Integrand size = 33, antiderivative size = 1636 \[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:
-2*(-d*g+e*f)^2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)/(e *x+d)^(1/2)-(e*(-a*e+b*d)*g^2-c*(3*d^2*g^2-4*d*e*f*g+2*e^2*f^2))*(g*x+f)^( 1/2)*(c*x^2+b*x+a)^(1/2)/c/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)-(2*c*f-(b+( -4*a*c+b^2)^(1/2))*g)*(e*(-a*e+b*d)*g^2-c*(3*d^2*g^2-4*d*e*f*g+2*e^2*f^2)) *(b-(-4*a*c+b^2)^(1/2)+2*c*x)*((-d*g+e*f)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2* c*f-(b+(-4*a*c+b^2)^(1/2))*g)/(e*x+d))^(1/2)*EllipticE((2*c*d-(b-(-4*a*c+b ^2)^(1/2))*e)^(1/2)*(g*x+f)^(1/2)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/( e*x+d)^(1/2),((2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)*(2*c*f-(b-(-4*a*c+b^2)^(1/2 ))*g)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g))^( 1/2))/c/e^2/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^ (1/2))*e)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/((-d*g+e*f)*(b-(-4*a*c+b^ 2)^(1/2)+2*c*x)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(e*x+d))^(1/2)/(c*x^2+b*x +a)^(1/2)+2*(-d*g+e*f)*(e*g*(3*b*e*f+3*(-4*a*c+b^2)^(1/2)*e*f-2*b*d*g-3*(- 4*a*c+b^2)^(1/2)*d*g-a*e*g)-c*(-3*d^2*g^2+2*d*e*f*g+2*e^2*f^2))*(b-(-4*a*c +b^2)^(1/2)+2*c*x)*((-d*g+e*f)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*f-(b+(-4* a*c+b^2)^(1/2))*g)/(e*x+d))^(1/2)*EllipticF((2*c*d-(b-(-4*a*c+b^2)^(1/2))* e)^(1/2)*(g*x+f)^(1/2)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(e*x+d)^(1/2 ),((2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)*(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)/(2*c* d-(b-(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g))^(1/2))/e^3/( 2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)/...
Leaf count is larger than twice the leaf count of optimal. \(14548\) vs. \(2(1636)=3272\).
Time = 36.98 (sec) , antiderivative size = 14548, normalized size of antiderivative = 8.89 \[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(f + g*x)^(5/2)/((d + e*x)^(3/2)*Sqrt[a + b*x + c*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.
\(\displaystyle \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1292 |
\(\displaystyle \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}}dx\) |
Input:
Int[(f + g*x)^(5/2)/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]),x]
Output:
$Aborted
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* (a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
Time = 12.58 (sec) , antiderivative size = 2908, normalized size of antiderivative = 1.78
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2908\) |
default | \(\text {Expression too large to display}\) | \(130840\) |
Input:
int((g*x+f)^(5/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOS E)
Output:
((g*x+f)*(e*x+d)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b *x+a)^(1/2)*(-2*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f)*(d^2 *g^2-2*d*e*f*g+e^2*f^2)/e^2/(a*e^2-b*d*e+c*d^2)/((x+d/e)*(c*e*g*x^3+b*e*g* x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f))^(1/2)+2*(g*(d^2*g^2-3*d*e*f*g+3*e^2* f^2)/e^3-(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^3*(a*e^2*g-b*d*e*g+b*e^2*f+c*d^2*g- c*d*e*f)/(a*e^2-b*d*e+c*d^2)+(a*e*g+b*e*f)*(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^2 /(a*e^2-b*d*e+c*d^2))*(1/2*(b+(-4*a*c+b^2)^(1/2))/c-f/g)*((d/e-1/2*(b+(-4* a*c+b^2)^(1/2))/c)*(x+f/g)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+f/g)/(x+d/e))^(1 /2)*(x+d/e)^2*((-d/e+f/g)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(1/2/c*(-b+(-4 *a*c+b^2)^(1/2))+f/g)/(x+d/e))^(1/2)*((-d/e+f/g)*(x+1/2*(b+(-4*a*c+b^2)^(1 /2))/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+f/g)/(x+d/e))^(1/2)/(d/e-1/2*(b+(-4 *a*c+b^2)^(1/2))/c)/(-d/e+f/g)/(c*e*g*(x+f/g)*(x+d/e)*(x-1/2/c*(-b+(-4*a*c +b^2)^(1/2)))*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*EllipticF(((d/e-1/2* (b+(-4*a*c+b^2)^(1/2))/c)*(x+f/g)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+f/g)/(x+d /e))^(1/2),((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(1/2*(b+(-4*a*c+b^2)^(1/2 ))/c-f/g)/(-1/2/c*(-b+(-4*a*c+b^2)^(1/2))-f/g)/(-d/e+1/2*(b+(-4*a*c+b^2)^( 1/2))/c))^(1/2))+2*(-1/e^2*g^2*(d*g-3*e*f)-(d^2*g^2-2*d*e*f*g+e^2*f^2)/e^2 *(b*e*g-c*d*g+c*e*f)/(a*e^2-b*d*e+c*d^2)+(2*b*e*g+2*c*e*f)*(d^2*g^2-2*d*e* f*g+e^2*f^2)/e^2/(a*e^2-b*d*e+c*d^2))*(1/2*(b+(-4*a*c+b^2)^(1/2))/c-f/g)*( (d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(x+f/g)/(-1/2*(b+(-4*a*c+b^2)^(1/2))...
Timed out. \[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)^(5/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((g*x+f)**(5/2)/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((f + g*x)**(5/2)/((d + e*x)**(3/2)*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^(5/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="ma xima")
Output:
integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
\[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {5}{2}}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^(5/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="gi ac")
Output:
integrate((g*x + f)^(5/2)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
Timed out. \[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((f + g*x)^(5/2)/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int((f + g*x)^(5/2)/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {(f+g x)^{5/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (g x +f \right )^{\frac {5}{2}}}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}d x \] Input:
int((g*x+f)^(5/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
int((g*x+f)^(5/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)