Integrand size = 44, antiderivative size = 136 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (4 c e f+2 c d g-3 b e g) (b+2 c x)}{3 e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{3 e^2 (2 c d-b e) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \] Output:
2/3*(-3*b*e*g+2*c*d*g+4*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^3/(d*(-b*e+c*d)-b* e^2*x-c*e^2*x^2)^(1/2)-2/3*(-d*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)/(d*(-b*e+c* d)-b*e^2*x-c*e^2*x^2)^(1/2)
Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.10 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {-8 c^2 \left (d^3 g+2 e^3 f x^2+d^2 e (-f+g x)+d e^2 x (2 f+g x)\right )+4 b c e \left (d^2 g+d e (-4 f+2 g x)+e^2 x (-2 f+3 g x)\right )+2 b^2 e^2 (2 d g+e (f+3 g x))}{3 e^2 (-2 c d+b e)^3 (d+e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2) ),x]
Output:
(-8*c^2*(d^3*g + 2*e^3*f*x^2 + d^2*e*(-f + g*x) + d*e^2*x*(2*f + g*x)) + 4 *b*c*e*(d^2*g + d*e*(-4*f + 2*g*x) + e^2*x*(-2*f + 3*g*x)) + 2*b^2*e^2*(2* d*g + e*(f + 3*g*x)))/(3*e^2*(-2*c*d + b*e)^3*(d + e*x)*Sqrt[(d + e*x)*(-( b*e) + c*(d - e*x))])
Time = 0.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1220, 1088}
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.
| \(\displaystyle \int \frac {f+g x}{(d+e x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-3 b e g+2 c d g+4 c e f) \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 e (2 c d-b e)}-\frac {2 (e f-d g)}{3 e^2 (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{3 e (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{3 e^2 (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
Input:
Int[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]
Output:
(2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(3*e*(2*c*d - b*e)^3*Sqrt[d* (c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(e*f - d*g))/(3*e^2*(2*c*d - b*e) *(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) Int[(d + e*x )^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Time = 2.34 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.68
| method | result | size |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (6 b c \,e^{3} g \,x^{2}-4 c^{2} d \,e^{2} g \,x^{2}-8 f \,c^{2} e^{3} x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -4 b c \,e^{3} f x -4 c^{2} d^{2} e g x -8 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +b^{2} e^{3} f +2 b c \,d^{2} e g -8 b c d \,e^{2} f -4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right )}{3 \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(228\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (6 b c \,e^{3} g \,x^{2}-4 c^{2} d \,e^{2} g \,x^{2}-8 f \,c^{2} e^{3} x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -4 b c \,e^{3} f x -4 c^{2} d^{2} e g x -8 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +b^{2} e^{3} f +2 b c \,d^{2} e g -8 b c d \,e^{2} f -4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right )}{3 \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +12 d^{2} e b \,c^{2}-8 d^{3} c^{3}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(228\) |
| trager | \(-\frac {2 \left (6 b c \,e^{3} g \,x^{2}-4 c^{2} d \,e^{2} g \,x^{2}-8 f \,c^{2} e^{3} x^{2}+3 b^{2} e^{3} g x +4 b c d \,e^{2} g x -4 b c \,e^{3} f x -4 c^{2} d^{2} e g x -8 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +b^{2} e^{3} f +2 b c \,d^{2} e g -8 b c d \,e^{2} f -4 c^{2} d^{3} g +4 c^{2} d^{2} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3 e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \left (e x +d \right )^{2} \left (b e -2 c d \right ) \left (c e x +b e -c d \right )}\) | \(233\) |
| default | \(\frac {2 g \left (-2 c \,e^{2} x -b \,e^{2}\right )}{e \left (-4 c \,e^{2} \left (-b d e +c \,d^{2}\right )-b^{2} e^{4}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}-\frac {\left (d g -e f \right ) \left (-\frac {2}{3 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right )}{3 \left (-b \,e^{2}+2 d e c \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{2}}\) | \(239\) |
Input:
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method=_RETUR NVERBOSE)
Output:
-2/3*(c*e*x+b*e-c*d)*(6*b*c*e^3*g*x^2-4*c^2*d*e^2*g*x^2-8*c^2*e^3*f*x^2+3* b^2*e^3*g*x+4*b*c*d*e^2*g*x-4*b*c*e^3*f*x-4*c^2*d^2*e*g*x-8*c^2*d*e^2*f*x+ 2*b^2*d*e^2*g+b^2*e^3*f+2*b*c*d^2*e*g-8*b*c*d*e^2*f-4*c^2*d^3*g+4*c^2*d^2* e*f)/(b^3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/e^2/(-c*e^2*x^2-b*e^ 2*x-b*d*e+c*d^2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (128) = 256\).
Time = 3.92 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.99 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f + {\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} g\right )} x^{2} - {\left (4 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} f + 2 \, {\left (2 \, c^{2} d^{3} - b c d^{2} e - b^{2} d e^{2}\right )} g + {\left (4 \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} f + {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} - 3 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, {\left (8 \, c^{4} d^{6} e^{2} - 20 \, b c^{3} d^{5} e^{3} + 18 \, b^{2} c^{2} d^{4} e^{4} - 7 \, b^{3} c d^{3} e^{5} + b^{4} d^{2} e^{6} - {\left (8 \, c^{4} d^{3} e^{5} - 12 \, b c^{3} d^{2} e^{6} + 6 \, b^{2} c^{2} d e^{7} - b^{3} c e^{8}\right )} x^{3} - {\left (8 \, c^{4} d^{4} e^{4} - 4 \, b c^{3} d^{3} e^{5} - 6 \, b^{2} c^{2} d^{2} e^{6} + 5 \, b^{3} c d e^{7} - b^{4} e^{8}\right )} x^{2} + {\left (8 \, c^{4} d^{5} e^{3} - 28 \, b c^{3} d^{4} e^{4} + 30 \, b^{2} c^{2} d^{3} e^{5} - 13 \, b^{3} c d^{2} e^{6} + 2 \, b^{4} d e^{7}\right )} x\right )}} \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algori thm="fricas")
Output:
2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (2*c^2*d* e^2 - 3*b*c*e^3)*g)*x^2 - (4*c^2*d^2*e - 8*b*c*d*e^2 + b^2*e^3)*f + 2*(2*c ^2*d^3 - b*c*d^2*e - b^2*d*e^2)*g + (4*(2*c^2*d*e^2 + b*c*e^3)*f + (4*c^2* d^2*e - 4*b*c*d*e^2 - 3*b^2*e^3)*g)*x)/(8*c^4*d^6*e^2 - 20*b*c^3*d^5*e^3 + 18*b^2*c^2*d^4*e^4 - 7*b^3*c*d^3*e^5 + b^4*d^2*e^6 - (8*c^4*d^3*e^5 - 12* b*c^3*d^2*e^6 + 6*b^2*c^2*d*e^7 - b^3*c*e^8)*x^3 - (8*c^4*d^4*e^4 - 4*b*c^ 3*d^3*e^5 - 6*b^2*c^2*d^2*e^6 + 5*b^3*c*d*e^7 - b^4*e^8)*x^2 + (8*c^4*d^5* e^3 - 28*b*c^3*d^4*e^4 + 30*b^2*c^2*d^3*e^5 - 13*b^3*c*d^2*e^6 + 2*b^4*d*e ^7)*x)
\[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
Output:
Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(d + e*x)), x)
Exception generated. \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algori thm="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
\[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \] Input:
integrate((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algori thm="giac")
Output:
integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d )), x)
Time = 11.58 (sec) , antiderivative size = 872, normalized size of antiderivative = 6.41 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((f + g*x)/((d + e*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2)),x)
Output:
(((2*b*g)/(3*e*(b*e - 2*c*d)^3) - (4*c*d*g)/(3*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) + (((2*d*g)/(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3) - (2*e*f)/(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12 *b*c*d*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (( x*(((e*(b*e - c*d) + c*d*e)*((4*c^3*e*(3*b*g - 2*c*f))/(3*(b*e - 2*c*d)^2* (4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (4*b*c^3*e*g)/(3*(b*e - 2*c*d)^2* (4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^3*g*(e*(b*e - c*d) + c*d*e)) /(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c*e^2) - ( 2*c^2*(8*b^2*e^2*g + 8*c^2*d^2*g - 10*b*c*e^2*f + 16*c^2*d*e*f - 16*b*c*d* e*g))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (2*b*c ^2*e*(3*b*g - 2*c*f))/(3*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2* d*e)) + (8*c^3*d*g*(b*e - c*d))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^ 2 - 4*b*c^2*d*e))) + (d*(b*e - c*d)*((4*c^3*e*(3*b*g - 2*c*f))/(3*(b*e - 2 *c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (4*b*c^3*e*g)/(3*(b*e - 2 *c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) - (8*c^3*g*(e*(b*e - c*d) + c*d*e))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e))))/(c* e^2) - (b*c*(8*b^2*e^2*g + 8*c^2*d^2*g - 10*b*c*e^2*f + 16*c^2*d*e*f - 16* b*c*d*e*g))/(3*e*(b*e - 2*c*d)^2*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)))*( c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/((d + e*x)*(b*e - c*d + c*e*x) )
Time = 0.24 (sec) , antiderivative size = 1113, normalized size of antiderivative = 8.18 \[ \int \frac {f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(2*i*( - 6*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*d**2*e**2*g - 12*sqrt(c )*sqrt( - b*e + c*d - c*e*x)*b**2*d*e**3*g*x - 6*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*e**4*g*x**2 + 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c*d** 3*e*g + 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c*d**2*e**2*f + 32*sqrt(c)* sqrt( - b*e + c*d - c*e*x)*b*c*d**2*e**2*g*x + 16*sqrt(c)*sqrt( - b*e + c* d - c*e*x)*b*c*d*e**3*f*x + 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c*d*e* *3*g*x**2 + 8*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c*e**4*f*x**2 - 8*sqrt( c)*sqrt( - b*e + c*d - c*e*x)*c**2*d**4*g - 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**2*d**3*e*f - 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**2*d**3*e* g*x - 32*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e**2*f*x - 8*sqrt(c) *sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e**2*g*x**2 - 16*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*c**2*d*e**3*f*x**2 + 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqr t( - b*e + 2*c*d)*b**2*d*e**2*g + sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b**2*e**3*f + 3*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b**2*e**3*g*x + 2*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2* c*d)*b*c*d**2*e*g - 8*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d) *b*c*d*e**2*f + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b*c *d*e**2*g*x - 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b*c*e **3*f*x + 6*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*b*c*e**3* g*x**2 - 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*c**2*d*...