Integrand size = 31, antiderivative size = 1282 \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Output:
(-2*b^3*e*g+2*c*(-2*a*c+b^2)*(d*g+e*f)-2*b*c*(-3*a*e*g+c*d*f)-2*c*(2*c^2*d *f+b^2*e*g-c*(2*a*e*g+b*d*g+b*e*f))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c *f^2-g*(-a*g+b*f))/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-2*e*(2*c^3*d^2*f*(-d* g+e*f)-b^2*e^2*g*(-a*e*g-b*d*g+2*b*e*f)-c*e*(4*a^2*e^2*g^2-a*b*e*g*(-3*d*g +7*e*f)-b^2*(-2*d^2*g^2+d*e*f*g+2*e^2*f^2))-c^2*(2*a*e*(-2*d^2*g^2+d*e*f*g +3*e^2*f^2)+b*d*(-d^2*g^2-d*e*f*g+2*e^2*f^2)))*(c*x^2+b*x+a)^(1/2)/(-4*a*c +b^2)/(a*e^2-b*d*e+c*d^2)^2/(-d*g+e*f)/(c*f^2-g*(-a*g+b*f))/(e*x+d)^(1/2)+ 2^(1/2)*(2*c^3*d^2*f*(-d*g+e*f)-b^2*e^2*g*(-a*e*g-b*d*g+2*b*e*f)-c*e*(4*a^ 2*e^2*g^2-a*b*e*g*(-3*d*g+7*e*f)-b^2*(-2*d^2*g^2+d*e*f*g+2*e^2*f^2))-c^2*( 2*a*e*(-2*d^2*g^2+d*e*f*g+3*e^2*f^2)+b*d*(-d^2*g^2-d*e*f*g+2*e^2*f^2)))*(e *x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2*c*x+ b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(- 4*a*c+b^2)^(1/2))*e))^(1/2))/(-4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^2/(-d* g+e*f)/(c*f^2-g*(-a*g+b*f))/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^( 1/2)/(c*x^2+b*x+a)^(1/2)-2*2^(1/2)*(2*c^2*d*f+b^2*e*g-c*(2*a*e*g+b*d*g+b*e *f))*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/ (-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2 ^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/( -4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)/(c*f^2-g*(-a*g+b*f))/(e*x+d)^(1/2)/( c*x^2+b*x+a)^(1/2)-4*2^(1/2)*(-4*a*c+b^2)^(1/2)*g^3*(c*(e*x+d)/(2*c*d-(...
Result contains complex when optimal does not.
Time = 43.38 (sec) , antiderivative size = 52650, normalized size of antiderivative = 41.07 \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[1/((d + e*x)^(3/2)*(f + g*x)*(a + b*x + c*x^2)^(3/2)),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.
\(\displaystyle \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1292 |
\(\displaystyle \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}}dx\) |
Input:
Int[1/((d + e*x)^(3/2)*(f + g*x)*(a + b*x + c*x^2)^(3/2)),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. \(3887\) vs. \(2(1204)=2408\).
Time = 13.85 (sec) , antiderivative size = 3888, normalized size of antiderivative = 3.03
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3888\) |
default | \(\text {Expression too large to display}\) | \(31581\) |
Input:
int(1/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2*c*e*(- (4*a^2*c*e^3*g^2-a*b^2*e^3*g^2+3*a*b*c*d*e^2*g^2-7*a*b*c*e^3*f*g-4*a*c^2*d ^2*e*g^2+2*a*c^2*d*e^2*f*g+6*a*c^2*e^3*f^2-b^3*d*e^2*g^2+2*b^3*e^3*f*g+2*b ^2*c*d^2*e*g^2-b^2*c*d*e^2*f*g-2*b^2*c*e^3*f^2-b*c^2*d^3*g^2-b*c^2*d^2*e*f *g+2*b*c^2*d*e^2*f^2+2*c^3*d^3*f*g-2*c^3*d^2*e*f^2)/(4*a^3*c*e^4-a^2*b^2*e ^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+2*a*b^2*c*d^2*e^2-8*a*b *c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^2*c^2*d^4)/(a*d*g^3-a*e *f*g^2-b*d*f*g^2+b*e*f^2*g+c*d*f^2*g-c*e*f^3)*x^2-(4*a^2*b*c*e^4*g^2-2*a^2 *c^2*d*e^3*g^2+2*a^2*c^2*e^4*f*g-a*b^3*e^4*g^2+4*a*b^2*c*d*e^3*g^2-8*a*b^2 *c*e^4*f*g-3*a*b*c^2*d^2*e^2*g^2+7*a*b*c^2*e^4*f^2-2*a*c^3*d^3*e*g^2+4*a*c ^3*d^2*e^2*f*g-2*a*c^3*d*e^3*f^2-b^4*d*e^3*g^2+2*b^4*e^4*f*g+b^3*c*d^2*e^2 *g^2-2*b^3*c*e^4*f^2+b^2*c^2*d^3*e*g^2-2*b^2*c^2*d^2*e^2*f*g+b^2*c^2*d*e^3 *f^2-b*c^3*d^4*g^2+b*c^3*d^2*e^2*f^2+2*c^4*d^4*f*g-2*c^4*d^3*e*f^2)/(4*a^3 *c*e^4-a^2*b^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+2*a*b^2 *c*d^2*e^2-8*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^2*c^2*d ^4)/(a*d*g^3-a*e*f*g^2-b*d*f*g^2+b*e*f^2*g+c*d*f^2*g-c*e*f^3)/c/e*x-(4*a^3 *c*e^4*g^2-a^2*b^2*e^4*g^2-4*a^2*b*c*e^4*f*g-2*a^2*c^2*d^2*e^2*g^2+2*a^2*c ^2*d*e^3*f*g+4*a^2*c^2*e^4*f^2+a*b^3*e^4*f*g+4*a*b^2*c*d^2*e^2*g^2-4*a*b^2 *c*d*e^3*f*g-a*b^2*c*e^4*f^2-6*a*b*c^2*d^3*e*g^2+3*a*b*c^2*d^2*e^2*f*g+3*a *b*c^2*d*e^3*f^2+2*a*c^3*d^4*g^2+2*a*c^3*d^3*e*f*g-4*a*c^3*d^2*e^2*f^2-...
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(e*x+d)**(3/2)/(g*x+f)/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral(1/((d + e*x)**(3/2)*(f + g*x)*(a + b*x + c*x**2)**(3/2)), x)
\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \] Input:
integrate(1/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima ")
Output:
integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)*(g*x + f)), x)
\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \] Input:
integrate(1/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)*(g*x + f)), x)
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (f+g\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int(1/((f + g*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x)
Output:
int(1/((f + g*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)), x)
\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}d x \] Input:
int(1/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(3/2),x)
Output:
int(1/(e*x+d)^(3/2)/(g*x+f)/(c*x^2+b*x+a)^(3/2),x)