Integrand size = 44, antiderivative size = 283 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {16 c (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (e f-d g)}{7 e^2 (2 c d-b e) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (10 c e f+4 c d g-7 b e g)}{35 e^2 (2 c d-b e)^2 (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {128 c^2 (10 c e f+4 c d g-7 b e g) (b+2 c x)}{105 e (2 c d-b e)^6 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \] Output:
16/105*c*(-7*b*e*g+4*c*d*g+10*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^4/(d*(-b*e+c *d)-b*e^2*x-c*e^2*x^2)^(3/2)-2/7*(-d*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^2/(d* (-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)-2/35*(-7*b*e*g+4*c*d*g+10*c*e*f)/e^2/( -b*e+2*c*d)^2/(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)+128/105*c^2*( -7*b*e*g+4*c*d*g+10*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^6/(d*(-b*e+c*d)-b*e^2* x-c*e^2*x^2)^(1/2)
Time = 0.64 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.65 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {-6 b^5 e^5 (5 e f+2 d g+7 e g x)+96 b^2 c^3 e^2 \left (17 d^4 g+d^2 e^2 x (65 f-54 g x)+2 e^4 x^3 (5 f-14 g x)+40 d e^3 x^2 (f-2 g x)+20 d^3 e (3 f+2 g x)\right )-64 c^5 \left (9 d^6 g-40 e^6 f x^5+4 d^2 e^4 x^3 (5 f-8 g x)-6 d^5 e (5 f-3 g x)-16 d e^5 x^4 (5 f+g x)+8 d^3 e^3 x^2 (15 f+g x)+3 d^4 e^2 x (15 f+16 g x)\right )+32 b c^4 e \left (6 d^5 g+8 d e^4 x^3 (45 f-8 g x)+8 e^5 x^4 (15 f-7 g x)-39 d^4 e (5 f-g x)+12 d^3 e^2 x (-5 f+24 g x)+4 d^2 e^3 x^2 (75 f+43 g x)\right )+4 b^4 c e^4 \left (43 d^2 g+e^2 x (15 f+28 g x)+2 d e (45 f+73 g x)\right )-16 b^3 c^2 e^3 \left (88 d^3 g+2 e^3 x^2 (5 f+21 g x)+2 d e^2 x (25 f+86 g x)+d^2 e (115 f+293 g x)\right )}{105 e^2 (-2 c d+b e)^6 (d+e x)^3 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[(f + g*x)/((d + e*x)^2*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/ 2)),x]
Output:
(-6*b^5*e^5*(5*e*f + 2*d*g + 7*e*g*x) + 96*b^2*c^3*e^2*(17*d^4*g + d^2*e^2 *x*(65*f - 54*g*x) + 2*e^4*x^3*(5*f - 14*g*x) + 40*d*e^3*x^2*(f - 2*g*x) + 20*d^3*e*(3*f + 2*g*x)) - 64*c^5*(9*d^6*g - 40*e^6*f*x^5 + 4*d^2*e^4*x^3* (5*f - 8*g*x) - 6*d^5*e*(5*f - 3*g*x) - 16*d*e^5*x^4*(5*f + g*x) + 8*d^3*e ^3*x^2*(15*f + g*x) + 3*d^4*e^2*x*(15*f + 16*g*x)) + 32*b*c^4*e*(6*d^5*g + 8*d*e^4*x^3*(45*f - 8*g*x) + 8*e^5*x^4*(15*f - 7*g*x) - 39*d^4*e*(5*f - g *x) + 12*d^3*e^2*x*(-5*f + 24*g*x) + 4*d^2*e^3*x^2*(75*f + 43*g*x)) + 4*b^ 4*c*e^4*(43*d^2*g + e^2*x*(15*f + 28*g*x) + 2*d*e*(45*f + 73*g*x)) - 16*b^ 3*c^2*e^3*(88*d^3*g + 2*e^3*x^2*(5*f + 21*g*x) + 2*d*e^2*x*(25*f + 86*g*x) + d^2*e*(115*f + 293*g*x)))/(105*e^2*(-2*c*d + b*e)^6*(d + e*x)^3*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.77 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1220, 1129, 1089, 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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^2 \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-7 b e g+4 c d g+10 c e f) \int \frac {1}{(d+e x) \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-7 b e g+4 c d g+10 c e f) \left (\frac {8 c \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{5 (2 c d-b e)}-\frac {2}{5 e (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1089 |
| \(\displaystyle \frac {(-7 b e g+4 c d g+10 c e f) \left (\frac {8 c \left (\frac {8 c \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 (2 c d-b e)^2}+\frac {2 (b+2 c x)}{3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{5 (2 c d-b e)}-\frac {2}{5 e (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {\left (\frac {8 c \left (\frac {16 c (b+2 c x)}{3 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (b+2 c x)}{3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{5 (2 c d-b e)}-\frac {2}{5 e (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right ) (-7 b e g+4 c d g+10 c e f)}{7 e (2 c d-b e)}-\frac {2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
Input:
Int[(f + g*x)/((d + e*x)^2*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
Output:
(-2*(e*f - d*g))/(7*e^2*(2*c*d - b*e)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + ((10*c*e*f + 4*c*d*g - 7*b*e*g)*(-2/(5*e*(2*c*d - b *e)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (8*c*((2*(b + 2*c*x))/(3*(2*c*d - b*e)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(b + 2*c*x))/(3*(2*c*d - b*e)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^ 2*x^2])))/(5*(2*c*d - b*e))))/(7*e*(2*c*d - b*e))
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
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 ]
Leaf count of result is larger than twice the leaf count of optimal. \(597\) vs. \(2(267)=534\).
Time = 2.50 (sec) , antiderivative size = 598, normalized size of antiderivative = 2.11
| method | result | size |
| default | \(\frac {g \left (-\frac {2}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right ) \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}}}+\frac {8 c \,e^{2} \left (-\frac {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 )^{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}}}-\frac {16 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 )^{4} \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 )}{5 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{2}}-\frac {\left (d g -e f \right ) \left (-\frac {2}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{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}}}+\frac {10 c \,e^{2} \left (-\frac {2}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right ) \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}}}+\frac {8 c \,e^{2} \left (-\frac {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 )^{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}}}-\frac {16 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 )^{4} \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 )}{5 \left (-b \,e^{2}+2 d e c \right )}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{3}}\) | \(598\) |
| trager | \(\frac {2 \left (896 b \,c^{4} e^{6} g \,x^{5}-512 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}+1344 b^{2} c^{3} e^{6} g \,x^{4}+1024 b \,c^{4} d \,e^{5} g \,x^{4}-1920 b \,c^{4} e^{6} f \,x^{4}-1024 c^{5} d^{2} e^{4} g \,x^{4}-2560 c^{5} d \,e^{5} f \,x^{4}+336 b^{3} c^{2} e^{6} g \,x^{3}+3840 b^{2} c^{3} d \,e^{5} g \,x^{3}-480 b^{2} c^{3} e^{6} f \,x^{3}-2752 b \,c^{4} d^{2} e^{4} g \,x^{3}-5760 b \,c^{4} d \,e^{5} f \,x^{3}+256 c^{5} d^{3} e^{3} g \,x^{3}+640 c^{5} d^{2} e^{4} f \,x^{3}-56 b^{4} c \,e^{6} g \,x^{2}+1376 b^{3} c^{2} d \,e^{5} g \,x^{2}+80 b^{3} c^{2} e^{6} f \,x^{2}+2592 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-1920 b^{2} c^{3} d \,e^{5} f \,x^{2}-4608 b \,c^{4} d^{3} e^{3} g \,x^{2}-4800 b \,c^{4} d^{2} e^{4} f \,x^{2}+1536 c^{5} d^{4} e^{2} g \,x^{2}+3840 c^{5} d^{3} e^{3} f \,x^{2}+21 b^{5} e^{6} g x -292 b^{4} c d \,e^{5} g x -30 b^{4} c \,e^{6} f x +2344 b^{3} c^{2} d^{2} e^{4} g x +400 b^{3} c^{2} d \,e^{5} f x -1920 b^{2} c^{3} d^{3} e^{3} g x -3120 b^{2} c^{3} d^{2} e^{4} f x -624 b \,c^{4} d^{4} e^{2} g x +960 b \,c^{4} d^{3} e^{3} f x +576 c^{5} d^{5} e g x +1440 c^{5} d^{4} e^{2} f x +6 b^{5} d \,e^{5} g +15 b^{5} e^{6} f -86 b^{4} c \,d^{2} e^{4} g -180 b^{4} c d \,e^{5} f +704 b^{3} c^{2} d^{3} e^{3} g +920 b^{3} c^{2} d^{2} e^{4} f -816 b^{2} c^{3} d^{4} e^{2} g -2880 b^{2} c^{3} d^{3} e^{3} f -96 b \,c^{4} d^{5} e g +3120 b \,c^{4} d^{4} e^{2} f +288 c^{5} d^{6} g -960 c^{5} d^{5} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{105 \left (b^{5} e^{5}-10 b^{4} c d \,e^{4}+40 b^{3} d^{2} e^{3} c^{2}-80 b^{2} c^{3} d^{3} e^{2}+80 b \,c^{4} d^{4} e -32 d^{5} c^{5}\right ) \left (b e -2 c d \right ) e^{2} \left (c e x +b e -c d \right )^{2} \left (e x +d \right )^{4}}\) | \(780\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (896 b \,c^{4} e^{6} g \,x^{5}-512 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}+1344 b^{2} c^{3} e^{6} g \,x^{4}+1024 b \,c^{4} d \,e^{5} g \,x^{4}-1920 b \,c^{4} e^{6} f \,x^{4}-1024 c^{5} d^{2} e^{4} g \,x^{4}-2560 c^{5} d \,e^{5} f \,x^{4}+336 b^{3} c^{2} e^{6} g \,x^{3}+3840 b^{2} c^{3} d \,e^{5} g \,x^{3}-480 b^{2} c^{3} e^{6} f \,x^{3}-2752 b \,c^{4} d^{2} e^{4} g \,x^{3}-5760 b \,c^{4} d \,e^{5} f \,x^{3}+256 c^{5} d^{3} e^{3} g \,x^{3}+640 c^{5} d^{2} e^{4} f \,x^{3}-56 b^{4} c \,e^{6} g \,x^{2}+1376 b^{3} c^{2} d \,e^{5} g \,x^{2}+80 b^{3} c^{2} e^{6} f \,x^{2}+2592 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-1920 b^{2} c^{3} d \,e^{5} f \,x^{2}-4608 b \,c^{4} d^{3} e^{3} g \,x^{2}-4800 b \,c^{4} d^{2} e^{4} f \,x^{2}+1536 c^{5} d^{4} e^{2} g \,x^{2}+3840 c^{5} d^{3} e^{3} f \,x^{2}+21 b^{5} e^{6} g x -292 b^{4} c d \,e^{5} g x -30 b^{4} c \,e^{6} f x +2344 b^{3} c^{2} d^{2} e^{4} g x +400 b^{3} c^{2} d \,e^{5} f x -1920 b^{2} c^{3} d^{3} e^{3} g x -3120 b^{2} c^{3} d^{2} e^{4} f x -624 b \,c^{4} d^{4} e^{2} g x +960 b \,c^{4} d^{3} e^{3} f x +576 c^{5} d^{5} e g x +1440 c^{5} d^{4} e^{2} f x +6 b^{5} d \,e^{5} g +15 b^{5} e^{6} f -86 b^{4} c \,d^{2} e^{4} g -180 b^{4} c d \,e^{5} f +704 b^{3} c^{2} d^{3} e^{3} g +920 b^{3} c^{2} d^{2} e^{4} f -816 b^{2} c^{3} d^{4} e^{2} g -2880 b^{2} c^{3} d^{3} e^{3} f -96 b \,c^{4} d^{5} e g +3120 b \,c^{4} d^{4} e^{2} f +288 c^{5} d^{6} g -960 c^{5} d^{5} e f \right )}{105 \left (e x +d \right ) \left (b^{6} e^{6}-12 b^{5} c d \,e^{5}+60 b^{4} c^{2} d^{2} e^{4}-160 b^{3} c^{3} d^{3} e^{3}+240 b^{2} c^{4} d^{4} e^{2}-192 b \,c^{5} d^{5} e +64 c^{6} d^{6}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(782\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (896 b \,c^{4} e^{6} g \,x^{5}-512 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}+1344 b^{2} c^{3} e^{6} g \,x^{4}+1024 b \,c^{4} d \,e^{5} g \,x^{4}-1920 b \,c^{4} e^{6} f \,x^{4}-1024 c^{5} d^{2} e^{4} g \,x^{4}-2560 c^{5} d \,e^{5} f \,x^{4}+336 b^{3} c^{2} e^{6} g \,x^{3}+3840 b^{2} c^{3} d \,e^{5} g \,x^{3}-480 b^{2} c^{3} e^{6} f \,x^{3}-2752 b \,c^{4} d^{2} e^{4} g \,x^{3}-5760 b \,c^{4} d \,e^{5} f \,x^{3}+256 c^{5} d^{3} e^{3} g \,x^{3}+640 c^{5} d^{2} e^{4} f \,x^{3}-56 b^{4} c \,e^{6} g \,x^{2}+1376 b^{3} c^{2} d \,e^{5} g \,x^{2}+80 b^{3} c^{2} e^{6} f \,x^{2}+2592 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-1920 b^{2} c^{3} d \,e^{5} f \,x^{2}-4608 b \,c^{4} d^{3} e^{3} g \,x^{2}-4800 b \,c^{4} d^{2} e^{4} f \,x^{2}+1536 c^{5} d^{4} e^{2} g \,x^{2}+3840 c^{5} d^{3} e^{3} f \,x^{2}+21 b^{5} e^{6} g x -292 b^{4} c d \,e^{5} g x -30 b^{4} c \,e^{6} f x +2344 b^{3} c^{2} d^{2} e^{4} g x +400 b^{3} c^{2} d \,e^{5} f x -1920 b^{2} c^{3} d^{3} e^{3} g x -3120 b^{2} c^{3} d^{2} e^{4} f x -624 b \,c^{4} d^{4} e^{2} g x +960 b \,c^{4} d^{3} e^{3} f x +576 c^{5} d^{5} e g x +1440 c^{5} d^{4} e^{2} f x +6 b^{5} d \,e^{5} g +15 b^{5} e^{6} f -86 b^{4} c \,d^{2} e^{4} g -180 b^{4} c d \,e^{5} f +704 b^{3} c^{2} d^{3} e^{3} g +920 b^{3} c^{2} d^{2} e^{4} f -816 b^{2} c^{3} d^{4} e^{2} g -2880 b^{2} c^{3} d^{3} e^{3} f -96 b \,c^{4} d^{5} e g +3120 b \,c^{4} d^{4} e^{2} f +288 c^{5} d^{6} g -960 c^{5} d^{5} e f \right )}{105 \left (e x +d \right ) \left (b^{6} e^{6}-12 b^{5} c d \,e^{5}+60 b^{4} c^{2} d^{2} e^{4}-160 b^{3} c^{3} d^{3} e^{3}+240 b^{2} c^{4} d^{4} e^{2}-192 b \,c^{5} d^{5} e +64 c^{6} d^{6}\right ) e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\) | \(782\) |
Input:
int((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RET URNVERBOSE)
Output:
g/e^2*(-2/5/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x +d/e))^(3/2)+8/5*c*e^2/(-b*e^2+2*c*d*e)*(-2/3*(-2*c*e^2*(x+d/e)-b*e^2+2*d* e*c)/(-b*e^2+2*c*d*e)^2/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)- 16/3*c*e^2/(-b*e^2+2*c*d*e)^4*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c)/(-c*e^2*(x+ d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)))-(d*g-e*f)/e^3*(-2/7/(-b*e^2+2*c*d *e)/(x+d/e)^2/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+10/7*c*e^2 /(-b*e^2+2*c*d*e)*(-2/5/(-b*e^2+2*c*d*e)/(x+d/e)/(-c*e^2*(x+d/e)^2+(-b*e^2 +2*c*d*e)*(x+d/e))^(3/2)+8/5*c*e^2/(-b*e^2+2*c*d*e)*(-2/3*(-2*c*e^2*(x+d/e )-b*e^2+2*d*e*c)/(-b*e^2+2*c*d*e)^2/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+ d/e))^(3/2)-16/3*c*e^2/(-b*e^2+2*c*d*e)^4*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c) /(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2))))
Timed out. \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algo rithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)/(e*x+d)**2/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x )
Output:
Timed out
Exception generated. \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),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. 22347 vs. \(2 (267) = 534\).
Time = 0.79 (sec) , antiderivative size = 22347, normalized size of antiderivative = 78.96 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algo rithm="giac")
Output:
-2/105*(128*(10*c^4*e*f + 4*c^4*d*g - 7*b*c^3*e*g)*sgn(1/(e*x + d))*sgn(e) /(64*sqrt(-c)*c^6*d^6*e - 192*b*sqrt(-c)*c^5*d^5*e^2 + 240*b^2*sqrt(-c)*c^ 4*d^4*e^3 - 160*b^3*sqrt(-c)*c^3*d^3*e^4 + 60*b^4*sqrt(-c)*c^2*d^2*e^5 - 1 2*b^5*sqrt(-c)*c*d*e^6 + b^6*sqrt(-c)*e^7) - (1030792151040*(c - 2*c*d/(e* x + d) + b*e/(e*x + d))^3*c^36*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))* d^36*e^7*f*sgn(1/(e*x + d))^6*sgn(e)^6 - 7215545057280*(c - 2*c*d/(e*x + d ) + b*e/(e*x + d))^2*c^37*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^36* e^7*f*sgn(1/(e*x + d))^6*sgn(e)^6 - 72155450572800*c^39*sqrt(-c + 2*c*d/(e *x + d) - b*e/(e*x + d))*d^36*e^7*f*sgn(1/(e*x + d))^6*sgn(e)^6 - 24051816 857600*c^38*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^36*e^7*f*sgn(1/ (e*x + d))^6*sgn(e)^6 - 18554258718720*b*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^3*c^35*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^35*e^8*f*sgn(1/(e *x + d))^6*sgn(e)^6 + 129879811031040*b*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^36*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^35*e^8*f*sgn(1/(e* x + d))^6*sgn(e)^6 + 1298798110310400*b*c^38*sqrt(-c + 2*c*d/(e*x + d) - b *e/(e*x + d))*d^35*e^8*f*sgn(1/(e*x + d))^6*sgn(e)^6 + 432932703436800*b*c ^37*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^35*e^8*f*sgn(1/(e*x + d ))^6*sgn(e)^6 + 162349763788800*b^2*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^ 3*c^34*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^34*e^9*f*sgn(1/(e*x + d))^6*sgn(e)^6 - 1136448346521600*b^2*(c - 2*c*d/(e*x + d) + b*e/(e*x +...
Time = 18.94 (sec) , antiderivative size = 11539, normalized size of antiderivative = 40.77 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
int((f + g*x)/((d + e*x)^2*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2)),x)
Output:
((800*c^6*d^4*g + 558*b^3*c^3*e^4*f - 222*b^4*c^2*e^4*g - 4192*c^6*d^3*e*f + 384*b*c^5*d^3*e*g + 6624*b*c^5*d^2*e^2*f - 3392*b^2*c^4*d*e^3*f + 1248* b^3*c^3*d*e^3*g - 1984*b^2*c^4*d^2*e^2*g)/(105*e^2*(b*e - 2*c*d)^8) - x*(( b*((b*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16 *b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8)))/c - (4*c^4*(40*c^2*d^2*g - 33*b^2*e^ 2*g + 62*b*c*e^2*f - 88*c^2*d*e*f + 16*b*c*d*e*g))/(105*(b*e - 2*c*d)^8) + (16*c^5*g*(c*d^2 - b*d*e))/(105*(b*e - 2*c*d)^8)))/c + (44*b^2*c^4*e^4*f - 30*b^3*c^3*e^4*g - 672*c^6*d^2*e^2*f + 224*c^6*d^3*e*g + 320*b*c^5*d*e^3 *f + 160*b*c^5*d^2*e^2*g - 128*b^2*c^4*d*e^3*g)/(105*e^2*(b*e - 2*c*d)^8) + ((c*d^2 - b*d*e)*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2* c*d)^8) + (16*b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8)))/(c*e^2)) + ((c*d^2 - b* d*e)*((b*((8*c^5*e*(4*c*d*g - 9*b*e*g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*b*c^5*e^2*g)/(105*(b*e - 2*c*d)^8)))/c - (4*c^4*(40*c^2*d^2*g - 33*b^2 *e^2*g + 62*b*c*e^2*f - 88*c^2*d*e*f + 16*b*c*d*e*g))/(105*(b*e - 2*c*d)^8 ) + (16*c^5*g*(c*d^2 - b*d*e))/(105*(b*e - 2*c*d)^8)))/(c*e^2))/(c*d^2 - c *e^2*x^2 - b*d*e - b*e^2*x)^(1/2) + (((8*c^2*g*(3*b*e - 4*c*d))/(105*e^2*( b*e - 2*c*d)^6) - (16*c^3*d*g)/(105*e^2*(b*e - 2*c*d)^6))*(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*(2*c*d*g - 7*b*e *g + 6*c*e*f))/(105*(b*e - 2*c*d)^8) + (16*c^5*d*g)/(105*(b*e - 2*c*d)^8)) )/e + (76*b^2*c^3*e^2*g - 176*c^5*d^2*g + 304*c^5*d*e*f - 200*b*c^4*e^2...
Time = 1.57 (sec) , antiderivative size = 3857, normalized size of antiderivative = 13.63 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((g*x+f)/(e*x+d)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
Output:
(2*i*(896*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c**2*d**4*e**3*g + 3584* sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c**2*d**3*e**4*g*x + 5376*sqrt(c)* sqrt( - b*e + c*d - c*e*x)*b**3*c**2*d**2*e**5*g*x**2 + 3584*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c**2*d*e**6*g*x**3 + 896*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**3*c**2*e**7*g*x**4 - 3200*sqrt(c)*sqrt( - b*e + c*d - c*e* x)*b**2*c**3*d**5*e**2*g - 1280*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c* *3*d**4*e**3*f - 11904*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d**4*e **3*g*x - 5120*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d**3*e**4*f*x - 15616*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d**3*e**4*g*x**2 - 76 80*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d**2*e**5*f*x**2 - 7424*sq rt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d**2*e**5*g*x**3 - 5120*sqrt(c) *sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d*e**6*f*x**3 + 384*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b**2*c**3*d*e**6*g*x**4 - 1280*sqrt(c)*sqrt( - b*e + c* d - c*e*x)*b**2*c**3*e**7*f*x**4 + 896*sqrt(c)*sqrt( - b*e + c*d - c*e*x)* b**2*c**3*e**7*g*x**5 + 3328*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**4*d** 6*e*g + 3840*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**4*d**5*e**2*f + 11008 *sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**4*d**5*e**2*g*x + 14080*sqrt(c)*s qrt( - b*e + c*d - c*e*x)*b*c**4*d**4*e**3*f*x + 10752*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**4*d**4*e**3*g*x**2 + 17920*sqrt(c)*sqrt( - b*e + c*d - c*e*x)*b*c**4*d**3*e**4*f*x**2 - 512*sqrt(c)*sqrt( - b*e + c*d - c*e*...