Integrand size = 24, antiderivative size = 731 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 c \left (128 c^2 d^3-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{21 e^5 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^3+3 a b e^3-c d e (13 b d-4 a e)+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{21 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 e^6 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{21 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2/21*(16*c^2*d^3+3*a*b*e^3-c*d*e*(-4*a*e+13*b*d)+e*(22*c^2*d^2+3*b^2*e^2- 2*c*e*(-5*a*e+11*b*d))*x)*(c*x^2+b*x+a)^(3/2)/e^3/(a*e^2-b*d*e+c*d^2)/(e*x +d)^(5/2)-2/7*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(7/2)+2/21*c*(128*c^2*d^3-4*c* d*e*(-29*a*e+44*b*d)+3*b*e^2*(-16*a*e+17*b*d)+e*(32*c^2*d^2+3*b^2*e^2-4*c* e*(-5*a*e+8*b*d))*x)*(c*x^2+b*x+a)^(1/2)/e^5/(a*e^2-b*d*e+c*d^2)/(e*x+d)^( 1/2)-1/21*(-b*e+2*c*d)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(-29*a*e+32*b*d))*Elli pticE(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^6 /(a*e^2-b*d*e+c*d^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)+4/21*(128*c^2*d^2+27*b^2*e^2-4*c*e*(-5*a*e+32*b*d))*Elli pticF(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)/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 22.98 (sec) , antiderivative size = 1446, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {\sqrt {d+e x} (a+x (b+c x))^{5/2} \left (\frac {2 c^2}{3 e^5}-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{7 e^5 (d+e x)^4}-\frac {6 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{7 e^5 (d+e x)^3}-\frac {2 \left (52 c^2 d^2-52 b c d e+9 b^2 e^2+16 a c e^2\right )}{21 e^5 (d+e x)^2}-\frac {2 (-2 c d+b e) \left (79 c^2 d^2-79 b c d e+3 b^2 e^2+67 a c e^2\right )}{21 e^5 \left (c d^2-b d e+a e^2\right ) (d+e x)}\right )}{\left (a+b x+c x^2\right )^2}-\frac {(d+e x)^{3/2} (a+x (b+c x))^{5/2} \left (-4 (-2 c d+b e) \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+3 b^2 e^2+4 c e (-32 b d+29 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} (-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (128 c^2 d^2+3 b^2 e^2+4 c e (-32 b d+29 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 (-3 b^4 e^4+b^3 e^3 \left (32 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )-2 b^2 c e^2 \left (16 c d^2+4 a e^2+67 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+8 c^2 \left (10 a^2 e^4-32 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (16 c d-29 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+4 b c e \left (96 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-32 c d+29 \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 )}{42 e^7 \left (c d^2-b d e+a e^2\right ) \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 )^{5/2} \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))^(5/2)*((2*c^2)/(3*e^5) - (2*(c*d^2 - b*d* e + a*e^2)^2)/(7*e^5*(d + e*x)^4) - (6*(-2*c*d + b*e)*(c*d^2 - b*d*e + a*e ^2))/(7*e^5*(d + e*x)^3) - (2*(52*c^2*d^2 - 52*b*c*d*e + 9*b^2*e^2 + 16*a* c*e^2))/(21*e^5*(d + e*x)^2) - (2*(-2*c*d + b*e)*(79*c^2*d^2 - 79*b*c*d*e + 3*b^2*e^2 + 67*a*c*e^2))/(21*e^5*(c*d^2 - b*d*e + a*e^2)*(d + e*x))))/(a + b*x + c*x^2)^2 - ((d + e*x)^(3/2)*(a + x*(b + c*x))^(5/2)*(-4*(-2*c*d + b*e)*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 + 3*b^2*e^2 + 4*c*e*(-32*b*d + 29*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*Sqr t[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(128*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-32*b*d + 29*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 + 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 - (2*d*e)/(d + e*x))) /(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sq rt[(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]*(-3*b^4*e^4 + b^3*e^3*( 32*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2]) - 2*b^2*c*e^2*(16*c*d^2 + 4*a*e^2 + 67 *d*Sqrt[(b^2 - 4*a*c)*e^2]) + 8*c^2*(10*a^2*e^4 - 32*c*d^3*Sqrt[(b^2 - ...
Time = 1.16 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1161, 1229, 27, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^{7/2}}dx}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5 \left (-\frac {2 \int -\frac {c \left (-13 d e b^2+16 c d^2 b+16 a e^2 b-12 a c d e+\left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^{3/2}}dx}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {c \int -\frac {\left (13 d e b^2-16 \left (c d^2+a e^2\right ) b+12 a c d e-\left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{3/2}}dx}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (-\frac {c \int \frac {\left (13 d e b^2-16 \left (c d^2+a e^2\right ) b+12 a c d e-\left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{3/2}}dx}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {5 \left (-\frac {c \left (-\frac {2 \int -\frac {51 d e^2 b^3-2 \left (27 a e^3+88 c d^2 e\right ) b^2+4 c d \left (32 c d^2+45 a e^2\right ) b-8 a c e \left (8 c d^2+5 a e^2\right )+(2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {c \left (\frac {\int \frac {51 d e^2 b^3-2 \left (27 a e^3+88 c d^2 e\right ) b^2+4 c d \left (32 c d^2+45 a e^2\right ) b-8 a c e \left (8 c d^2+5 a e^2\right )+(2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (-\frac {c \left (\frac {\frac {(2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2+27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {5 \left (-\frac {2 \left (16 c^2 d^3-c e (13 b d-4 a e) d+3 a b e^3+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac {c \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d+27 b^2 e^2+20 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{3 e^2}-\frac {2 \left (128 c^2 d^3-4 c e (44 b d-29 a e) d+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{7 e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {5 \left (-\frac {2 \left (16 c^2 d^3-c e (13 b d-4 a e) d+3 a b e^3+e \left (22 c^2 d^2+3 b^2 e^2-2 c e (11 b d-5 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{15 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{5/2}}-\frac {c \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d+27 b^2 e^2+20 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{3 e^2}-\frac {2 \left (128 c^2 d^3-4 c e (44 b d-29 a e) d+3 b e^2 (17 b d-16 a e)+e \left (32 c^2 d^2+3 b^2 e^2-4 c e (8 b d-5 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{7 e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {5 \left (-\frac {c \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2+27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 e^2}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-4 c e (8 b d-5 a e)+3 b^2 e^2+32 c^2 d^2\right )-4 c d e (44 b d-29 a e)+3 b e^2 (17 b d-16 a e)+128 c^2 d^3\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (11 b d-5 a e)+3 b^2 e^2+22 c^2 d^2\right )-c d e (13 b d-4 a e)+3 a b e^3+16 c^2 d^3\right )}{15 e^2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\right )}{7 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}\) |
(-2*(a + b*x + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) + (5*((-2*(16*c^2*d^3 + 3*a*b*e^3 - c*d*e*(13*b*d - 4*a*e) + e*(22*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(1 1*b*d - 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(15*e^2*(c*d^2 - b*d*e + a*e^2 )*(d + e*x)^(5/2)) - (c*((-2*(128*c^2*d^3 - 4*c*d*e*(44*b*d - 29*a*e) + 3* b*e^2*(17*b*d - 16*a*e) + e*(32*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(8*b*d - 5*a*e ))*x)*Sqrt[a + b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) + ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*S qrt[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*Sqr t[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]) - (4*Sqr t[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - 128*b*c*d*e + 27*b^2*e^2 + 20*a*c*e^2)*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[Sqr t[(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]*Sqr t[a + b*x + c*x^2]))/(3*e^2)))/(5*e^2*(c*d^2 - b*d*e + a*e^2))))/(7*e)
3.25.59.3.1 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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[p/(e*(m + 1)) Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] && !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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. \(1523\) vs. \(2(661)=1322\).
Time = 4.65 (sec) , antiderivative size = 1524, normalized size of antiderivative = 2.08
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1524\) |
| risch | \(\text {Expression too large to display}\) | \(6256\) |
| default | \(\text {Expression too large to display}\) | \(25728\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/7*(a^2 *e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-2*b*c*d^3*e+c^2*d^4)/e^9*(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)^4-6/7*(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^8*(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)^3-2/21*(16*a*c*e^2+9*b^2*e^2-52*b*c*d*e+52*c^2*d^2)/ e^7*(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)^2-2/21*(c*e*x^ 2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)/e^6*(67*a*b*c*e^3-134*a*c^2*d*e^2+3*b^3*e ^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)/((x+d/e)*(c*e*x^2+b*e*x+a*e ))^(1/2)+2/3*c^2/e^5*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(c* (3*a*c*e^2+3*b^2*e^2-12*b*c*d*e+10*c^2*d^2)/e^6-1/21*c*(16*a*c*e^2+9*b^2*e ^2-52*b*c*d*e+52*c^2*d^2)/e^6-1/21/e^6*(b*e-c*d)*(67*a*b*c*e^3-134*a*c^2*d *e^2+3*b^3*e^3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)/(a*e^2-b*d*e+c* d^2)+1/21*b/e^5/(a*e^2-b*d*e+c*d^2)*(67*a*b*c*e^3-134*a*c^2*d*e^2+3*b^3*e^ 3-85*b^2*c*d*e^2+237*b*c^2*d^2*e-158*c^3*d^3)-2/3*c^2/e^5*(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*(c^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 1719, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\text {Too large to display} \]
2/63*((256*c^4*d^8 - 512*b*c^3*d^7*e + 2*(139*b^2*c^2 + 212*a*c^3)*d^6*e^2 - 2*(11*b^3*c + 212*a*b*c^2)*d^5*e^3 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2) *d^4*e^4 + (256*c^4*d^4*e^4 - 512*b*c^3*d^3*e^5 + 2*(139*b^2*c^2 + 212*a*c ^3)*d^2*e^6 - 2*(11*b^3*c + 212*a*b*c^2)*d*e^7 - (3*b^4 - 46*a*b^2*c - 120 *a^2*c^2)*e^8)*x^4 + 4*(256*c^4*d^5*e^3 - 512*b*c^3*d^4*e^4 + 2*(139*b^2*c ^2 + 212*a*c^3)*d^3*e^5 - 2*(11*b^3*c + 212*a*b*c^2)*d^2*e^6 - (3*b^4 - 46 *a*b^2*c - 120*a^2*c^2)*d*e^7)*x^3 + 6*(256*c^4*d^6*e^2 - 512*b*c^3*d^5*e^ 3 + 2*(139*b^2*c^2 + 212*a*c^3)*d^4*e^4 - 2*(11*b^3*c + 212*a*b*c^2)*d^3*e ^5 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*d^2*e^6)*x^2 + 4*(256*c^4*d^7*e - 512*b*c^3*d^6*e^2 + 2*(139*b^2*c^2 + 212*a*c^3)*d^5*e^3 - 2*(11*b^3*c + 21 2*a*b*c^2)*d^4*e^4 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2)*d^3*e^5)*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*(256*c^4* d^7*e - 384*b*c^3*d^6*e^2 + 2*(67*b^2*c^2 + 116*a*c^3)*d^5*e^3 - (3*b^3*c + 116*a*b*c^2)*d^4*e^4 + (256*c^4*d^3*e^5 - 384*b*c^3*d^2*e^6 + 2*(67*b^2* c^2 + 116*a*c^3)*d*e^7 - (3*b^3*c + 116*a*b*c^2)*e^8)*x^4 + 4*(256*c^4*d^4 *e^4 - 384*b*c^3*d^3*e^5 + 2*(67*b^2*c^2 + 116*a*c^3)*d^2*e^6 - (3*b^3*c + 116*a*b*c^2)*d*e^7)*x^3 + 6*(256*c^4*d^5*e^3 - 384*b*c^3*d^4*e^4 + 2*(67* b^2*c^2 + 116*a*c^3)*d^3*e^5 - (3*b^3*c + 116*a*b*c^2)*d^2*e^6)*x^2 + 4...
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]