Integrand size = 33, antiderivative size = 1505 \[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:
-2*(-d*g+e*f)^2*(c*x^2+b*x+a)^(1/2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)/(g *x+f)^(1/2)+2*(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)*(-d*g+e*f)*(c*f^2-g*(-a*g+b *f))*(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*d-(b+(-4*a*c+b^2)^(1/2))*e)/(g*x+f))^(1/2)*EllipticE((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)^(1 /2)/(g*x+f)^(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/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/(a*e^2-b*d*e+c*d^2)/(2 *c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)/(-(- d*g+e*f)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(g* x+f))^(1/2)/(c*x^2+b*x+a)^(1/2)-2*(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)*(-d*g+e *f)*(c*f^2-g*(-a*g+b*f))*(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*d-(b+(-4*a*c+b^2)^(1/2))*e)/(g*x+f))^(1/2)*Elli pticF((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)^(1/2)/(g*x+f)^(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/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/(a *e^2-b*d*e+c*d^2)/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(2*c*f-(b+(-4*a*c +b^2)^(1/2))*g)/(-(-d*g+e*f)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*d-(b-(-4*a* c+b^2)^(1/2))*e)/(g*x+f))^(1/2)/(c*x^2+b*x+a)^(1/2)-2*g*(-d*g+e*f)*(b-(...
Leaf count is larger than twice the leaf count of optimal. \(6304\) vs. \(2(1505)=3010\).
Time = 35.45 (sec) , antiderivative size = 6304, normalized size of antiderivative = 4.19 \[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(f + g*x)^(3/2)/((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.
\(\displaystyle \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1292 |
\(\displaystyle \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}}dx\) |
Input:
Int[(f + g*x)^(3/2)/((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]
Leaf count of result is larger than twice the leaf count of optimal. \(2795\) vs. \(2(1358)=2716\).
Time = 12.12 (sec) , antiderivative size = 2796, normalized size of antiderivative = 1.86
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(2796\) |
default | \(\text {Expression too large to display}\) | \(40476\) |
Input:
int((g*x+f)^(3/2)/(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*g- e*f)/e/(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*g-2*e*f)/e^2+(d*g-e*f)/e^2*(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*g-e*f)/e/( 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*(g^2/e+(d*g-e*f)/e*(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*g-e*f)/e/(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)*((-...
Timed out. \[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)^(3/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fr icas")
Output:
Timed out
\[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((g*x+f)**(3/2)/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((f + g*x)**(3/2)/((d + e*x)**(3/2)*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^(3/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="ma xima")
Output:
integrate((g*x + f)^(3/2)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
\[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (g x + f\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*x+f)^(3/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="gi ac")
Output:
integrate((g*x + f)^(3/2)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/2)), x)
Timed out. \[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((f + g*x)^(3/2)/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int((f + g*x)^(3/2)/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {(f+g x)^{3/2}}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (g x +f \right )^{\frac {3}{2}}}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}d x \] Input:
int((g*x+f)^(3/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
int((g*x+f)^(3/2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)