Integrand size = 44, antiderivative size = 137 \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (2 c d-b e) (d+e x)^2}-\frac {2 (2 c e f+4 c d g-3 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (2 c d-b e)^2 (d+e x)} \] Output:
-2/3*(-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)^2-2/3*(-3*b*e*g+4*c*d*g+2*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)
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65 \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (2 c \left (d^2 g+e^2 f x+2 d e (f+g x)\right )-b e (2 d g+e (f+3 g x))\right )}{3 e^2 (-2 c d+b e)^2 (d+e x)^2} \] Input:
Integrate[(f + g*x)/((d + e*x)^2*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2] ),x]
Output:
(-2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(2*c*(d^2*g + e^2*f*x + 2*d*e*( f + g*x)) - b*e*(2*d*g + e*(f + 3*g*x))))/(3*e^2*(-2*c*d + b*e)^2*(d + e*x )^2)
Time = 0.57 (sec) , antiderivative size = 137, 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, 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.
\(\displaystyle \int \frac {f+g x}{(d+e x)^2 \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle \frac {(-3 b e g+4 c d g+2 c e f) \int \frac {1}{(d+e x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (d+e x)^2 (2 c d-b e)}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle -\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 e^2 (d+e x)^2 (2 c d-b e)}-\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g+4 c d g+2 c e f)}{3 e^2 (d+e x) (2 c d-b e)^2}\) |
Input:
Int[(f + g*x)/((d + e*x)^2*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]
Output:
(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*e^2*(2*c*d - b*e)*(d + e*x)^2) - (2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*e^2*(2*c*d - b*e)^2*(d + e*x))
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_))*((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 = 3.00 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.84
method | result | size |
trager | \(\frac {2 \left (3 b \,e^{2} g x -4 c d e g x -2 c \,e^{2} f x +2 b d e g +b \,e^{2} f -2 c \,d^{2} g -4 c d e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \left (e x +d \right )^{2} e^{2}}\) | \(115\) |
gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (3 b \,e^{2} g x -4 c d e g x -2 c \,e^{2} f x +2 b d e g +b \,e^{2} f -2 c \,d^{2} g -4 c d e f \right )}{3 \left (e x +d \right ) e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\) | \(127\) |
orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (3 b \,e^{2} g x -4 c d e g x -2 c \,e^{2} f x +2 b d e g +b \,e^{2} f -2 c \,d^{2} g -4 c d e f \right )}{3 \left (e x +d \right ) e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\) | \(127\) |
default | \(-\frac {2 g \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{e^{2} \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (d g -e f \right ) \left (-\frac {2 \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {4 c \,e^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{3 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )}\right )}{e^{3}}\) | \(213\) |
Input:
int((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RET URNVERBOSE)
Output:
2/3*(3*b*e^2*g*x-4*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+b*e^2*f-2*c*d^2*g-4*c*d *e*f)/(b^2*e^2-4*b*c*d*e+4*c^2*d^2)/(e*x+d)^2/e^2*(-c*e^2*x^2-b*e^2*x-b*d* e+c*d^2)^(1/2)
Time = 0.92 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.33 \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (4 \, c d e - b e^{2}\right )} f + 2 \, {\left (c d^{2} - b d e\right )} g + {\left (2 \, c e^{2} f + {\left (4 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )}}{3 \, {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4} + {\left (4 \, c^{2} d^{2} e^{4} - 4 \, b c d e^{5} + b^{2} e^{6}\right )} x^{2} + 2 \, {\left (4 \, c^{2} d^{3} e^{3} - 4 \, b c d^{2} e^{4} + b^{2} d e^{5}\right )} x\right )}} \] Input:
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo rithm="fricas")
Output:
-2/3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((4*c*d*e - b*e^2)*f + 2*( c*d^2 - b*d*e)*g + (2*c*e^2*f + (4*c*d*e - 3*b*e^2)*g)*x)/(4*c^2*d^4*e^2 - 4*b*c*d^3*e^3 + b^2*d^2*e^4 + (4*c^2*d^2*e^4 - 4*b*c*d*e^5 + b^2*e^6)*x^2 + 2*(4*c^2*d^3*e^3 - 4*b*c*d^2*e^4 + b^2*d*e^5)*x)
\[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\int \frac {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{2}}\, dx \] Input:
integrate((g*x+f)/(e*x+d)**2/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x )
Output:
Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**2), x)
Exception generated. \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (129) = 258\).
Time = 0.43 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.20 \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 \, {\left (\frac {{\left (2 \, \sqrt {-c} c e f + 4 \, \sqrt {-c} c d g - 3 \, b \sqrt {-c} e g\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}} - \frac {\frac {{\left (3 \, c \sqrt {-c + \frac {2 \, c d}{e x + d} - \frac {b e}{e x + d}} + {\left (-c + \frac {2 \, c d}{e x + d} - \frac {b e}{e x + d}\right )}^{\frac {3}{2}}\right )} e^{2} f}{2 \, c d e - b e^{2}} - \frac {{\left (3 \, c \sqrt {-c + \frac {2 \, c d}{e x + d} - \frac {b e}{e x + d}} + {\left (-c + \frac {2 \, c d}{e x + d} - \frac {b e}{e x + d}\right )}^{\frac {3}{2}}\right )} d e g}{2 \, c d e - b e^{2}} + 3 \, \sqrt {-c + \frac {2 \, c d}{e x + d} - \frac {b e}{e x + d}} g}{2 \, c d e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - b e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}\right )}}{3 \, {\left | e \right |}} \] Input:
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo rithm="giac")
Output:
2/3*((2*sqrt(-c)*c*e*f + 4*sqrt(-c)*c*d*g - 3*b*sqrt(-c)*e*g)*sgn(1/(e*x + d))*sgn(e)/(4*c^2*d^2*e - 4*b*c*d*e^2 + b^2*e^3) - ((3*c*sqrt(-c + 2*c*d/ (e*x + d) - b*e/(e*x + d)) + (-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)) *e^2*f/(2*c*d*e - b*e^2) - (3*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) + (-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2))*d*e*g/(2*c*d*e - b*e^2) + 3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*g)/(2*c*d*e*sgn(1/(e*x + d)) *sgn(e) - b*e^2*sgn(1/(e*x + d))*sgn(e)))/abs(e)
Time = 6.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.74 \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {2\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (2\,c\,d^2\,g-b\,e^2\,f-3\,b\,e^2\,g\,x+2\,c\,e^2\,f\,x-2\,b\,d\,e\,g+4\,c\,d\,e\,f+4\,c\,d\,e\,g\,x\right )}{3\,e^2\,{\left (b\,e-2\,c\,d\right )}^2\,{\left (d+e\,x\right )}^2} \] Input:
int((f + g*x)/((d + e*x)^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)),x)
Output:
-(2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(2*c*d^2*g - b*e^2*f - 3*b *e^2*g*x + 2*c*e^2*f*x - 2*b*d*e*g + 4*c*d*e*f + 4*c*d*e*g*x))/(3*e^2*(b*e - 2*c*d)^2*(d + e*x)^2)
Time = 0.24 (sec) , antiderivative size = 627, normalized size of antiderivative = 4.58 \[ \int \frac {f+g x}{(d+e x)^2 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {2 i \left (2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, b d e g +\sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, b \,e^{2} f +3 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, b \,e^{2} g x -2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c \,d^{2} g -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c d e f -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c d e g x -2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c \,e^{2} f x +\sqrt {c}\, b^{2} d^{2} e^{2} g +2 \sqrt {c}\, b^{2} d \,e^{3} g x +\sqrt {c}\, b^{2} e^{4} g \,x^{2}-2 \sqrt {c}\, b c \,d^{3} e g -2 \sqrt {c}\, b c \,d^{2} e^{2} f -4 \sqrt {c}\, b c \,d^{2} e^{2} g x -4 \sqrt {c}\, b c d \,e^{3} f x -2 \sqrt {c}\, b c d \,e^{3} g \,x^{2}-2 \sqrt {c}\, b c \,e^{4} f \,x^{2}+4 \sqrt {c}\, c^{2} d^{3} e f +8 \sqrt {c}\, c^{2} d^{2} e^{2} f x +4 \sqrt {c}\, c^{2} d \,e^{3} f \,x^{2}\right )}{3 e^{2} \left (b^{3} e^{5} x^{2}-6 b^{2} c d \,e^{4} x^{2}+12 b \,c^{2} d^{2} e^{3} x^{2}-8 c^{3} d^{3} e^{2} x^{2}+2 b^{3} d \,e^{4} x -12 b^{2} c \,d^{2} e^{3} x +24 b \,c^{2} d^{3} e^{2} x -16 c^{3} d^{4} e x +b^{3} d^{2} e^{3}-6 b^{2} c \,d^{3} e^{2}+12 b \,c^{2} d^{4} e -8 c^{3} d^{5}\right )} \] Input:
int((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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*d*e*g + 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*e**2*f + 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*e**2*g*x - 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*d* *2*g - 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)*c*d*e*f - 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)*c*d*e*g*x - 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*e**2*f*x + sqrt(c)* b**2*d**2*e**2*g + 2*sqrt(c)*b**2*d*e**3*g*x + sqrt(c)*b**2*e**4*g*x**2 - 2*sqrt(c)*b*c*d**3*e*g - 2*sqrt(c)*b*c*d**2*e**2*f - 4*sqrt(c)*b*c*d**2*e* *2*g*x - 4*sqrt(c)*b*c*d*e**3*f*x - 2*sqrt(c)*b*c*d*e**3*g*x**2 - 2*sqrt(c )*b*c*e**4*f*x**2 + 4*sqrt(c)*c**2*d**3*e*f + 8*sqrt(c)*c**2*d**2*e**2*f*x + 4*sqrt(c)*c**2*d*e**3*f*x**2))/(3*e**2*(b**3*d**2*e**3 + 2*b**3*d*e**4* x + b**3*e**5*x**2 - 6*b**2*c*d**3*e**2 - 12*b**2*c*d**2*e**3*x - 6*b**2*c *d*e**4*x**2 + 12*b*c**2*d**4*e + 24*b*c**2*d**3*e**2*x + 12*b*c**2*d**2*e **3*x**2 - 8*c**3*d**5 - 16*c**3*d**4*e*x - 8*c**3*d**3*e**2*x**2))