Integrand size = 24, antiderivative size = 816 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{1155 c^3 e^3}+\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-21 a e)+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b d e-20 a^2 e^2\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
2/231*(c^2*d^2-6*b^2*e^2+c*e*(-3*a*e+13*b*d)+14*c*e*(-b*e+2*c*d)*x)*(c*x^2 +b*x+a)^(3/2)*(e*x+d)^(1/2)/c^2/e+2/11*e*(c*x^2+b*x+a)^(5/2)*(e*x+d)^(1/2) /c+2/1155*(8*c^4*d^4+8*b^4*e^4-c^3*d^2*e*(-42*a*e+19*b*d)-b^2*c*e^3*(21*a* e+19*b*d)+3*c^2*e^2*(-10*a^2*e^2+17*a*b*d*e+2*b^2*d^2)-3*c*e*(-b*e+2*c*d)* (c^2*d^2+8*b^2*e^2-c*e*(31*a*e+b*d))*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/ c^3/e^3-8/1155*(-b*e+2*c*d)*(c^2*d^2-2*b^2*e^2-c*e*(-9*a*e+b*d))*(c^2*d^2+ b^2*e^2-c*e*(3*a*e+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)/c^4/e^4/(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)+2/1155*(a*e^2-b*d*e+c*d^2)*(16*c^4*d^4-8*b^ 4*e^4-4*c^3*d^2*e*(-21*a*e+8*b*d)+b^2*c*e^3*(51*a*e+13*b*d)+3*c^2*e^2*(-20 *a^2*e^2-28*a*b*d*e+b^2*d^2))*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^4/e^ 4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 34.95 (sec) , antiderivative size = 10848, normalized size of antiderivative = 13.29 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Result too large to show} \]
Time = 1.50 (sec) , antiderivative size = 855, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1166, 27, 1231, 27, 1231, 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.
| \(\displaystyle \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {2 \int \frac {\left (11 c d^2-e (5 b d+a e)+6 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 \sqrt {d+e x}}dx}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (11 c d^2-e (5 b d+a e)+6 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{\sqrt {d+e x}}dx}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (c e (13 b d-3 a e)-6 b^2 e^2+14 c e x (2 c d-b e)+c^2 d^2\right )}{21 c e}-\frac {2 \int -\frac {3 e \left (6 d e^2 b^3-\left (13 c d^2 e-2 a e^3\right ) b^2-c d \left (c d^2+27 a e^2\right ) b+2 a c e \left (29 c d^2-3 a e^2\right )-(2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (6 d e^2 b^3-\left (13 c d^2 e-2 a e^3\right ) b^2-c d \left (c d^2+27 a e^2\right ) b+2 a c e \left (29 c d^2-3 a e^2\right )-(2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{7 c e}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (c e (13 b d-3 a e)-6 b^2 e^2+14 c e x (2 c d-b e)+c^2 d^2\right )}{21 c e}}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c^2 e^2 \left (-10 a^2 e^2+17 a b d e+2 b^2 d^2\right )-3 c e x (2 c d-b e) \left (-c e (31 a e+b d)+8 b^2 e^2+c^2 d^2\right )-b^2 c e^3 (21 a e+19 b d)-c^3 d^2 e (19 b d-42 a e)+8 b^4 e^4+8 c^4 d^4\right )}{15 c e^2}-\frac {2 \int \frac {2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac {1}{2} b d (4 c d-b e)-\frac {1}{2} a e (2 c d+b e)\right )+5 c e (b d-2 a e) \left (6 d e^2 b^3-\left (13 c d^2 e-2 a e^3\right ) b^2-c d \left (c d^2+27 a e^2\right ) b+2 a c e \left (29 c d^2-3 a e^2\right )\right )+8 (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{7 c e}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (c e (13 b d-3 a e)-6 b^2 e^2+14 c e x (2 c d-b e)+c^2 d^2\right )}{21 c e}}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c^2 e^2 \left (-10 a^2 e^2+17 a b d e+2 b^2 d^2\right )-3 c e x (2 c d-b e) \left (-c e (31 a e+b d)+8 b^2 e^2+c^2 d^2\right )-b^2 c e^3 (21 a e+19 b d)-c^3 d^2 e (19 b d-42 a e)+8 b^4 e^4+8 c^4 d^4\right )}{15 c e^2}-\frac {\int \frac {(2 c d-b e) \left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right ) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right )+5 c e (b d-2 a e) \left (6 d e^2 b^3-\left (13 c d^2 e-2 a e^3\right ) b^2-c d \left (c d^2+27 a e^2\right ) b+2 a c e \left (29 c d^2-3 a e^2\right )\right )+8 (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{7 c e}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (c e (13 b d-3 a e)-6 b^2 e^2+14 c e x (2 c d-b e)+c^2 d^2\right )}{21 c e}}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (3 c^2 e^2 \left (-10 a^2 e^2+17 a b d e+2 b^2 d^2\right )-3 c e x (2 c d-b e) \left (-c e (31 a e+b d)+8 b^2 e^2+c^2 d^2\right )-b^2 c e^3 (21 a e+19 b d)-c^3 d^2 e (19 b d-42 a e)+8 b^4 e^4+8 c^4 d^4\right )}{15 c e^2}-\frac {\frac {8 (2 c d-b e) \left (-c e (b d-9 a e)-2 b^2 e^2+c^2 d^2\right ) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-60 a^2 c^2 e^4+51 a b^2 c e^4-84 a b c^2 d e^3+84 a c^3 d^2 e^2-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{15 c e^2}}{7 c e}+\frac {2 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \left (c e (13 b d-3 a e)-6 b^2 e^2+14 c e x (2 c d-b e)+c^2 d^2\right )}{21 c e}}{11 c}+\frac {2 e \sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 e \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac {\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{21 c e}+\frac {\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-32 b c^3 e d^3+84 a c^3 e^2 d^2+3 b^2 c^2 e^2 d^2-84 a b c^2 e^3 d+13 b^3 c e^3 d-8 b^4 e^4-60 a^2 c^2 e^4+51 a b^2 c e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{7 c e}}{11 c}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 e \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac {\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{21 c e}+\frac {\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-32 b c^3 e d^3+84 a c^3 e^2 d^2+3 b^2 c^2 e^2 d^2-84 a b c^2 e^3 d+13 b^3 c e^3 d-8 b^4 e^4-60 a^2 c^2 e^4+51 a b^2 c e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{7 c e}}{11 c}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e \sqrt {d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac {\frac {2 \sqrt {d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{21 c e}+\frac {\frac {2 \sqrt {d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{15 c e^2}-\frac {\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-32 b c^3 e d^3+84 a c^3 e^2 d^2+3 b^2 c^2 e^2 d^2-84 a b c^2 e^3 d+13 b^3 c e^3 d-8 b^4 e^4-60 a^2 c^2 e^4+51 a b^2 c e^4\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}}{15 c e^2}}{7 c e}}{11 c}\) |
(2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*c) + ((2*Sqrt[d + e*x]*(c^ 2*d^2 - 6*b^2*e^2 + c*e*(13*b*d - 3*a*e) + 14*c*e*(2*c*d - b*e)*x)*(a + b* x + c*x^2)^(3/2))/(21*c*e) + ((2*Sqrt[d + e*x]*(8*c^4*d^4 + 8*b^4*e^4 - c^ 3*d^2*e*(19*b*d - 42*a*e) - b^2*c*e^3*(19*b*d + 21*a*e) + 3*c^2*e^2*(2*b^2 *d^2 + 17*a*b*d*e - 10*a^2*e^2) - 3*c*e*(2*c*d - b*e)*(c^2*d^2 + 8*b^2*e^2 - c*e*(b*d + 31*a*e))*x)*Sqrt[a + b*x + c*x^2])/(15*c*e^2) - ((8*Sqrt[2]* Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c^2*d^2 - 2*b^2*e^2 - c*e*(b*d - 9*a*e))* (c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*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 + S qrt[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)*(16*c^4*d^4 - 32*b*c^3*d^3*e + 3*b^2*c^2*d^2*e^2 + 84*a*c^3* d^2*e^2 + 13*b^3*c*d*e^3 - 84*a*b*c^2*d*e^3 - 8*b^4*e^4 + 51*a*b^2*c*e^4 - 60*a^2*c^2*e^4)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*S qrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqr t[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]))/(15*c*e^2))/(7*c*e))/(11*c)
3.25.47.3.1 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[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[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)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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. \(2887\) vs. \(2(746)=1492\).
Time = 1.55 (sec) , antiderivative size = 2888, normalized size of antiderivative = 3.54
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2888\) |
| risch | \(\text {Expression too large to display}\) | \(4982\) |
| default | \(\text {Expression too large to display}\) | \(11933\) |
(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d )*((e*x+d)*(c*x^2+b*x+a))^(1/2)*(2/11*c*e*x^4*(c*e*x^3+b*e*x^2+c*d*x^2+a*e *x+b*d*x+a*d)^(1/2)+2/9*(2*b*c*e^2+2*c^2*d*e-2/11*c*e*(5*b*e+5*c*d))/c/e*x ^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/7*(2*a*c*e^2+b^2*e^2+ 4*b*c*d*e+c^2*d^2-2/11*c*e*(9/2*a*e+9/2*b*d)-2/9*(2*b*c*e^2+2*c^2*d*e-2/11 *c*e*(5*b*e+5*c*d))/c/e*(4*b*e+4*c*d))/c/e*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a* e*x+b*d*x+a*d)^(1/2)+2/5*(2*a*b*e^2+36/11*a*c*d*e+2*b^2*d*e+2*b*c*d^2-2/9* (2*b*c*e^2+2*c^2*d*e-2/11*c*e*(5*b*e+5*c*d))/c/e*(7/2*a*e+7/2*b*d)-2/7*(2* a*c*e^2+b^2*e^2+4*b*c*d*e+c^2*d^2-2/11*c*e*(9/2*a*e+9/2*b*d)-2/9*(2*b*c*e^ 2+2*c^2*d*e-2/11*c*e*(5*b*e+5*c*d))/c/e*(4*b*e+4*c*d))/c/e*(3*b*e+3*c*d))/ c/e*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*(a^2*e^2+4*a*b*d *e+2*a*d^2*c+b^2*d^2-2/3*(2*b*c*e^2+2*c^2*d*e-2/11*c*e*(5*b*e+5*c*d))/c/e* a*d-2/7*(2*a*c*e^2+b^2*e^2+4*b*c*d*e+c^2*d^2-2/11*c*e*(9/2*a*e+9/2*b*d)-2/ 9*(2*b*c*e^2+2*c^2*d*e-2/11*c*e*(5*b*e+5*c*d))/c/e*(4*b*e+4*c*d))/c/e*(5/2 *a*e+5/2*b*d)-2/5*(2*a*b*e^2+36/11*a*c*d*e+2*b^2*d*e+2*b*c*d^2-2/9*(2*b*c* e^2+2*c^2*d*e-2/11*c*e*(5*b*e+5*c*d))/c/e*(7/2*a*e+7/2*b*d)-2/7*(2*a*c*e^2 +b^2*e^2+4*b*c*d*e+c^2*d^2-2/11*c*e*(9/2*a*e+9/2*b*d)-2/9*(2*b*c*e^2+2*c^2 *d*e-2/11*c*e*(5*b*e+5*c*d))/c/e*(4*b*e+4*c*d))/c/e*(3*b*e+3*c*d))/c/e*(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*(a^2*d^2 -2/5*(2*a*b*e^2+36/11*a*c*d*e+2*b^2*d*e+2*b*c*d^2-2/9*(2*b*c*e^2+2*c^2*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 906, normalized size of antiderivative = 1.11 \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 3 \, {\left (11 \, b^{2} c^{4} + 36 \, a c^{5}\right )} d^{4} e^{2} + 2 \, {\left (7 \, b^{3} c^{3} - 108 \, a b c^{4}\right )} d^{3} e^{3} + 3 \, {\left (11 \, b^{4} c^{2} - 102 \, a b^{2} c^{3} + 312 \, a^{2} c^{4}\right )} d^{2} e^{4} - 6 \, {\left (8 \, b^{5} c - 69 \, a b^{3} c^{2} + 156 \, a^{2} b c^{3}\right )} d e^{5} + {\left (16 \, b^{6} - 144 \, a b^{4} c + 369 \, a^{2} b^{2} c^{2} - 180 \, a^{3} c^{3}\right )} e^{6}\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 ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, {\left (b^{2} c^{4} + 6 \, a c^{5}\right )} d^{3} e^{3} + 2 \, {\left (b^{3} c^{3} - 9 \, a b c^{4}\right )} d^{2} e^{4} - {\left (5 \, b^{4} c^{2} - 36 \, a b^{2} c^{3} + 54 \, a^{2} c^{4}\right )} d e^{5} + {\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 27 \, a^{2} b c^{3}\right )} e^{6}\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 (105 \, c^{6} e^{6} x^{4} + 8 \, c^{6} d^{4} e^{2} - 19 \, b c^{5} d^{3} e^{3} + {\left (6 \, b^{2} c^{4} + 47 \, a c^{5}\right )} d^{2} e^{4} - {\left (19 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} d e^{5} + {\left (8 \, b^{4} c^{2} - 51 \, a b^{2} c^{3} + 60 \, a^{2} c^{4}\right )} e^{6} + 140 \, {\left (c^{6} d e^{5} + b c^{5} e^{6}\right )} x^{3} + 5 \, {\left (c^{6} d^{2} e^{4} + 41 \, b c^{5} d e^{5} + {\left (b^{2} c^{4} + 39 \, a c^{5}\right )} e^{6}\right )} x^{2} - 2 \, {\left (3 \, c^{6} d^{3} e^{3} - 7 \, b c^{5} d^{2} e^{4} - {\left (7 \, b^{2} c^{4} + 163 \, a c^{5}\right )} d e^{5} + {\left (3 \, b^{3} c^{3} - 16 \, a b c^{4}\right )} e^{6}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{3465 \, c^{5} e^{5}} \]
2/3465*((16*c^6*d^6 - 48*b*c^5*d^5*e + 3*(11*b^2*c^4 + 36*a*c^5)*d^4*e^2 + 2*(7*b^3*c^3 - 108*a*b*c^4)*d^3*e^3 + 3*(11*b^4*c^2 - 102*a*b^2*c^3 + 312 *a^2*c^4)*d^2*e^4 - 6*(8*b^5*c - 69*a*b^3*c^2 + 156*a^2*b*c^3)*d*e^5 + (16 *b^6 - 144*a*b^4*c + 369*a^2*b^2*c^2 - 180*a^3*c^3)*e^6)*sqrt(c*e)*weierst rassPInverse(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)) + 24*(2*c^6*d^5*e - 5*b* c^5*d^4*e^2 + 2*(b^2*c^4 + 6*a*c^5)*d^3*e^3 + 2*(b^3*c^3 - 9*a*b*c^4)*d^2* e^4 - (5*b^4*c^2 - 36*a*b^2*c^3 + 54*a^2*c^4)*d*e^5 + (2*b^5*c - 15*a*b^3* c^2 + 27*a^2*b*c^3)*e^6)*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*(105*c^6*e^6*x^4 + 8*c^6*d^4*e^ 2 - 19*b*c^5*d^3*e^3 + (6*b^2*c^4 + 47*a*c^5)*d^2*e^4 - (19*b^3*c^3 - 116* a*b*c^4)*d*e^5 + (8*b^4*c^2 - 51*a*b^2*c^3 + 60*a^2*c^4)*e^6 + 140*(c^6*d* e^5 + b*c^5*e^6)*x^3 + 5*(c^6*d^2*e^4 + 41*b*c^5*d*e^5 + (b^2*c^4 + 39*a*c ^5)*e^6)*x^2 - 2*(3*c^6*d^3*e^3 - 7*b*c^5*d^2*e^4 - (7*b^2*c^4 + 163*a*c^5 )*d*e^5 + (3*b^3*c^3 - 16*a*b*c^4)*e^6)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e...
\[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
\[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]