Integrand size = 34, antiderivative size = 1786 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx =\text {Too large to display} \] Output:
2/7293*(3*c^3*d^3+80*b^3*e^3+c^2*d*e*(-61*a*e+298*b*d)-19*b*c*e^2*(a*e+15* b*d)+11*c*e*(55*c^2*d^2+16*b^2*e^2-c*e*(9*a*e+55*b*d))*x)*(a*d+(a*e+b*d)*x +(b*e+c*d)*x^2+c*e*x^3)^(5/2)/c^3/e/(e*x+d)^2+2/153153*(128*c^7*d^7+128*b^ 7*e^7-4*c^6*d^5*e*(-225*a*e+124*b*d)-16*b^5*c*e^6*(57*a*e+31*b*d)+3*b^3*c^ 2*e^5*(421*a^2*e^2+1126*a*b*d*e+197*b^2*d^2)+3*c^5*d^3*e^2*(1240*a^2*e^2-8 60*a*b*d*e+197*b^2*d^2)-b*c^3*e^4*(-1236*a^3*e^3+3897*a^2*b*d*e^2+4140*a*b ^2*d^2*e+95*b^3*d^3)-c^4*d*e^3*(5244*a^3*e^3-5040*a^2*b*d*e^2-1050*a*b^2*d ^2*e+95*b^3*d^3)-3*c*e*(32*c^6*d^6-128*b^6*e^6-4*c^5*d^4*e*(-55*a*e+24*b*d )+16*b^4*c*e^5*(67*a*e+29*b*d)+5*c^4*d^2*e^2*(-1416*a^2*e^2-88*a*b*d*e+13* b^2*d^2)-b^2*c^2*e^4*(2463*a^2*e^2+3650*a*b*d*e+495*b^2*d^2)+6*c^3*e^3*(15 4*a^3*e^3+1180*a^2*b*d*e^2+645*a*b^2*d^2*e+5*b^3*d^3))*x)*(a*d+(a*e+b*d)*x +(b*e+c*d)*x^2+c*e*x^3)^(5/2)/c^5/e^5/(e*x+d)^2/(c*x^2+b*x+a)^2+10/153153* (16*c^5*d^5-96*b^5*e^5-c^4*d^3*e*(-96*a*e+47*b*d)+32*b^3*c*e^4*(12*a*e+11* b*d)+6*c^3*d*e^2*(-72*a^2*e^2+235*a*b*d*e+5*b^2*d^2)-b*c^2*e^3*(15*a^2*e^2 +1266*a*b*d*e+383*b^2*d^2)-14*c*e*(c^4*d^4+16*b^4*e^4-8*b^2*c*e^3*(9*a*e+7 *b*d)-2*c^3*d^2*e*(111*a*e+b*d)+3*c^2*e^2*(11*a^2*e^2+74*a*b*d*e+19*b^2*d^ 2))*x)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2)/c^4/e^3/(e*x+d)^2/(c* x^2+b*x+a)+4/51*e*(-b*e+2*c*d)*(c*x^2+b*x+a)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^ 2+c*e*x^3)^(5/2)/c^2/(e*x+d)^2+2/17*e*(c*x^2+b*x+a)*(a*d+(a*e+b*d)*x+(b*e+ c*d)*x^2+c*e*x^3)^(5/2)/c/(e*x+d)-2/153153*2^(1/2)*(-4*a*c+b^2)^(1/2)*(...
Result contains complex when optimal does not.
Time = 37.05 (sec) , antiderivative size = 23293, normalized size of antiderivative = 13.04 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\text {Result too large to show} \] Input:
Integrate[(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3)^(5/2),x]
Output:
Result too large to show
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.
| \(\displaystyle \int \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 2481 |
| \(\displaystyle \int \left (\frac {(2 c d-b e) \left (-c e (b d-9 a e)-2 b^2 e^2+c^2 d^2\right )}{27 c^2 e^2}+\frac {\left (3 c e (a e+b d)-(b e+c d)^2\right ) \left (\frac {b e+c d}{3 c e}+x\right )}{3 c e}+c e \left (\frac {b e+c d}{3 c e}+x\right )^3\right )^{5/2}d\left (\frac {b e+c d}{3 c e}+x\right )\) |
Input:
Int[(a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3)^(5/2),x]
Output:
$Aborted
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 *d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
Leaf count of result is larger than twice the leaf count of optimal. \(9999\) vs. \(2(1722)=3444\).
Time = 7.70 (sec) , antiderivative size = 10000, normalized size of antiderivative = 5.60
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(10000\) |
| default | \(\text {Expression too large to display}\) | \(18854\) |
| elliptic | \(\text {Expression too large to display}\) | \(18854\) |
Input:
int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Time = 0.16 (sec) , antiderivative size = 1776, normalized size of antiderivative = 0.99 \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2),x, algorithm="fric as")
Output:
2/459459*((256*c^9*d^9 - 1152*b*c^8*d^8*e + 6*(293*b^2*c^7 + 364*a*c^8)*d^ 7*e^2 - 21*(37*b^3*c^6 + 364*a*b*c^7)*d^6*e^3 - 3*(71*b^4*c^5 - 2122*a*b^2 *c^6 - 3400*a^2*c^7)*d^5*e^4 - 3*(71*b^5*c^4 - 1065*a*b^3*c^5 + 8500*a^2*b *c^6)*d^4*e^5 - 3*(259*b^6*c^3 - 3392*a*b^4*c^4 + 15698*a^2*b^2*c^5 - 3226 4*a^3*c^6)*d^3*e^6 + 3*(586*b^7*c^2 - 7427*a*b^5*c^3 + 32047*a^2*b^3*c^4 - 48396*a^3*b*c^5)*d^2*e^7 - 3*(384*b^8*c - 4972*a*b^6*c^2 + 22405*a^2*b^4* c^3 - 38382*a^3*b^2*c^4 + 14184*a^4*c^5)*d*e^8 + (256*b^9 - 3456*a*b^7*c + 16734*a^2*b^5*c^2 - 33375*a^3*b^3*c^3 + 21276*a^4*b*c^4)*e^9)*sqrt(c*e)*w eierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9* a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 6*(128*c^9*d^8*e - 512*b*c^8*d^7*e^2 + (647*b^2*c^7 + 996*a*c^8)*d^6*e^3 - (149*b^3*c^6 + 2988*a*b*c^7)*d^5*e^4 - 5*(20*b^4*c^5 - 309*a*b^2*c^6 - 876*a^2*c^7)*d^4*e ^5 - (149*b^5*c^4 - 1890*a*b^3*c^5 + 8760*a^2*b*c^6)*d^3*e^6 + (647*b^6*c^ 3 - 7317*a*b^4*c^4 + 26433*a^2*b^2*c^5 - 26484*a^3*c^6)*d^2*e^7 - (512*b^7 *c^2 - 5874*a*b^5*c^3 + 22053*a^2*b^3*c^4 - 26484*a^3*b*c^5)*d*e^8 + (128* b^8*c - 1536*a*b^6*c^2 + 6279*a^2*b^4*c^3 - 9393*a^3*b^2*c^4 + 2772*a^4*c^ 5)*e^9)*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 ...
Timed out. \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3)**(5/2),x)
Output:
Timed out
\[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\int { {\left (c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2),x, algorithm="maxi ma")
Output:
integrate((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)^(5/2), x)
\[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\int { {\left (c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2),x, algorithm="giac ")
Output:
integrate((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)^(5/2), x)
Timed out. \[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\int {\left (c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+\left (a\,e+b\,d\right )\,x+a\,d\right )}^{5/2} \,d x \] Input:
int((a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^(5/2),x)
Output:
int((a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^(5/2), x)
\[ \int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{5/2} \, dx=\int \left (a d +\left (a e +b d \right ) x +\left (b e +c d \right ) x^{2}+c e \,x^{3}\right )^{\frac {5}{2}}d x \] Input:
int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2),x)
Output:
int((a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(5/2),x)