Integrand size = 24, antiderivative size = 923 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=-\frac {2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2/63*(16*c^2*d^3-b*e^2*(-5*a*e+2*b*d)-c*d*e*(-4*a*e+11*b*d)+e*(26*c^2*d^2 +3*b^2*e^2-2*c*e*(-7*a*e+13*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)^(7/2)-2/9*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(9/2)-2/63*(128*c^4 *d^5-2*a*b^3*e^5-4*c^3*d^3*e*(-49*a*e+60*b*d)-b*c*e^3*(-24*a^2*e^2+9*a*b*d *e+b^2*d^2)+3*c^2*d*e^2*(12*a^2*e^2-52*a*b*d*e+37*b^2*d^2)+e*(160*c^4*d^4- 2*b^4*e^4-4*c^3*d^2*e*(-69*a*e+80*b*d)-b^2*c*e^3*(-27*a*e+11*b*d)+3*c^2*e^ 2*(28*a^2*e^2-92*a*b*d*e+57*b^2*d^2))*x)*(c*x^2+b*x+a)^(1/2)/e^5/(a*e^2-b* d*e+c*d^2)^2/(e*x+d)^(3/2)+2/63*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-57*a*e+ 64*b*d)-b^2*c*e^3*(-15*a*e+7*b*d)+3*c^2*e^2*(28*a^2*e^2-76*a*b*d*e+45*b^2* 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)*(-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)^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)-2/63*(-b*e+2*c*d)*(128*c^2*d^2-b^2*e^2-4*c*e *(-33*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)/e^6/(a*e^2-b*d*e+ c*d^2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 36.14 (sec) , antiderivative size = 8108, normalized size of antiderivative = 8.78 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Result too large to show} \]
Time = 1.44 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1161, 1229, 27, 1229, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1161 |
| \(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^{9/2}}dx}{9 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5 \left (-\frac {6 \int \frac {\left (2 e^2 b^3+11 c d e b^2-8 c \left (2 c d^2+3 a e^2\right ) b+20 a c^2 d e-c \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^{5/2}}dx}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (13 b d-7 a e)+3 b^2 e^2+26 c^2 d^2\right )-c d e (11 b d-4 a e)-b e^2 (2 b d-5 a e)+16 c^2 d^3\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\right )}{9 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {3 \int \frac {\left (2 e^2 b^3+11 c d e b^2-8 c \left (2 c d^2+3 a e^2\right ) b+20 a c^2 d e-c \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^{5/2}}dx}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (13 b d-7 a e)+3 b^2 e^2+26 c^2 d^2\right )-c d e (11 b d-4 a e)-b e^2 (2 b d-5 a e)+16 c^2 d^3\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\right )}{9 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {2 \sqrt {a+b x+c x^2} \left (3 c^2 d e^2 \left (12 a^2 e^2-52 a b d e+37 b^2 d^2\right )-b c e^3 \left (-24 a^2 e^2+9 a b d e+b^2 d^2\right )+e x \left (3 c^2 e^2 \left (28 a^2 e^2-92 a b d e+57 b^2 d^2\right )-b^2 c e^3 (11 b d-27 a e)-4 c^3 d^2 e (80 b d-69 a e)-2 b^4 e^4+160 c^4 d^4\right )-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)+128 c^4 d^5\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int -\frac {c \left (d e^3 b^4-\left (111 c d^2 e^2-a e^4\right ) b^3+12 c d e \left (20 c d^2+19 a e^2\right ) b^2-4 c \left (32 c^2 d^4+81 a c e^2 d^2+33 a^2 e^4\right ) b+32 a c^2 d e \left (2 c d^2+3 a e^2\right )-2 \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}\right )}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (13 b d-7 a e)+3 b^2 e^2+26 c^2 d^2\right )-c d e (11 b d-4 a e)-b e^2 (2 b d-5 a e)+16 c^2 d^3\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\right )}{9 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {c \int \frac {d e^3 b^4-\left (111 c d^2 e^2-a e^4\right ) b^3+12 c d e \left (20 c d^2+19 a e^2\right ) b^2-4 c \left (32 c^2 d^4+81 a c e^2 d^2+33 a^2 e^4\right ) b+32 a c^2 d e \left (2 c d^2+3 a e^2\right )-2 \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} \left (3 c^2 d e^2 \left (12 a^2 e^2-52 a b d e+37 b^2 d^2\right )-b c e^3 \left (-24 a^2 e^2+9 a b d e+b^2 d^2\right )+e x \left (3 c^2 e^2 \left (28 a^2 e^2-92 a b d e+57 b^2 d^2\right )-b^2 c e^3 (11 b d-27 a e)-4 c^3 d^2 e (80 b d-69 a e)-2 b^4 e^4+160 c^4 d^4\right )-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)+128 c^4 d^5\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (13 b d-7 a e)+3 b^2 e^2+26 c^2 d^2\right )-c d e (11 b d-4 a e)-b e^2 (2 b d-5 a e)+16 c^2 d^3\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\right )}{9 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {5 \left (-\frac {3 \left (\frac {c \left (\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (132 a c e^2-b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {a+b x+c x^2} \left (3 c^2 d e^2 \left (12 a^2 e^2-52 a b d e+37 b^2 d^2\right )-b c e^3 \left (-24 a^2 e^2+9 a b d e+b^2 d^2\right )+e x \left (3 c^2 e^2 \left (28 a^2 e^2-92 a b d e+57 b^2 d^2\right )-b^2 c e^3 (11 b d-27 a e)-4 c^3 d^2 e (80 b d-69 a e)-2 b^4 e^4+160 c^4 d^4\right )-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)+128 c^4 d^5\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\right )}{35 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (a+b x+c x^2\right )^{3/2} \left (e x \left (-2 c e (13 b d-7 a e)+3 b^2 e^2+26 c^2 d^2\right )-c d e (11 b d-4 a e)-b e^2 (2 b d-5 a e)+16 c^2 d^3\right )}{35 e^2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )}\right )}{9 e}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {5 \left (-\frac {2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac {3 \left (\frac {2 \sqrt {c x^2+b x+a} \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right )}{3 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d-b^2 e^2+132 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}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 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}}\right )}{3 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{35 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{9 e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {5 \left (-\frac {2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac {3 \left (\frac {2 \sqrt {c x^2+b x+a} \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right )}{3 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d-b^2 e^2+132 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}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 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}}\right )}{3 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{35 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{9 e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {5 \left (-\frac {2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{35 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac {3 \left (\frac {2 \sqrt {c x^2+b x+a} \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right )}{3 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}+\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b e d+a e^2\right ) \left (128 c^2 d^2-128 b c e d-b^2 e^2+132 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}\right )}{3 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{35 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{9 e}-\frac {2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}\) |
(-2*(a + b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (5*((-2*(16*c^2*d^3 - b*e^2*(2*b*d - 5*a*e) - c*d*e*(11*b*d - 4*a*e) + e*(26*c^2*d^2 + 3*b^2*e^ 2 - 2*c*e*(13*b*d - 7*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(35*e^2*(c*d^2 - b *d*e + a*e^2)*(d + e*x)^(7/2)) - (3*((2*(128*c^4*d^5 - 2*a*b^3*e^5 - 4*c^3 *d^3*e*(60*b*d - 49*a*e) - b*c*e^3*(b^2*d^2 + 9*a*b*d*e - 24*a^2*e^2) + 3* c^2*d*e^2*(37*b^2*d^2 - 52*a*b*d*e + 12*a^2*e^2) + e*(160*c^4*d^4 - 2*b^4* e^4 - 4*c^3*d^2*e*(80*b*d - 69*a*e) - b^2*c*e^3*(11*b*d - 27*a*e) + 3*c^2* e^2*(57*b^2*d^2 - 92*a*b*d*e + 28*a^2*e^2))*x)*Sqrt[a + b*x + c*x^2])/(3*e ^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) + (c*((-2*Sqrt[2]*Sqrt[b^2 - 4 *a*c]*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*( 7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*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]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - 128 *b*c*d*e - b^2*e^2 + 132*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[Arc Sin[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 ...
3.25.60.3.1 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[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. \(1743\) vs. \(2(853)=1706\).
Time = 2.61 (sec) , antiderivative size = 1744, normalized size of antiderivative = 1.89
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1744\) |
| default | \(\text {Expression too large to display}\) | \(44994\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/9*(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^10*(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)^5-38/63*(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^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)^4-2/63*(28*a*c*e^2+15*b^2*e^2-88*b*c*d*e+88*c^2*d ^2)/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)^3-2/63*(57 *a*b*c*e^3-114*a*c^2*d*e^2+b^3*e^3-63*b^2*c*d*e^2+183*b*c^2*d^2*e-122*c^3* d^3)/e^7/(a*e^2-b*d*e+c*d^2)*(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)^2-2/63*(c*e*x^2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)^2/e^6*(105*a^2*c ^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3 *c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^3*d^3*e+193*c^4*d^4)/((x+d/e)*(c*e*x^ 2+b*e*x+a*e))^(1/2)+2*(c^2*(3*b*e-5*c*d)/e^6-1/63*c*(57*a*b*c*e^3-114*a*c^ 2*d*e^2+b^3*e^3-63*b^2*c*d*e^2+183*b*c^2*d^2*e-122*c^3*d^3)/e^6/(a*e^2-b*d *e+c*d^2)-1/63/e^6*(b*e-c*d)*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d *e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b* c^3*d^3*e+193*c^4*d^4)/(a*e^2-b*d*e+c*d^2)^2+1/63*b/e^5/(a*e^2-b*d*e+c*d^2 )^2*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2* b^4*e^4-14*b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^3*d^3*e+193*c^4*d^4))*( 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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.36 (sec) , antiderivative size = 2808, normalized size of antiderivative = 3.04 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Too large to display} \]
-2/189*((256*c^5*d^10 - 640*b*c^4*d^9*e + 2*(239*b^2*c^3 + 324*a*c^4)*d^8* e^2 - (77*b^3*c^2 + 972*a*b*c^3)*d^7*e^3 - (13*b^4*c - 258*a*b^2*c^2 - 456 *a^2*c^3)*d^6*e^4 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^5*e^5 + (256*c^ 5*d^5*e^5 - 640*b*c^4*d^4*e^6 + 2*(239*b^2*c^3 + 324*a*c^4)*d^3*e^7 - (77* b^3*c^2 + 972*a*b*c^3)*d^2*e^8 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)* d*e^9 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*e^10)*x^5 + 5*(256*c^5*d^6*e^ 4 - 640*b*c^4*d^5*e^5 + 2*(239*b^2*c^3 + 324*a*c^4)*d^4*e^6 - (77*b^3*c^2 + 972*a*b*c^3)*d^3*e^7 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^2*e^8 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d*e^9)*x^4 + 10*(256*c^5*d^7*e^3 - 640*b*c^4*d^6*e^4 + 2*(239*b^2*c^3 + 324*a*c^4)*d^5*e^5 - (77*b^3*c^2 + 97 2*a*b*c^3)*d^4*e^6 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^3*e^7 - (2 *b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^2*e^8)*x^3 + 10*(256*c^5*d^8*e^2 - 64 0*b*c^4*d^7*e^3 + 2*(239*b^2*c^3 + 324*a*c^4)*d^6*e^4 - (77*b^3*c^2 + 972* a*b*c^3)*d^5*e^5 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^4*e^6 - (2*b ^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^3*e^7)*x^2 + 5*(256*c^5*d^9*e - 640*b*c ^4*d^8*e^2 + 2*(239*b^2*c^3 + 324*a*c^4)*d^7*e^3 - (77*b^3*c^2 + 972*a*b*c ^3)*d^6*e^4 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^5*e^5 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d^4*e^6)*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...
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{11/2}} \,d x \]