Integrand size = 24, antiderivative size = 1195 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Output:
1/5*(2*b*e^2+5*c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/e^4+1/5*c*x^3*(c*x^4+b*x^2+a )^(1/2)/e^2+1/5*(12*a*c*e^4+b^2*e^4+25*b*c*d^2*e^2+30*c^2*d^4)*x*(c*x^4+b* x^2+a)^(1/2)/c^(1/2)/e^6/(a^(1/2)+c^(1/2)*x^2)+(a*e^4+b*d^2*e^2+c*d^4)*x*( c*x^4+b*x^2+a)^(1/2)/e^4/(-e^2*x^2+d^2)-3/4*d*(2*c*e^2*x^2+3*b*e^2+4*c*d^2 )*(c*x^4+b*x^2+a)^(1/2)/e^5-d*(c*x^4+b*x^2+a)^(3/2)/e/(-e^2*x^2+d^2)-3/2*d *(b*e^2+2*c*d^2)*(a*e^4+b*d^2*e^2+c*d^4)^(1/2)*arctanh((a*e^4+b*d^2*e^2+c* d^4)^(1/2)*x/d/e/(c*x^4+b*x^2+a)^(1/2))/e^7-3/8*d*(4*a*c*e^4+b^2*e^4+8*b*c *d^2*e^2+8*c^2*d^4)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2)) /c^(1/2)/e^7+3/2*d*(b*e^2+2*c*d^2)*(a*e^4+b*d^2*e^2+c*d^4)^(1/2)*arctanh(1 /2*(b*d^2+2*a*e^2+(b*e^2+2*c*d^2)*x^2)/(a*e^4+b*d^2*e^2+c*d^4)^(1/2)/(c*x^ 4+b*x^2+a)^(1/2))/e^7-1/5*a^(1/4)*(12*a*c*e^4+b^2*e^4+25*b*c*d^2*e^2+30*c^ 2*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/ 2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1 /2))/c^(3/4)/e^6/(c*x^4+b*x^2+a)^(1/2)+1/10*a^(1/4)*(60*c^(5/2)*d^6+20*a^( 1/2)*c^2*d^4*e^2+a^(1/2)*b^2*e^6+8*b*c^(1/2)*e^4*(a*e^2+2*b*d^2)+6*a^(1/2) *c*e^4*(2*a*e^2+3*b*d^2)+c^(3/2)*(32*a*d^2*e^4+70*b*d^4*e^2))*(a^(1/2)+c^( 1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM( 2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(3/4)/e^6/( c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+b*x^2+a)^(1/2)+3/4*(c^(1/2)*d^2-a^(1/2)*e^ 2)*(b*e^2+2*c*d^2)*(a*e^4+b*d^2*e^2+c*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^...
Result contains complex when optimal does not.
Time = 22.70 (sec) , antiderivative size = 11129, normalized size of antiderivative = 9.31 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(3/2)/(d + e*x)^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.
\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2}dx\) |
Input:
Int[(a + b*x^2 + c*x^4)^(3/2)/(d + e*x)^2,x]
Output:
$Aborted
Time = 5.63 (sec) , antiderivative size = 1199, normalized size of antiderivative = 1.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(1199\) |
elliptic | \(\text {Expression too large to display}\) | \(1199\) |
risch | \(\text {Expression too large to display}\) | \(1746\) |
Input:
int((c*x^4+b*x^2+a)^(3/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-(a*e^4+b*d^2*e^2+c*d^4)/e^5*(c*x^4+b*x^2+a)^(1/2)/(e*x+d)+1/5*c*x^3*(c*x^ 4+b*x^2+a)^(1/2)/e^2-1/2*c*d/e^3*x^2*(c*x^4+b*x^2+a)^(1/2)+1/3*(c/e^4*(2*b *e^2+3*c*d^2)-4/5*c/e^2*b)/c*x*(c*x^4+b*x^2+a)^(1/2)+1/2*(-4*c*d/e^5*(b*e^ 2+c*d^2)+3/2*c*d/e^3*b)/c*(c*x^4+b*x^2+a)^(1/2)+1/4*((2*a*b*e^6+6*a*c*d^2* e^4+3*b^2*d^2*e^4+10*b*c*d^4*e^2+7*c^2*d^6)/e^8-d^2/e^8*c*(a*e^4+b*d^2*e^2 +c*d^4)-1/3*(c/e^4*(2*b*e^2+3*c*d^2)-4/5*c/e^2*b)/c*a)*2^(1/2)/((-b+(-4*a* c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+( -4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1 /2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a /c)^(1/2))+1/2*(-2*d/e^7*(2*a*c*e^4+b^2*e^4+4*b*c*d^2*e^2+3*c^2*d^4)+c*d/e ^3*a-1/2*(-4*c*d/e^5*(b*e^2+c*d^2)+3/2*c*d/e^3*b)/c*b)*ln((2*c*x^2+b)/c^(1 /2)+2*(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-1/2*(1/e^6*(2*a*c*e^4+b^2*e^4+6*b*c*d ^2*e^2+5*c^2*d^4)+(a*e^4+b*d^2*e^2+c*d^4)/e^6*c-3/5*c/e^2*a-2/3*(c/e^4*(2* b*e^2+3*c*d^2)-4/5*c/e^2*b)/c*b)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/ 2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a *x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x* 2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2 ))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1 /2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))-3*d/e^9*(a*b*e^6+2*a*c*d^2* e^4+b^2*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)*(-1/2/(c*d^4/e^4+b*d^2/e^2+a)^...
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/(e*x+d)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(3/2)/(e*x+d)**2,x)
Output:
Integral((a + b*x**2 + c*x**4)**(3/2)/(d + e*x)**2, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/(e*x+d)^2,x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/(e*x + d)^2, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/(e*x + d)^2, x)
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(3/2)/(d + e*x)^2,x)
Output:
int((a + b*x^2 + c*x^4)^(3/2)/(d + e*x)^2, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e x +d \right )^{2}}d x \] Input:
int((c*x^4+b*x^2+a)^(3/2)/(e*x+d)^2,x)
Output:
int((c*x^4+b*x^2+a)^(3/2)/(e*x+d)^2,x)