Integrand size = 44, antiderivative size = 138 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (2 c d-b e) (d+e x)^6}+\frac {2 (7 b e g-2 c (e f+6 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 e^2 (2 c d-b e)^2 (d+e x)^5} \] Output:
-2/7*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(-b*e+2*c*d)/(e *x+d)^6+2/35*(7*b*e*g-2*c*(6*d*g+e*f))*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5 /2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^5
Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-b e (5 e f+2 d g+7 e g x)+2 c \left (d^2 g+e^2 f x+6 d e (f+g x)\right )\right )}{35 e^2 (-2 c d+b e)^2 (d+e x)^4} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x )^6,x]
Output:
(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e *(5*e*f + 2*d*g + 7*e*g*x)) + 2*c*(d^2*g + e^2*f*x + 6*d*e*(f + g*x))))/(3 5*e^2*(-2*c*d + b*e)^2*(d + e*x)^4)
Time = 0.60 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99, 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.
\(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle \frac {(-7 b e g+12 c d g+2 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^5}dx}{7 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (d+e x)^6 (2 c d-b e)}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-7 b e g+12 c d g+2 c e f)}{35 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (d+e x)^6 (2 c d-b e)}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^6,x]
Output:
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*e^2*(2*c*d - b*e)*(d + e*x)^6) - (2*(2*c*e*f + 12*c*d*g - 7*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^5)
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 = 6.59 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (7 b \,e^{2} g x -12 c d e g x -2 c \,e^{2} f x +2 b d e g +5 b \,e^{2} f -2 c \,d^{2} g -12 c d e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{5} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) | \(128\) |
orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (7 b \,e^{2} g x -12 c d e g x -2 c \,e^{2} f x +2 b d e g +5 b \,e^{2} f -2 c \,d^{2} g -12 c d e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{5} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\) | \(128\) |
default | \(-\frac {2 g \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5 e^{6} \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{5}}\right )}{e^{7}}\) | \(213\) |
trager | \(\frac {2 \left (7 b \,c^{2} e^{4} g \,x^{3}-12 c^{3} d \,e^{3} g \,x^{3}-2 c^{3} e^{4} f \,x^{3}+14 b^{2} c \,e^{4} g \,x^{2}-36 b \,c^{2} d \,e^{3} g \,x^{2}+b \,c^{2} e^{4} f \,x^{2}+22 c^{3} d^{2} e^{2} g \,x^{2}-8 c^{3} d \,e^{3} f \,x^{2}+7 b^{3} e^{4} g x -22 b^{2} c d \,e^{3} g x +8 b^{2} c \,e^{4} f x +23 b \,c^{2} d^{2} e^{2} g x -30 b \,c^{2} d \,e^{3} f x -8 c^{3} d^{3} e g x +22 c^{3} d^{2} e^{2} f x +2 b^{3} d \,e^{3} g +5 b^{3} e^{4} f -6 b^{2} c \,d^{2} e^{2} g -22 b^{2} c d \,e^{3} f +6 b \,c^{2} d^{3} e g +29 b \,c^{2} d^{2} e^{2} f -2 c^{3} d^{4} g -12 c^{3} d^{3} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{35 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2} \left (e x +d \right )^{4}}\) | \(341\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6,x,method=_RET URNVERBOSE)
Output:
-2/35*(c*e*x+b*e-c*d)*(7*b*e^2*g*x-12*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+5*b* e^2*f-2*c*d^2*g-12*c*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d) ^5/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. 464 vs. \(2 (130) = 260\).
Time = 25.68 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.36 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c^{3} e^{4} f + {\left (12 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + {\left ({\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f - 2 \, {\left (11 \, c^{3} d^{2} e^{2} - 18 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} + {\left (12 \, c^{3} d^{3} e - 29 \, b c^{2} d^{2} e^{2} + 22 \, b^{2} c d e^{3} - 5 \, b^{3} e^{4}\right )} f + 2 \, {\left (c^{3} d^{4} - 3 \, b c^{2} d^{3} e + 3 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} g - {\left (2 \, {\left (11 \, c^{3} d^{2} e^{2} - 15 \, b c^{2} d e^{3} + 4 \, b^{2} c e^{4}\right )} f - {\left (8 \, c^{3} d^{3} e - 23 \, b c^{2} d^{2} e^{2} + 22 \, b^{2} c d e^{3} - 7 \, b^{3} e^{4}\right )} g\right )} x\right )}}{35 \, {\left (4 \, c^{2} d^{6} e^{2} - 4 \, b c d^{5} e^{3} + b^{2} d^{4} e^{4} + {\left (4 \, c^{2} d^{2} e^{6} - 4 \, b c d e^{7} + b^{2} e^{8}\right )} x^{4} + 4 \, {\left (4 \, c^{2} d^{3} e^{5} - 4 \, b c d^{2} e^{6} + b^{2} d e^{7}\right )} x^{3} + 6 \, {\left (4 \, c^{2} d^{4} e^{4} - 4 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )} x^{2} + 4 \, {\left (4 \, c^{2} d^{5} e^{3} - 4 \, b c d^{4} e^{4} + b^{2} d^{3} e^{5}\right )} x\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6,x, algo rithm="fricas")
Output:
-2/35*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^3*e^4*f + (12*c^3*d *e^3 - 7*b*c^2*e^4)*g)*x^3 + ((8*c^3*d*e^3 - b*c^2*e^4)*f - 2*(11*c^3*d^2* e^2 - 18*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 + (12*c^3*d^3*e - 29*b*c^2*d^2* e^2 + 22*b^2*c*d*e^3 - 5*b^3*e^4)*f + 2*(c^3*d^4 - 3*b*c^2*d^3*e + 3*b^2*c *d^2*e^2 - b^3*d*e^3)*g - (2*(11*c^3*d^2*e^2 - 15*b*c^2*d*e^3 + 4*b^2*c*e^ 4)*f - (8*c^3*d^3*e - 23*b*c^2*d^2*e^2 + 22*b^2*c*d*e^3 - 7*b^3*e^4)*g)*x) /(4*c^2*d^6*e^2 - 4*b*c*d^5*e^3 + b^2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e ^7 + b^2*e^8)*x^4 + 4*(4*c^2*d^3*e^5 - 4*b*c*d^2*e^6 + b^2*d*e^7)*x^3 + 6* (4*c^2*d^4*e^4 - 4*b*c*d^3*e^5 + b^2*d^2*e^6)*x^2 + 4*(4*c^2*d^5*e^3 - 4*b *c*d^4*e^4 + b^2*d^3*e^5)*x)
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{6}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**6,x )
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**6, x )
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6,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. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6,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,[0,4,0,0]%%%},[8,1]%%%}+%%%{%%{[%%%{-8,[0,3,1,0]%%%} ,0]:[1,0,
Time = 11.12 (sec) , antiderivative size = 3763, normalized size of antiderivative = 27.27 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Too large to display} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^6,x)
Output:
(((d*((d*((16*c^4*(6*b*e*g - 10*c*d*g + c*e*f))/(105*(b*e - 2*c*d)^4) - (1 6*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (608*c^5*d^2*g + 196*b^2*c^3*e^2*g - 160*c^5*d*e*f + 96*b*c^4*e^2*f - 688*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^4 )))/e + (4*b*c^2*(19*b^2*e^2*g + 76*c^2*d^2*g + 11*b*c*e^2*f - 20*c^2*d*e* f - 76*b*c*d*e*g))/(105*e*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b *e^2*x)^(1/2))/(d + e*x) - (((d*((d*((8*c^4*(7*b*e*g - 10*c*d*g + 2*c*e*f) )/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (208*c^ 5*d^2*g + 76*b^2*c^3*e^2*g - 80*c^5*d*e*f + 56*b*c^4*e^2*f - 248*b*c^4*d*e *g)/(105*e*(b*e - 2*c*d)^4)))/e + (2*b*c^2*(13*b^2*e^2*g + 52*c^2*d^2*g + 12*b*c*e^2*f - 20*c^2*d*e*f - 52*b*c*d*e*g))/(105*e*(b*e - 2*c*d)^4))*(c*d ^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((d*((d*((16*c^4*(7* b*e*g - 12*c*d*g + c*e*f))/(105*(b*e - 2*c*d)^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (768*c^5*d^2*g + 244*b^2*c^3*e^2*g - 192*c^5*d*e*f + 112 *b*c^4*e^2*f - 864*b*c^4*d*e*g)/(105*e*(b*e - 2*c*d)^4)))/e + (4*b*c^2*(24 *b^2*e^2*g + 96*c^2*d^2*g + 13*b*c*e^2*f - 24*c^2*d*e*f - 96*b*c*d*e*g))/( 105*e*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((8*c^4*(19*b*e*g - 34*c*d*g + 2*c*e*f))/(105*(b*e - 2*c*d )^4) - (16*c^5*d*g)/(105*(b*e - 2*c*d)^4)))/e - (1728*c^5*d^2*g + 504*b^2* c^3*e^2*g - 272*c^5*d*e*f + 152*b*c^4*e^2*f - 1864*b*c^4*d*e*g)/(105*e*(b* e - 2*c*d)^4)))/e + (8*b*c^2*(27*b^2*e^2*g + 108*c^2*d^2*g + 9*b*c*e^2*...
Time = 0.75 (sec) , antiderivative size = 1837, normalized size of antiderivative = 13.31 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^6,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**3*d*e**3*g + 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**3*e**4*f + 7*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**3*e**4*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)*b**2*c*d**2*e**2*g - 22*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*c*d*e**3*f - 22*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*c* d*e**3*g*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)*b**2*c*e**4*f*x + 14*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*c*e**4*g*x**2 + 6*sqr t(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**2*d**3*e*g + 29*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**2*d**2*e**2*f + 23*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**2*d**2*e* *2*g*x - 30*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**2*d*e**3*f*x - 36*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*s qrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b*c**2*d*e**3*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 )*b*c**2*e**4*f*x**2 + 7*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + ...