Integrand size = 44, antiderivative size = 210 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (2 c d-b e) (d+e x)^3}-\frac {2 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^2 (d+e x)^2}-\frac {4 c (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{15 e^2 (2 c d-b e)^3 (d+e x)} \]
-2/5*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)/(e *x+d)^3-2/15*(-5*b*e*g+6*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 1/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^2-4/15*c*(-5*b*e*g+6*c*d*g+4*c*e*f)*(d*(-b *e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)
Time = 0.17 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.79 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (-c d+b e+c e x) \left (b^2 e^2 (3 e f+2 d g+5 e g x)-2 b c e \left (7 d^2 g+e^2 x (2 f+5 g x)+2 d e (4 f+9 g x)\right )+4 c^2 \left (3 d^3 g+2 e^3 f x^2+3 d e^2 x (2 f+g x)+d^2 e (7 f+9 g x)\right )\right )}{15 e^2 (-2 c d+b e)^3 (d+e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \]
(-2*(-(c*d) + b*e + c*e*x)*(b^2*e^2*(3*e*f + 2*d*g + 5*e*g*x) - 2*b*c*e*(7 *d^2*g + e^2*x*(2*f + 5*g*x) + 2*d*e*(4*f + 9*g*x)) + 4*c^2*(3*d^3*g + 2*e ^3*f*x^2 + 3*d*e^2*x*(2*f + g*x) + d^2*e*(7*f + 9*g*x))))/(15*e^2*(-2*c*d + b*e)^3*(d + e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.41 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1220, 1129, 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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^3 \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-5 b e g+6 c d g+4 c e f) \int \frac {1}{(d+e x)^2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-5 b e g+6 c d g+4 c e f) \left (\frac {2 c \int \frac {1}{(d+e x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 (2 c d-b e)}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^2 (2 c d-b e)}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {\left (-\frac {4 c \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x) (2 c d-b e)^2}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e (d+e x)^2 (2 c d-b e)}\right ) (-5 b e g+6 c d g+4 c e f)}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 e^2 (d+e x)^3 (2 c d-b e)}\) |
(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*e^2*(2*c*d - b*e)*(d + e*x)^3) + ((4*c*e*f + 6*c*d*g - 5*b*e*g)*((-2*Sqrt[d*(c*d - b*e ) - b*e^2*x - c*e^2*x^2])/(3*e*(2*c*d - b*e)*(d + e*x)^2) - (4*c*Sqrt[d*(c *d - b*e) - b*e^2*x - c*e^2*x^2])/(3*e*(2*c*d - b*e)^2*(d + e*x))))/(5*e*( 2*c*d - b*e))
3.23.14.3.1 Defintions of rubi rules used
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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 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 = 1.07 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07
| method | result | size |
| trager | \(\frac {2 \left (-10 b c \,e^{3} g \,x^{2}+12 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+5 b^{2} e^{3} g x -36 b c d \,e^{2} g x -4 b c \,e^{3} f x +36 c^{2} d^{2} e g x +24 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +3 b^{2} e^{3} f -14 b c \,d^{2} e g -16 b c d \,e^{2} f +12 c^{2} d^{3} g +28 c^{2} d^{2} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \left (e x +d \right )^{3}}\) | \(224\) |
| gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-10 b c \,e^{3} g \,x^{2}+12 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+5 b^{2} e^{3} g x -36 b c d \,e^{2} g x -4 b c \,e^{3} f x +36 c^{2} d^{2} e g x +24 c^{2} d \,e^{2} f x +2 b^{2} d \,e^{2} g +3 b^{2} e^{3} f -14 b c \,d^{2} e g -16 b c d \,e^{2} f +12 c^{2} d^{3} g +28 c^{2} d^{2} e f \right )}{15 \left (e x +d \right )^{2} \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\) | \(236\) |
| default | \(\frac {g \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{e^{3}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{5 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 c \,e^{2} \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{5 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{4}}\) | \(365\) |
2/15*(-10*b*c*e^3*g*x^2+12*c^2*d*e^2*g*x^2+8*c^2*e^3*f*x^2+5*b^2*e^3*g*x-3 6*b*c*d*e^2*g*x-4*b*c*e^3*f*x+36*c^2*d^2*e*g*x+24*c^2*d*e^2*f*x+2*b^2*d*e^ 2*g+3*b^2*e^3*f-14*b*c*d^2*e*g-16*b*c*d*e^2*f+12*c^2*d^3*g+28*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/(e*x+d)^3*(-c*e^2*x^ 2-b*e^2*x-b*d*e+c*d^2)^(1/2)
Time = 7.70 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.75 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, c^{2} e^{3} f + {\left (6 \, c^{2} d e^{2} - 5 \, b c e^{3}\right )} g\right )} x^{2} + {\left (28 \, c^{2} d^{2} e - 16 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + 2 \, {\left (6 \, c^{2} d^{3} - 7 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (6 \, c^{2} d e^{2} - b c e^{3}\right )} f + {\left (36 \, c^{2} d^{2} e - 36 \, b c d e^{2} + 5 \, b^{2} e^{3}\right )} g\right )} x\right )}}{15 \, {\left (8 \, c^{3} d^{6} e^{2} - 12 \, b c^{2} d^{5} e^{3} + 6 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5} + {\left (8 \, c^{3} d^{3} e^{5} - 12 \, b c^{2} d^{2} e^{6} + 6 \, b^{2} c d e^{7} - b^{3} e^{8}\right )} x^{3} + 3 \, {\left (8 \, c^{3} d^{4} e^{4} - 12 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )} x^{2} + 3 \, {\left (8 \, c^{3} d^{5} e^{3} - 12 \, b c^{2} d^{4} e^{4} + 6 \, b^{2} c d^{3} e^{5} - b^{3} d^{2} e^{6}\right )} x\right )}} \]
-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*c^2*e^3*f + (6*c^2* d*e^2 - 5*b*c*e^3)*g)*x^2 + (28*c^2*d^2*e - 16*b*c*d*e^2 + 3*b^2*e^3)*f + 2*(6*c^2*d^3 - 7*b*c*d^2*e + b^2*d*e^2)*g + (4*(6*c^2*d*e^2 - b*c*e^3)*f + (36*c^2*d^2*e - 36*b*c*d*e^2 + 5*b^2*e^3)*g)*x)/(8*c^3*d^6*e^2 - 12*b*c^2 *d^5*e^3 + 6*b^2*c*d^4*e^4 - b^3*d^3*e^5 + (8*c^3*d^3*e^5 - 12*b*c^2*d^2*e ^6 + 6*b^2*c*d*e^7 - b^3*e^8)*x^3 + 3*(8*c^3*d^4*e^4 - 12*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)*x^2 + 3*(8*c^3*d^5*e^3 - 12*b*c^2*d^4*e^4 + 6 *b^2*c*d^3*e^5 - b^3*d^2*e^6)*x)
\[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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,[6,3,13,0]%%%}+%%%{-12,[5,4,12,1]%%%}+%%%{60,[4,5,11 ,2]%%%}+%
Time = 12.21 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.24 \[ \int \frac {f+g x}{(d+e x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {\left (\frac {8\,c^2\,d\,g+16\,c^2\,e\,f-8\,b\,c\,e\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^2\,d\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x}-\frac {\left (\frac {2\,b\,g}{5\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}-\frac {4\,c\,d\,g}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {12\,c\,d\,g-12\,b\,e\,g+8\,c\,e\,f}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}+\frac {4\,c\,d\,g}{5\,e\,\left (3\,b\,e^2-6\,c\,d\,e\right )\,\left (b\,e-2\,c\,d\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {2\,f}{5\,b\,e^2-10\,c\,d\,e}-\frac {2\,d\,g}{e\,\left (5\,b\,e^2-10\,c\,d\,e\right )}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {4\,c\,g\,\left (3\,b\,e-4\,c\,d\right )}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}-\frac {8\,c^2\,d\,g}{15\,e^2\,{\left (b\,e-2\,c\,d\right )}^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{d+e\,x} \]
(((8*c^2*d*g + 16*c^2*e*f - 8*b*c*e*g)/(15*e^2*(b*e - 2*c*d)^3) - (8*c^2*d *g)/(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) - (((2*b*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)) - (4*c*d*g)/( 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 - (((12*c*d*g - 12*b*e*g + 8*c*e*f)/(5*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)) + (4*c*d*g)/(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 + (((2*f)/(5* b*e^2 - 10*c*d*e) - (2*d*g)/(e*(5*b*e^2 - 10*c*d*e)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 - (((4*c*g*(3*b*e - 4*c*d))/(15*e^2*( b*e - 2*c*d)^3) - (8*c^2*d*g)/(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)