^2\right ) b^2-c \left (22 a B d e+3 A \left (c d^2+a e^2\right )\right ) b+2 a c \left (15 B c d^2+12 A c e d-5 a B e^2\right )+\left (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b-2 c^2 \left (3 A c d^2-20 a B e d-9 a A e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {e \left (\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c}-\frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{3 c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (B e b^2-c (B d+A e) b+2 c (A c d-a B e)\right ) x\right ) (d+e x)^{3/2}}{c \left (b^2-4 a c\right ) \sqrt {c x^2+b x+a}}+\frac {e \left (\frac {2 \left (4 B e b^2-3 c (B d+A e) b+2 c (3 A c d-5 a B e)\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (4 B e b^2-3 c (B d+A e) b+2 c (3 A c d-5 a B e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 B e^2 b^3-c e (13 B d+6 A e) b^2+c \left (3 B c d^2+6 A c e d-29 a B e^2\right ) b-2 c^2 \left (3 A c d^2-20 a B e d-9 a A e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 c}\right )}{c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {e \left (\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {e \left (\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right )}{3 c}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (2 c (3 A c d-5 a B e)-3 b c (A e+B d)+4 b^2 B e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b c \left (-29 a B e^2+6 A c d e+3 B c d^2\right )-2 c^2 \left (-9 a A e^2-20 a B d e+3 A c d^2\right )-b^2 c e (6 A e+13 B d)+8 b^3 B e^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 c}\right )}{c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^{3/2} \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\) |
(2*(d + e*x)^(3/2)*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c *(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c* x^2]) + (e*((2*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*S qrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) - ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8 *b^3*B*e^2 - b^2*c*e*(13*B*d + 6*A*e) - 2*c^2*(3*A*c*d^2 - 20*a*B*d*e - 9* a*A*e^2) + b*c*(3*B*c*d^2 + 6*A*c*d*e - 29*a*B*e^2))*Sqrt[d + e*x]*Sqrt[-( (c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2 *c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a* c]*(c*d^2 - b*d*e + a*e^2)*(4*b^2*B*e - 3*b*c*(B*d + A*e) + 2*c*(3*A*c*d - 5*a*B*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-( (c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2 *c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^ 2]))/(3*c)))/(c*(b^2 - 4*a*c))
3.27.35.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
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])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1569\) vs. \(2(616)=1232\).
Time = 9.40 (sec) , antiderivative size = 1570, normalized size of antiderivative = 2.32
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1570\) |
| risch | \(\text {Expression too large to display}\) | \(2217\) |
| default | \(\text {Expression too large to display}\) | \(10385\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2*(c*e*x +c*d)*((2*A*a*c^2*e^2-A*b^2*c*e^2+2*A*b*c^2*d*e-2*A*c^3*d^2-3*B*a*b*c*e^2+ 4*B*a*c^2*d*e+B*b^3*e^2-2*B*b^2*c*d*e+B*b*c^2*d^2)/c^3/(4*a*c-b^2)*x-(A*a* b*c*e^2-4*A*a*c^2*d*e+A*b*c^2*d^2+2*B*a^2*c*e^2-B*a*b^2*e^2+2*B*a*b*c*d*e- 2*B*a*c^2*d^2)/c^3/(4*a*c-b^2))/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2/3*B/ c^2*e^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(-e*(A*b*c*e^2-3 *A*c^2*d*e+B*a*c*e^2-B*b^2*e^2+3*B*b*c*d*e-3*B*c^2*d^2)/c^3+(4*A*a*b*c^2*e ^3-12*A*a*c^3*d*e^2-A*b^3*c*e^3+4*A*b^2*c^2*d*e^2-4*A*b*c^3*d^2*e+4*A*c^4* d^3+4*B*a^2*c^2*e^3-5*B*a*b^2*c*e^3+14*B*a*b*c^2*d*e^2-12*B*a*c^3*d^2*e+B* b^4*e^3-4*B*b^3*c*d*e^2+5*B*b^2*c^2*d^2*e-2*B*b*c^3*d^3)/(4*a*c-b^2)/c^3-1 /c^2*e*(A*a*b*c*e^2-4*A*a*c^2*d*e+A*b*c^2*d^2+2*B*a^2*c*e^2-B*a*b^2*e^2+2* B*a*b*c*d*e-2*B*a*c^2*d^2)/(4*a*c-b^2)+2/c^2*d*(2*A*a*c^2*e^2-A*b^2*c*e^2+ 2*A*b*c^2*d*e-2*A*c^3*d^2-3*B*a*b*c*e^2+4*B*a*c^2*d*e+B*b^3*e^2-2*B*b^2*c* d*e+B*b*c^2*d^2)/(4*a*c-b^2)-2/3*B/c^2*e^2*(1/2*a*e+1/2*b*d))*(d/e-1/2*(b+ (-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)* ((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^( 1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c) )^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e) /(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2 ))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(1/c^2*e^2*(A*c*e-...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1325, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
2/9*((3*(B*a*b*c^3 - 2*A*a*c^4)*d^3 + (8*B*a*b^2*c^2 - (50*B*a^2 - 9*A*a*b )*c^3)*d^2*e - (17*B*a*b^3*c + 54*A*a^2*c^3 - (77*B*a^2*b + 9*A*a*b^2)*c^2 )*d*e^2 + (8*B*a*b^4 + 3*(10*B*a^3 + 9*A*a^2*b)*c^2 - (41*B*a^2*b^2 + 6*A* a*b^3)*c)*e^3 + (3*(B*b*c^4 - 2*A*c^5)*d^3 + (8*B*b^2*c^3 - (50*B*a - 9*A* b)*c^4)*d^2*e - (17*B*b^3*c^2 + 54*A*a*c^4 - (77*B*a*b + 9*A*b^2)*c^3)*d*e ^2 + (8*B*b^4*c + 3*(10*B*a^2 + 9*A*a*b)*c^3 - (41*B*a*b^2 + 6*A*b^3)*c^2) *e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^3 + (8*B*b^3*c^2 - (50*B*a*b - 9* A*b^2)*c^3)*d^2*e - (17*B*b^4*c + 54*A*a*b*c^3 - (77*B*a*b^2 + 9*A*b^3)*c^ 2)*d*e^2 + (8*B*b^5 + 3*(10*B*a^2*b + 9*A*a*b^2)*c^2 - (41*B*a*b^3 + 6*A*b ^4)*c)*e^3)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6* a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e) /(c*e)) + 3*(3*(B*a*b*c^3 - 2*A*a*c^4)*d^2*e - (13*B*a*b^2*c^2 - 2*(20*B*a ^2 + 3*A*a*b)*c^3)*d*e^2 + (8*B*a*b^3*c + 18*A*a^2*c^3 - (29*B*a^2*b + 6*A *a*b^2)*c^2)*e^3 + (3*(B*b*c^4 - 2*A*c^5)*d^2*e - (13*B*b^2*c^3 - 2*(20*B* a + 3*A*b)*c^4)*d*e^2 + (8*B*b^3*c^2 + 18*A*a*c^4 - (29*B*a*b + 6*A*b^2)*c ^3)*e^3)*x^2 + (3*(B*b^2*c^3 - 2*A*b*c^4)*d^2*e - (13*B*b^3*c^2 - 2*(20*B* a*b + 3*A*b^2)*c^3)*d*e^2 + (8*B*b^4*c + 18*A*a*b*c^3 - (29*B*a*b^2 + 6*A* b^3)*c^2)*e^3)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - ...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]