Integrand size = 22, antiderivative size = 1224 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac {3 \left (10 b c d \left (c d^2+3 a e^2\right )-8 a c e \left (2 c d^2+3 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{5/3} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{5/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac {3^{3/4} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} c^{5/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}} \]
-3/4*(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(4/3) +3/4*(10*b*c*d*(3*a*e^2+c*d^2)-8*a*c*e*(3*a*e^2+2*c*d^2)-b^2*(-a*e^3+11*c* d^2*e)+(-b*e+2*c*d)*(10*c^2*d^2-b^2*e^2-2*c*e*(-7*a*e+5*b*d))*x)/c/(-4*a*c +b^2)^2/(c*x^2+b*x+a)^(1/3)-3/4*(-b*e+2*c*d)*(5*c^2*d^2-b^2*e^2-c*e*(-9*a* e+5*b*d))*(2*c*x+b)*2^(2/3)/c^(5/3)/(-4*a*c+b^2)^2/(2^(2/3)*c^(1/3)*(c*x^2 +b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))-1/2*3^(3/4)*(-b*e+2*c*d)*(5* c^2*d^2-b^2*e^2-c*e*(-9*a*e+5*b*d))*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c *x^2+b*x+a)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+ b^2)^(1/3)*(1-3^(1/2)))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^ (1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(((-4*a*c+b^2)^(2/3)-2^(2/3)*c^(1/3)*(-4 *a*c+b^2)^(1/3)*(c*x^2+b*x+a)^(1/3)+2*2^(1/3)*c^(2/3)*(c*x^2+b*x+a)^(2/3)) /(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^( 1/2)*2^(1/6)/c^(5/3)/(-4*a*c+b^2)^(5/3)/(2*c*x+b)/((-4*a*c+b^2)^(1/3)*((-4 *a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3))/(2^(2/3)*c^(1/3)*(c*x ^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)+3/8*3^(1/4)*(-b*e +2*c*d)*(5*c^2*d^2-b^2*e^2-c*e*(-9*a*e+5*b*d))*((-4*a*c+b^2)^(1/3)+2^(2/3) *c^(1/3)*(c*x^2+b*x+a)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/ 3)+(-4*a*c+b^2)^(1/3)*(1-3^(1/2)))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(- 4*a*c+b^2)^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((( -4*a*c+b^2)^(2/3)-2^(2/3)*c^(1/3)*(-4*a*c+b^2)^(1/3)*(c*x^2+b*x+a)^(1/3...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.00 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.33 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\frac {3 c \left (b^4 e^3 x^2+b^2 \left (a^2 e^3+a c e \left (-9 d^2+42 d e x-11 e^2 x^2\right )+c^2 d x \left (8 d^2-45 d e x+6 e^2 x^2\right )\right )+4 c \left (-6 a^3 e^3+5 c^3 d^3 x^3+a^2 c e \left (-6 d^2+3 d e x-8 e^2 x^2\right )+a c^2 d x \left (7 d^2+9 e^2 x^2\right )\right )+2 b c \left (a^2 e^2 (15 d-19 e x)+15 c^2 d^2 x^2 (d-e x)+a c \left (7 d^3-21 d^2 e x+27 d e^2 x^2-9 e^3 x^3\right )\right )+b^3 \left (2 a e^3 x-c \left (d^3+12 d^2 e x-9 d e^2 x^2-2 e^3 x^3\right )\right )\right )-2^{2/3} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2+c e (-5 b d+9 a e)\right ) (b+2 c x) (a+x (b+c x)) \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]
(3*c*(b^4*e^3*x^2 + b^2*(a^2*e^3 + a*c*e*(-9*d^2 + 42*d*e*x - 11*e^2*x^2) + c^2*d*x*(8*d^2 - 45*d*e*x + 6*e^2*x^2)) + 4*c*(-6*a^3*e^3 + 5*c^3*d^3*x^ 3 + a^2*c*e*(-6*d^2 + 3*d*e*x - 8*e^2*x^2) + a*c^2*d*x*(7*d^2 + 9*e^2*x^2) ) + 2*b*c*(a^2*e^2*(15*d - 19*e*x) + 15*c^2*d^2*x^2*(d - e*x) + a*c*(7*d^3 - 21*d^2*e*x + 27*d*e^2*x^2 - 9*e^3*x^3)) + b^3*(2*a*e^3*x - c*(d^3 + 12* d^2*e*x - 9*d*e^2*x^2 - 2*e^3*x^3))) - 2^(2/3)*(2*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 + c*e*(-5*b*d + 9*a*e))*(b + 2*c*x)*(a + x*(b + c*x))*((c*(a + x*( b + c*x)))/(-b^2 + 4*a*c))^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b + 2*c *x)^2/(b^2 - 4*a*c)])/(4*c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3))
Time = 1.10 (sec) , antiderivative size = 1265, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1164, 27, 1224, 1095, 832, 759, 2416}
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.
| \(\displaystyle \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx\) |
| \(\Big \downarrow \) 1164 |
| \(\displaystyle -\frac {3 \int \frac {(d+e x) \left (10 c d^2-11 b e d+12 a e^2-e (2 c d-b e) x\right )}{3 \left (c x^2+b x+a\right )^{4/3}}dx}{4 \left (b^2-4 a c\right )}-\frac {3 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(d+e x) \left (10 c d^2-e (11 b d-12 a e)-e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^{4/3}}dx}{4 \left (b^2-4 a c\right )}-\frac {3 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 1224 |
| \(\displaystyle -\frac {\frac {2 (2 c d-b e) \left (-c e (5 b d-9 a e)-b^2 e^2+5 c^2 d^2\right ) \int \frac {1}{\sqrt [3]{c x^2+b x+a}}dx}{c \left (b^2-4 a c\right )}-\frac {3 \left (x (2 c d-b e) \left (-2 c e (5 b d-7 a e)-b^2 e^2+10 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-a e^3\right )\right )+10 b c d \left (3 a e^2+c d^2\right )-8 a c e \left (3 a e^2+2 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{4 \left (b^2-4 a c\right )}-\frac {3 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 1095 |
| \(\displaystyle -\frac {\frac {6 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-c e (5 b d-9 a e)-b^2 e^2+5 c^2 d^2\right ) \int \frac {\sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{c \left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 \left (x (2 c d-b e) \left (-2 c e (5 b d-7 a e)-b^2 e^2+10 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-a e^3\right )\right )+10 b c d \left (3 a e^2+c d^2\right )-8 a c e \left (3 a e^2+2 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{4 \left (b^2-4 a c\right )}-\frac {3 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle -\frac {\frac {6 \sqrt {(b+2 c x)^2} (2 c d-b e) \left (-c e (5 b d-9 a e)-b^2 e^2+5 c^2 d^2\right ) \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}\right )}{c \left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 \left (x (2 c d-b e) \left (-2 c e (5 b d-7 a e)-b^2 e^2+10 c^2 d^2\right )-\left (b^2 \left (11 c d^2 e-a e^3\right )\right )+10 b c d \left (3 a e^2+c d^2\right )-8 a c e \left (3 a e^2+2 c d^2\right )\right )}{c \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{4 \left (b^2-4 a c\right )}-\frac {3 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {3 (b d-2 a e+(2 c d-b e) x) (d+e x)^2}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}-\frac {\frac {6 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}\right )}{c \left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 \left (-\left (\left (11 c d^2 e-a e^3\right ) b^2\right )+10 c d \left (c d^2+3 a e^2\right ) b-8 a c e \left (2 c d^2+3 a e^2\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt [3]{c x^2+b x+a}}}{4 \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle -\frac {3 (b d-2 a e+(2 c d-b e) x) (d+e x)^2}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}-\frac {\frac {6 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \sqrt {(b+2 c x)^2} \left (\frac {\frac {\sqrt [3]{2} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}{\sqrt [3]{c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}\right )}{c \left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 \left (-\left (\left (11 c d^2 e-a e^3\right ) b^2\right )+10 c d \left (c d^2+3 a e^2\right ) b-8 a c e \left (2 c d^2+3 a e^2\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt [3]{c x^2+b x+a}}}{4 \left (b^2-4 a c\right )}\) |
(-3*(d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) - ((-3*(10*b*c*d*(c*d^2 + 3*a*e^2) - 8*a*c*e*(2*c*d^2 + 3 *a*e^2) - b^2*(11*c*d^2*e - a*e^3) + (2*c*d - b*e)*(10*c^2*d^2 - b^2*e^2 - 2*c*e*(5*b*d - 7*a*e))*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3)) + (6 *(2*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 - c*e*(5*b*d - 9*a*e))*Sqrt[(b + 2*c*x )^2]*(((2^(1/3)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])/(c^(1/3)*((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*(b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^( 2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)* c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3 )*(a + b*x + c*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c )^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4 *a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/ (2^(2/3)*c^(1/3)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)* c^(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^( 2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)]))/(2^(2/3)*c^(1/3)) - ((1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a *c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))* Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*...
3.25.89.3.1 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_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && PosQ[a]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[3*(Sqrt[(b + 2*c*x)^2]/(b + 2*c*x)) Subst[Int[x^(3*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 *c*x^3], x], x, (a + b*x + c*x^2)^(1/3)], x] /; FreeQ[{a, b, c}, x] && Inte gerQ[3*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c *(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 1] && !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
\[\int \frac {\left (e x +d \right )^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]
\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]
integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(c*x^2 + b*x + a)^(2/3) /(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b *c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^2), x)
Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]
\[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]