Integrand size = 44, antiderivative size = 137 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 e^2 (2 c d-b e) (d+e x)^4}-\frac {2 (2 c e f+8 c d g-5 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 e^2 (2 c d-b e)^2 (d+e x)^3} \] Output:
-2/5*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/(e *x+d)^4-2/15*(-5*b*e*g+8*c*d*g+2*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 3/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^3
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (-b e (3 e f+2 d g+5 e g x)+2 c \left (d^2 g+e^2 f x+4 d e (f+g x)\right )\right )}{15 e^2 (-2 c d+b e)^2 (d+e x)^3} \] Input:
Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 4,x]
Output:
(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e*(3 *e*f + 2*d*g + 5*e*g*x)) + 2*c*(d^2*g + e^2*f*x + 4*d*e*(f + g*x))))/(15*e ^2*(-2*c*d + b*e)^2*(d + e*x)^3)
Time = 0.67 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1216, 1218, 1123}
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.
\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^4} \, dx\) |
\(\Big \downarrow \) 1216 |
\(\displaystyle \int \frac {(f+g x) \left (\frac {c d^2-b d e}{d}-c e x\right )^4}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 1218 |
\(\displaystyle -\frac {(5 b e g-2 c (4 d g+e f)) \int \frac {(c d-b e-c e x)^3}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{5 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^4}{5 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {2 (-b e+c d-c e x)^3 (5 b e g-2 c (4 d g+e f))}{15 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g) (-b e+c d-c e x)^4}{5 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}\) |
Input:
Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^4,x]
Output:
(-2*(e*f - d*g)*(c*d - b*e - c*e*x)^4)/(5*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2)) + (2*(5*b*e*g - 2*c*(e*f + 4*d*g))*(c*d - b *e - c*e*x)^3)/(15*e^2*(2*c*d - b*e)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^ 2)^(3/2))
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2], x_Symbol] :> Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + 1/2))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c* d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IntegerQ[n]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 4.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (5 b \,e^{2} g x -8 c d e g x -2 c \,e^{2} f x +2 b d e g +3 b \,e^{2} f -2 c \,d^{2} g -8 c d e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15 \left (e x +d \right )^{3} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) | \(128\) |
orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (5 b \,e^{2} g x -8 c d e g x -2 c \,e^{2} f x +2 b d e g +3 b \,e^{2} f -2 c \,d^{2} g -8 c d e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15 \left (e x +d \right )^{3} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) | \(128\) |
trager | \(-\frac {2 \left (5 b c \,e^{3} g \,x^{2}-8 c^{2} d \,e^{2} g \,x^{2}-2 f \,c^{2} e^{3} x^{2}+5 b^{2} e^{3} g x -11 b c d \,e^{2} g x +b c \,e^{3} f x +6 c^{2} d^{2} e g x -6 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +3 b^{2} e^{3} f -4 b c \,d^{2} e g -11 b c d \,e^{2} f +2 c^{2} d^{3} g +8 c^{2} d^{2} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{15 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2} \left (e x +d \right )^{3}}\) | \(209\) |
default | \(-\frac {2 g \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{4} \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{5}}\) | \(213\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x,method=_RET URNVERBOSE)
Output:
-2/15*(c*e*x+b*e-c*d)*(5*b*e^2*g*x-8*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+3*b*e ^2*f-2*c*d^2*g-8*c*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^3 /e^2/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (129) = 258\).
Time = 3.98 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.24 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{2} e^{3} f + {\left (8 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} - {\left (8 \, c^{2} d^{2} e - 11 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f - 2 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left ({\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (6 \, c^{2} d^{2} e - 11 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \, {\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} + {\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x, algo rithm="fricas")
Output:
2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^2*e^3*f + (8*c^2*d*e ^2 - 5*b*c*e^3)*g)*x^2 - (8*c^2*d^2*e - 11*b*c*d*e^2 + 3*b^2*e^3)*f - 2*(c ^2*d^3 - 2*b*c*d^2*e + b^2*d*e^2)*g + ((6*c^2*d*e^2 - b*c*e^3)*f - (6*c^2* d^2*e - 11*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b ^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^ 4 - 4*b*c*d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^2*d^4*e^3 - 4*b*c*d^3*e^4 + b^ 2*d^2*e^5)*x)
\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{4}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**4,x )
Output:
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**4, x)
Exception generated. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x, algo rithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-2*c*d>0)', see `assume?` for more deta
Exception generated. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x, algo rithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[0,3,0,0]%%%},[6,1]%%%}+%%%{%%{[%%%{-6,[0,2,1,0]%%%} ,0]:[1,0,
Time = 7.30 (sec) , antiderivative size = 1022, normalized size of antiderivative = 7.46 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^4,x)
Output:
(((48*c^3*d^2*g - 16*c^3*d*e*f + 12*b*c^2*e^2*f + 20*b^2*c*e^2*g - 64*b*c^ 2*d*e*g)/(15*e^2*(b*e - 2*c*d)^3) - (d*((4*c^2*(7*b*e*g - 12*c*d*g + 2*c*e *f))/(15*e*(b*e - 2*c*d)^3) - (8*c^3*d*g)/(15*e*(b*e - 2*c*d)^3)))/e)*(c*d ^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((2*f*(b*e - c*d))/( 5*b*e^2 - 10*c*d*e) - (d*((2*b*e*g - 2*c*d*g + 2*c*e*f)/(5*b*e^2 - 10*c*d* e) - (2*c*d*g)/(5*b*e^2 - 10*c*d*e)))/e)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^ 2*x)^(1/2))/(d + e*x)^3 - (((d*((4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*g)/(5*(3* b*e^2 - 6*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d))))/e - (2*b*(b*e*g - 2*c*d*g + c*e*f))/(5*(3*b*e^2 - 6*c*d*e)*(b* e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (( (d*((8*c^2*(6*b*e*g - 11*c*d*g + c*e*f))/(15*e*(b*e - 2*c*d)^3) - (8*c^3*d *g)/(15*e*(b*e - 2*c*d)^3)))/e - (8*c*(b*e - c*d)*(5*b*e*g - 10*c*d*g + c* e*f))/(15*e^2*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2 ))/(d + e*x) + (((d*((4*c*(4*b*e*g - 7*c*d*g + c*e*f))/(5*(3*b*e^2 - 6*c*d *e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d))))/e - (4*(b*e - c*d)*(3*b*e*g - 6*c*d*g + c*e*f))/(5*e*(3*b*e^2 - 6*c*d*e)*(b *e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + ( ((d*((8*c^3*e*f - 24*c^3*d*g + 16*b*c^2*e*g)/(15*e*(b*e - 2*c*d)^3) - (8*c ^3*d*g)/(15*e*(b*e - 2*c*d)^3)))/e - (2*b*c*(3*b*e*g - 6*c*d*g + 2*c*e*f)) /(15*e*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(...
Time = 0.34 (sec) , antiderivative size = 1197, normalized size of antiderivative = 8.74 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^4,x)
Output:
(2*i*( - 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b* e + c*d - c*e*x)*b**2*d*e**2*g - 3*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*f - 5*sqrt(d + e*x)*sqr t(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*g *x + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b*c*d**2*e*g + 11*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*f + 11*sqrt(d + e*x)*sqrt( b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*g* x - sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b*c*e**3*f*x - 5*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2 *c*d)*sqrt( - b*e + c*d - c*e*x)*b*c*e**3*g*x**2 - 2*sqrt(d + e*x)*sqrt(b* e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*d**3*g - 8 *sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e*f - 6*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c *d)*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e*g*x + 6*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*d*e**2*f*x + 8*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*d*e**2*g*x**2 + 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b* e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*e**3*f*x**2 + sqrt(c)*b**2*c*d* *3*e**2*g + 3*sqrt(c)*b**2*c*d**2*e**3*g*x + 3*sqrt(c)*b**2*c*d*e**4*g*...