Integrand size = 30, antiderivative size = 573 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (128 c^2 d^2+15 b^2 e^2-4 c e (28 b d-9 a e)+16 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (16 c d-5 b e+6 c e x) \left (a+b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 e^5 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+15 b^2 e^2-4 c e (32 b d-17 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
2/15*(6*c*e*x-5*b*e+16*c*d)*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)^(3/2)-2/15*(12 8*c^2*d^2+15*b^2*e^2-4*c*e*(-9*a*e+28*b*d)+16*c*e*(-b*e+2*c*d)*x)*(c*x^2+b *x+a)^(1/2)/e^4/(e*x+d)^(1/2)+2/15*(128*c^2*d^2+23*b^2*e^2-4*c*e*(-9*a*e+3 2*b*d))*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)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2 ))^(1/2)/e^5/(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)-2/15*(-b*e+2*c*d)*(128*c^2*d^2+15*b^2*e^2-4*c*e*(-17*a*e+32*b*d) )*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)*(-4*a*c+b^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/e^5/(e*x+d)^(1/2)/(c*x^2+b*x+a)^ (1/2)
Result contains complex when optimal does not.
Time = 17.46 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.18 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {d+e x} (a+x (b+c x))^{3/2} \left (-\frac {2 c (28 c d-17 b e)}{15 e^4}+\frac {4 c^2 x}{5 e^3}-\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^2}-\frac {4 \left (11 c^2 d^2-11 b c d e+2 b^2 e^2+3 a c e^2\right )}{3 e^4 (d+e x)}\right )}{a+b x+c x^2}-\frac {(d+e x)^{3/2} (a+x (b+c x))^{3/2} \left (-4 \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}}} \left (128 c^2 d^2+23 b^2 e^2+4 c e (-32 b d+9 a e)\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 {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (128 c^2 d^2+23 b^2 e^2+4 c e (-32 b d+9 a e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} 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 )}{\sqrt {d+e x}}-\frac {i \sqrt {2} \left (-8 b^3 e^3+b^2 e^2 \left (16 c d+23 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+32 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (32 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-16 c d+9 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \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 )}{\sqrt {d+e x}}\right )}{15 e^6 \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}}} \left (a+b x+c x^2\right )^{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 + x*(b + c*x))^(3/2)*((-2*c*(28*c*d - 17*b*e))/(15*e^4) + (4*c^2*x)/(5*e^3) - (2*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^4*(d + e*x)^2) - (4*(11*c^2*d^2 - 11*b*c*d*e + 2*b^2*e^2 + 3*a*c*e^2))/(3*e^4* (d + e*x))))/(a + b*x + c*x^2) - ((d + e*x)^(3/2)*(a + x*(b + c*x))^(3/2)* (-4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2 ])]*(128*c^2*d^2 + 23*b^2*e^2 + 4*c*e*(-32*b*d + 9*a*e))*(c*(-1 + d/(d + e *x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + (I*Sqrt[ 2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(128*c^2*d^2 + 23*b^2*e^2 + 4*c *e*(-32*b*d + 9*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sq rt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x ) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*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])])/Sqrt[d + e*x]], -((-2 *c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^ 2]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-8*b^3*e^3 + b^2*e^2*(16*c*d + 23*Sqrt[ (b^2 - 4*a*c)*e^2]) + 32*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4 *c*(32*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-16*c*d + 9*Sqrt[(b^2 - 4*a* c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 ...
Time = 0.92 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1230, 27, 1230, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {\left (-5 e b^2+16 c d b-12 a c e+16 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^{3/2}}dx}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\int \frac {\left (-5 e b^2+16 c d b-12 a c e+16 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{3/2}}dx}{5 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}-\frac {2 \int -\frac {-15 e^2 b^3+112 c d e b^2-4 c \left (32 c d^2+17 a e^2\right ) b+64 a c^2 d e-2 c \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\int \frac {-15 e^2 b^3+112 c d e b^2-4 c \left (32 c d^2+17 a e^2\right ) b+64 a c^2 d e-2 c \left (128 c^2 d^2+23 b^2 e^2-4 c e (32 b d-9 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {(2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 c \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-5 b e+16 c d+6 c e x)}{15 e^2 (d+e x)^{3/2}}-\frac {\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-17 a e)+15 b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (32 b d-9 a e)+23 b^2 e^2+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{3 e^2}+\frac {2 \sqrt {a+b x+c x^2} \left (-4 c e (28 b d-9 a e)+15 b^2 e^2+16 c e x (2 c d-b e)+128 c^2 d^2\right )}{3 e^2 \sqrt {d+e x}}}{5 e^2}\) |
(2*(16*c*d - 5*b*e + 6*c*e*x)*(a + b*x + c*x^2)^(3/2))/(15*e^2*(d + e*x)^( 3/2)) - ((2*(128*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(28*b*d - 9*a*e) + 16*c*e*(2 *c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) + ((-2*Sqrt[2] *Sqrt[b^2 - 4*a*c]*(128*c^2*d^2 + 23*b^2*e^2 - 4*c*e*(32*b*d - 9*a*e))*Sqr t[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[S qrt[(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)])/(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]*(2*c*d - b*e)*(128*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(32*b*d - 17*a*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*e^2))/(5*e^2)
3.17.35.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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. \(1401\) vs. \(2(509)=1018\).
Time = 3.55 (sec) , antiderivative size = 1402, normalized size of antiderivative = 2.45
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1402\) |
| risch | \(\text {Expression too large to display}\) | \(2842\) |
| default | \(\text {Expression too large to display}\) | \(9599\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/3*(a*b *e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^6*(c*e*x^3+b*e*x^2+c*d *x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^2-4/3*(c*e*x^2+b*e*x+a*e)*(3*a*c*e^2+2 *b^2*e^2-11*b*c*d*e+11*c^2*d^2)/e^5/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+4/ 5*c^2/e^3*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(c^2/e^3*( 5*b*e-4*c*d)-4/5*c^2/e^3*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x +b*d*x+a*d)^(1/2)+2*((6*a*b*c*e^3-8*a*c^2*d*e^2+b^3*e^3-8*b^2*c*d*e^2+15*b *c^2*d^2*e-8*c^3*d^3)/e^5-1/3*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2 *c^2*d^3)/e^5*c-2/3*(3*a*c*e^2+2*b^2*e^2-11*b*c*d*e+11*c^2*d^2)/e^5*(b*e-c *d)+2/3*b/e^4*(3*a*c*e^2+2*b^2*e^2-11*b*c*d*e+11*c^2*d^2)-4/5*c^2/e^3*a*d- 2/3*(c^2/e^3*(5*b*e-4*c*d)-4/5*c^2/e^3*(2*b*e+2*c*d))/c/e*(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)*Ell ipticF(((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*(2*c/e ^4*(2*a*c*e^2+2*b^2*e^2-5*b*c*d*e+3*c^2*d^2)+2/3*(3*a*c*e^2+2*b^2*e^2-11*b *c*d*e+11*c^2*d^2)/e^4*c-4/5*c^2/e^3*(3/2*a*e+3/2*b*d)-2/3*(c^2/e^3*(5*b*e -4*c*d)-4/5*c^2/e^3*(2*b*e+2*c*d))/c/e*(b*e+c*d))*(d/e-1/2*(b+(-4*a*c+b...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.23 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.44 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left ({\left (256 \, c^{3} d^{5} - 384 \, b c^{2} d^{4} e + 6 \, {\left (21 \, b^{2} c + 44 \, a c^{2}\right )} d^{3} e^{2} + {\left (b^{3} - 132 \, a b c\right )} d^{2} e^{3} + {\left (256 \, c^{3} d^{3} e^{2} - 384 \, b c^{2} d^{2} e^{3} + 6 \, {\left (21 \, b^{2} c + 44 \, a c^{2}\right )} d e^{4} + {\left (b^{3} - 132 \, a b c\right )} e^{5}\right )} x^{2} + 2 \, {\left (256 \, c^{3} d^{4} e - 384 \, b c^{2} d^{3} e^{2} + 6 \, {\left (21 \, b^{2} c + 44 \, a c^{2}\right )} d^{2} e^{3} + {\left (b^{3} - 132 \, a b c\right )} d e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (128 \, c^{3} d^{4} e - 128 \, b c^{2} d^{3} e^{2} + {\left (23 \, b^{2} c + 36 \, a c^{2}\right )} d^{2} e^{3} + {\left (128 \, c^{3} d^{2} e^{3} - 128 \, b c^{2} d e^{4} + {\left (23 \, b^{2} c + 36 \, a c^{2}\right )} e^{5}\right )} x^{2} + 2 \, {\left (128 \, c^{3} d^{3} e^{2} - 128 \, b c^{2} d^{2} e^{3} + {\left (23 \, b^{2} c + 36 \, a c^{2}\right )} d e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (6 \, c^{3} e^{5} x^{3} - 128 \, c^{3} d^{3} e^{2} + 112 \, b c^{2} d^{2} e^{3} - 5 \, a b c e^{5} - 5 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d e^{4} - {\left (16 \, c^{3} d e^{4} - 17 \, b c^{2} e^{5}\right )} x^{2} - 2 \, {\left (80 \, c^{3} d^{2} e^{3} - 72 \, b c^{2} d e^{4} + 5 \, {\left (2 \, b^{2} c + 3 \, a c^{2}\right )} e^{5}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{45 \, {\left (c e^{8} x^{2} + 2 \, c d e^{7} x + c d^{2} e^{6}\right )}} \]
-2/45*((256*c^3*d^5 - 384*b*c^2*d^4*e + 6*(21*b^2*c + 44*a*c^2)*d^3*e^2 + (b^3 - 132*a*b*c)*d^2*e^3 + (256*c^3*d^3*e^2 - 384*b*c^2*d^2*e^3 + 6*(21*b ^2*c + 44*a*c^2)*d*e^4 + (b^3 - 132*a*b*c)*e^5)*x^2 + 2*(256*c^3*d^4*e - 3 84*b*c^2*d^3*e^2 + 6*(21*b^2*c + 44*a*c^2)*d^2*e^3 + (b^3 - 132*a*b*c)*d*e ^4)*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)) + 6*(128*c^3*d^4*e - 128*b*c^2*d^3*e^2 + (23*b^2*c + 36*a*c^2)*d^2*e^3 + ( 128*c^3*d^2*e^3 - 128*b*c^2*d*e^4 + (23*b^2*c + 36*a*c^2)*e^5)*x^2 + 2*(12 8*c^3*d^3*e^2 - 128*b*c^2*d^2*e^3 + (23*b^2*c + 36*a*c^2)*d*e^4)*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 - 6*a*c^2)*d*e^2 + (2*b^3 - 9 *a*b*c)*e^3)/(c^3*e^3), 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*(6*c^3*e^5*x^3 - 128*c^3*d^3*e^2 + 112*b*c^2*d^2*e^3 - 5*a*b*c *e^5 - 5*(3*b^2*c + 4*a*c^2)*d*e^4 - (16*c^3*d*e^4 - 17*b*c^2*e^5)*x^2 - 2 *(80*c^3*d^2*e^3 - 72*b*c^2*d*e^4 + 5*(2*b^2*c + 3*a*c^2)*e^5)*x)*sqrt(c*x ^2 + b*x + a)*sqrt(e*x + d))/(c*e^8*x^2 + 2*c*d*e^7*x + c*d^2*e^6)
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]