3.2229 \(\int \frac{f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=283 \[ \frac{128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(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*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (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)) - (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)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2)) + (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])

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Rubi [A]  time = 0.83018, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{128 c^2 (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 c (b+2 c x) (-7 b e g+4 c d g+10 c e f)}{105 e (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-7 b e g+4 c d g+10 c e f)}{35 e^2 (d+e x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{7 e^2 (d+e x)^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  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]

[Out]

(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*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (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)) - (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)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2)) + (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])

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Rubi in Sympy [A]  time = 84.7035, size = 270, normalized size = 0.95 \[ - \frac{64 c^{2} \left (2 b + 4 c x\right ) \left (7 b e g - 4 c d g - 10 c e f\right )}{105 e \left (b e - 2 c d\right )^{6} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{16 c \left (b + 2 c x\right ) \left (7 b e g - 4 c d g - 10 c e f\right )}{105 e \left (b e - 2 c d\right )^{4} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} + \frac{2 \left (7 b e g - 4 c d g - 10 c e f\right )}{35 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )^{2} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 \left (d g - e f\right )}{7 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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)

[Out]

-64*c**2*(2*b + 4*c*x)*(7*b*e*g - 4*c*d*g - 10*c*e*f)/(105*e*(b*e - 2*c*d)**6*sq
rt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - 16*c*(b + 2*c*x)*(7*b*e*g - 4*c*
d*g - 10*c*e*f)/(105*e*(b*e - 2*c*d)**4*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d
))**(3/2)) + 2*(7*b*e*g - 4*c*d*g - 10*c*e*f)/(35*e**2*(d + e*x)*(b*e - 2*c*d)**
2*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)) - 2*(d*g - e*f)/(7*e**2*(d
+ e*x)**2*(b*e - 2*c*d)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2))

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Mathematica [A]  time = 1.26812, size = 262, normalized size = 0.93 \[ \frac{2 (d+e x)^3 (c (d-e x)-b e)^3 \left (-\frac{35 c^3 (-11 b e g+8 c d g+14 c e f)}{b e-c d+c e x}-\frac{35 c^3 (b e-2 c d) (-b e g+c d g+c e f)}{(b e-c d+c e x)^2}-\frac{c^2 (-511 b e g+232 c d g+790 c e f)}{d+e x}+\frac{c (b e-2 c d) (-98 b e g+11 c d g+185 c e f)}{(d+e x)^2}+\frac{3 (b e-2 c d)^2 (7 b e g+6 c d g-20 c e f)}{(d+e x)^3}+\frac{15 (b e-2 c d)^3 (e f-d g)}{(d+e x)^4}\right )}{105 e^2 (b e-2 c d)^6 ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

[In]  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]

[Out]

(2*(d + e*x)^3*(-(b*e) + c*(d - e*x))^3*((15*(-2*c*d + b*e)^3*(e*f - d*g))/(d +
e*x)^4 + (3*(-2*c*d + b*e)^2*(-20*c*e*f + 6*c*d*g + 7*b*e*g))/(d + e*x)^3 + (c*(
-2*c*d + b*e)*(185*c*e*f + 11*c*d*g - 98*b*e*g))/(d + e*x)^2 - (c^2*(790*c*e*f +
232*c*d*g - 511*b*e*g))/(d + e*x) - (35*c^3*(-2*c*d + b*e)*(c*e*f + c*d*g - b*e
*g))/(-(c*d) + b*e + c*e*x)^2 - (35*c^3*(14*c*e*f + 8*c*d*g - 11*b*e*g))/(-(c*d)
+ b*e + c*e*x)))/(105*e^2*(-2*c*d + b*e)^6*((d + e*x)*(-(b*e) + c*(d - e*x)))^(
5/2))

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Maple [B]  time = 0.02, size = 782, normalized size = 2.8 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 896\,b{c}^{4}{e}^{6}g{x}^{5}-512\,{c}^{5}d{e}^{5}g{x}^{5}-1280\,{c}^{5}{e}^{6}f{x}^{5}+1344\,{b}^{2}{c}^{3}{e}^{6}g{x}^{4}+1024\,b{c}^{4}d{e}^{5}g{x}^{4}-1920\,b{c}^{4}{e}^{6}f{x}^{4}-1024\,{c}^{5}{d}^{2}{e}^{4}g{x}^{4}-2560\,{c}^{5}d{e}^{5}f{x}^{4}+336\,{b}^{3}{c}^{2}{e}^{6}g{x}^{3}+3840\,{b}^{2}{c}^{3}d{e}^{5}g{x}^{3}-480\,{b}^{2}{c}^{3}{e}^{6}f{x}^{3}-2752\,b{c}^{4}{d}^{2}{e}^{4}g{x}^{3}-5760\,b{c}^{4}d{e}^{5}f{x}^{3}+256\,{c}^{5}{d}^{3}{e}^{3}g{x}^{3}+640\,{c}^{5}{d}^{2}{e}^{4}f{x}^{3}-56\,{b}^{4}c{e}^{6}g{x}^{2}+1376\,{b}^{3}{c}^{2}d{e}^{5}g{x}^{2}+80\,{b}^{3}{c}^{2}{e}^{6}f{x}^{2}+2592\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}g{x}^{2}-1920\,{b}^{2}{c}^{3}d{e}^{5}f{x}^{2}-4608\,b{c}^{4}{d}^{3}{e}^{3}g{x}^{2}-4800\,b{c}^{4}{d}^{2}{e}^{4}f{x}^{2}+1536\,{c}^{5}{d}^{4}{e}^{2}g{x}^{2}+3840\,{c}^{5}{d}^{3}{e}^{3}f{x}^{2}+21\,{b}^{5}{e}^{6}gx-292\,{b}^{4}cd{e}^{5}gx-30\,{b}^{4}c{e}^{6}fx+2344\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}gx+400\,{b}^{3}{c}^{2}d{e}^{5}fx-1920\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}gx-3120\,{b}^{2}{c}^{3}{d}^{2}{e}^{4}fx-624\,b{c}^{4}{d}^{4}{e}^{2}gx+960\,b{c}^{4}{d}^{3}{e}^{3}fx+576\,{c}^{5}{d}^{5}egx+1440\,{c}^{5}{d}^{4}{e}^{2}fx+6\,{b}^{5}d{e}^{5}g+15\,{b}^{5}{e}^{6}f-86\,{b}^{4}c{d}^{2}{e}^{4}g-180\,{b}^{4}cd{e}^{5}f+704\,{b}^{3}{c}^{2}{d}^{3}{e}^{3}g+920\,{b}^{3}{c}^{2}{d}^{2}{e}^{4}f-816\,{b}^{2}{c}^{3}{d}^{4}{e}^{2}g-2880\,{b}^{2}{c}^{3}{d}^{3}{e}^{3}f-96\,b{c}^{4}{d}^{5}eg+3120\,b{c}^{4}{d}^{4}{e}^{2}f+288\,{c}^{5}{d}^{6}g-960\,{c}^{5}{d}^{5}ef \right ) }{ \left ( 105\,ex+105\,d \right ) \left ({b}^{6}{e}^{6}-12\,{b}^{5}cd{e}^{5}+60\,{b}^{4}{c}^{2}{d}^{2}{e}^{4}-160\,{b}^{3}{c}^{3}{d}^{3}{e}^{3}+240\,{b}^{2}{c}^{4}{d}^{4}{e}^{2}-192\,b{c}^{5}{d}^{5}e+64\,{c}^{6}{d}^{6} \right ){e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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)

[Out]

-2/105*(c*e*x+b*e-c*d)*(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+8
0*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)/(e*x+d)/(b^6*e^6-1
2*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)/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 30.534, size = 1904, normalized size = 6.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

-2/105*(128*(10*c^5*e^6*f + (4*c^5*d*e^5 - 7*b*c^4*e^6)*g)*x^5 + 64*(10*(4*c^5*d
*e^5 + 3*b*c^4*e^6)*f + (16*c^5*d^2*e^4 - 16*b*c^4*d*e^5 - 21*b^2*c^3*e^6)*g)*x^
4 - 16*(10*(4*c^5*d^2*e^4 - 36*b*c^4*d*e^5 - 3*b^2*c^3*e^6)*f + (16*c^5*d^3*e^3
- 172*b*c^4*d^2*e^4 + 240*b^2*c^3*d*e^5 + 21*b^3*c^2*e^6)*g)*x^3 - 8*(10*(48*c^5
*d^3*e^3 - 60*b*c^4*d^2*e^4 - 24*b^2*c^3*d*e^5 + b^3*c^2*e^6)*f + (192*c^5*d^4*e
^2 - 576*b*c^4*d^3*e^3 + 324*b^2*c^3*d^2*e^4 + 172*b^3*c^2*d*e^5 - 7*b^4*c*e^6)*
g)*x^2 + 5*(192*c^5*d^5*e - 624*b*c^4*d^4*e^2 + 576*b^2*c^3*d^3*e^3 - 184*b^3*c^
2*d^2*e^4 + 36*b^4*c*d*e^5 - 3*b^5*e^6)*f - 2*(144*c^5*d^6 - 48*b*c^4*d^5*e - 40
8*b^2*c^3*d^4*e^2 + 352*b^3*c^2*d^3*e^3 - 43*b^4*c*d^2*e^4 + 3*b^5*d*e^5)*g - (1
0*(144*c^5*d^4*e^2 + 96*b*c^4*d^3*e^3 - 312*b^2*c^3*d^2*e^4 + 40*b^3*c^2*d*e^5 -
3*b^4*c*e^6)*f + (576*c^5*d^5*e - 624*b*c^4*d^4*e^2 - 1920*b^2*c^3*d^3*e^3 + 23
44*b^3*c^2*d^2*e^4 - 292*b^4*c*d*e^5 + 21*b^5*e^6)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2
*x + c*d^2 - b*d*e)/(64*c^8*d^12*e^2 - 320*b*c^7*d^11*e^3 + 688*b^2*c^6*d^10*e^4
- 832*b^3*c^5*d^9*e^5 + 620*b^4*c^4*d^8*e^6 - 292*b^5*c^3*d^7*e^7 + 85*b^6*c^2*
d^6*e^8 - 14*b^7*c*d^5*e^9 + b^8*d^4*e^10 + (64*c^8*d^6*e^8 - 192*b*c^7*d^5*e^9
+ 240*b^2*c^6*d^4*e^10 - 160*b^3*c^5*d^3*e^11 + 60*b^4*c^4*d^2*e^12 - 12*b^5*c^3
*d*e^13 + b^6*c^2*e^14)*x^6 + 2*(64*c^8*d^7*e^7 - 128*b*c^7*d^6*e^8 + 48*b^2*c^6
*d^5*e^9 + 80*b^3*c^5*d^4*e^10 - 100*b^4*c^4*d^3*e^11 + 48*b^5*c^3*d^2*e^12 - 11
*b^6*c^2*d*e^13 + b^7*c*e^14)*x^5 - (64*c^8*d^8*e^6 - 576*b*c^7*d^7*e^7 + 1328*b
^2*c^6*d^6*e^8 - 1408*b^3*c^5*d^5*e^9 + 780*b^4*c^4*d^4*e^10 - 212*b^5*c^3*d^3*e
^11 + 13*b^6*c^2*d^2*e^12 + 6*b^7*c*d*e^13 - b^8*e^14)*x^4 - 4*(64*c^8*d^9*e^5 -
256*b*c^7*d^8*e^6 + 368*b^2*c^6*d^7*e^7 - 208*b^3*c^5*d^6*e^8 - 20*b^4*c^4*d^5*
e^9 + 88*b^5*c^3*d^4*e^10 - 47*b^6*c^2*d^3*e^11 + 11*b^7*c*d^2*e^12 - b^8*d*e^13
)*x^3 - (64*c^8*d^10*e^4 + 64*b*c^7*d^9*e^5 - 912*b^2*c^6*d^8*e^6 + 1952*b^3*c^5
*d^7*e^7 - 2020*b^4*c^4*d^6*e^8 + 1188*b^5*c^3*d^5*e^9 - 407*b^6*c^2*d^4*e^10 +
76*b^7*c*d^3*e^11 - 6*b^8*d^2*e^12)*x^2 + 2*(64*c^8*d^11*e^3 - 384*b*c^7*d^10*e^
4 + 944*b^2*c^6*d^9*e^5 - 1264*b^3*c^5*d^8*e^6 + 1020*b^4*c^4*d^7*e^7 - 512*b^5*
c^3*d^6*e^8 + 157*b^6*c^2*d^5*e^9 - 27*b^7*c*d^4*e^10 + 2*b^8*d^3*e^11)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

Timed out