3.2230 \(\int \frac{f+g x}{(d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=358 \[ \frac{256 c^3 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 c^2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x)^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{9 e^2 (d+e x)^3 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
[Out]
(32*c^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^5*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(e*f - d*g))/(9*e^2*(2*c*d - b*e)*(d
+ e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(4*c*e*f + 2*c*d*g -
3*b*e*g))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(3/2)) - (4*c*(4*c*e*f + 2*c*d*g - 3*b*e*g))/(21*e^2*(2*c*d - b*e)^3*(d + e*
x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (256*c^3*(4*c*e*f + 2*c*d*g -
3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^7*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2])
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Rubi [A] time = 1.07857, antiderivative size = 358, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091
\[ \frac{256 c^3 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{32 c^2 (b+2 c x) (-3 b e g+2 c d g+4 c e f)}{63 e (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 c (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (-3 b e g+2 c d g+4 c e f)}{21 e^2 (d+e x)^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{2 (e f-d g)}{9 e^2 (d+e x)^3 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)/((d + e*x)^3*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
(32*c^2*(4*c*e*f + 2*c*d*g - 3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^5*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(e*f - d*g))/(9*e^2*(2*c*d - b*e)*(d
+ e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(4*c*e*f + 2*c*d*g -
3*b*e*g))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(3/2)) - (4*c*(4*c*e*f + 2*c*d*g - 3*b*e*g))/(21*e^2*(2*c*d - b*e)^3*(d + e*
x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (256*c^3*(4*c*e*f + 2*c*d*g -
3*b*e*g)*(b + 2*c*x))/(63*e*(2*c*d - b*e)^7*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2])
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Rubi in Sympy [A] time = 116.773, size = 345, normalized size = 0.96 \[ \frac{128 c^{3} \left (2 b + 4 c x\right ) \left (3 b e g - 2 c d g - 4 c e f\right )}{63 e \left (b e - 2 c d\right )^{7} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{32 c^{2} \left (b + 2 c x\right ) \left (3 b e g - 2 c d g - 4 c e f\right )}{63 e \left (b e - 2 c d\right )^{5} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{4 c \left (3 b e g - 2 c d g - 4 c e f\right )}{21 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )^{3} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} + \frac{2 \left (3 b e g - 2 c d g - 4 c e f\right )}{21 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )^{2} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 \left (d g - e f\right )}{9 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)/(e*x+d)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
128*c**3*(2*b + 4*c*x)*(3*b*e*g - 2*c*d*g - 4*c*e*f)/(63*e*(b*e - 2*c*d)**7*sqrt
(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) + 32*c**2*(b + 2*c*x)*(3*b*e*g - 2*c
*d*g - 4*c*e*f)/(63*e*(b*e - 2*c*d)**5*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)
)**(3/2)) - 4*c*(3*b*e*g - 2*c*d*g - 4*c*e*f)/(21*e**2*(d + e*x)*(b*e - 2*c*d)**
3*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)) + 2*(3*b*e*g - 2*c*d*g - 4*
c*e*f)/(21*e**2*(d + e*x)**2*(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)/(9*e**2*(d + e*x)**3*(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 = 1.56442, size = 300, normalized size = 0.84 \[ \frac{2 (d+e x)^3 (c (d-e x)-b e)^3 \left (\frac{21 c^4 (-14 b e g+11 c d g+17 c e f)}{b e-c d+c e x}+\frac{21 c^4 (b e-2 c d) (-b e g+c d g+c e f)}{(b e-c d+c e x)^2}+\frac{c^3 (-474 b e g+281 c d g+667 c e f)}{d+e x}+\frac{c^2 (2 c d-b e) (-111 b e g+46 c d g+176 c e f)}{(d+e x)^2}+\frac{3 c (b e-2 c d)^2 (-12 b e g+c d g+23 c e f)}{(d+e x)^3}-\frac{(2 c d-b e)^3 (9 b e g+8 c d g-26 c e f)}{(d+e x)^4}+\frac{7 (b e-2 c d)^4 (e f-d g)}{(d+e x)^5}\right )}{63 e^2 (b e-2 c d)^7 ((d+e x) (c (d-e x)-b e))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)/((d + e*x)^3*(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*((7*(-2*c*d + b*e)^4*(e*f - d*g))/(d + e
*x)^5 - ((2*c*d - b*e)^3*(-26*c*e*f + 8*c*d*g + 9*b*e*g))/(d + e*x)^4 + (3*c*(-2
*c*d + b*e)^2*(23*c*e*f + c*d*g - 12*b*e*g))/(d + e*x)^3 + (c^2*(2*c*d - b*e)*(1
76*c*e*f + 46*c*d*g - 111*b*e*g))/(d + e*x)^2 + (c^3*(667*c*e*f + 281*c*d*g - 47
4*b*e*g))/(d + e*x) + (21*c^4*(-2*c*d + b*e)*(c*e*f + c*d*g - b*e*g))/(-(c*d) +
b*e + c*e*x)^2 + (21*c^4*(17*c*e*f + 11*c*d*g - 14*b*e*g))/(-(c*d) + b*e + c*e*x
)))/(63*e^2*(-2*c*d + b*e)^7*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))
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Maple [B] time = 0.022, size = 1036, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)/(e*x+d)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
[Out]
-2/63*(c*e*x+b*e-c*d)*(-768*b*c^5*e^7*g*x^6+512*c^6*d*e^6*g*x^6+1024*c^6*e^7*f*x
^6-1152*b^2*c^4*e^7*g*x^5-1536*b*c^5*d*e^6*g*x^5+1536*b*c^5*e^7*f*x^5+1536*c^6*d
^2*e^5*g*x^5+3072*c^6*d*e^6*f*x^5-288*b^3*c^3*e^7*g*x^4-4416*b^2*c^4*d*e^6*g*x^4
+384*b^2*c^4*e^7*f*x^4+1920*b*c^5*d^2*e^5*g*x^4+6144*b*c^5*d*e^6*f*x^4+768*c^6*d
^3*e^4*g*x^4+1536*c^6*d^2*e^5*f*x^4+48*b^4*c^2*e^7*g*x^3-1472*b^3*c^3*d*e^6*g*x^
3-64*b^3*c^3*e^7*f*x^3-5376*b^2*c^4*d^2*e^5*g*x^3+1920*b^2*c^4*d*e^6*f*x^3+6912*
b*c^5*d^3*e^4*g*x^3+8448*b*c^5*d^2*e^5*f*x^3-1792*c^6*d^4*e^3*g*x^3-3584*c^6*d^3
*e^4*f*x^3-18*b^5*c*e^7*g*x^2+300*b^4*c^2*d*e^6*g*x^2+24*b^4*c^2*e^7*f*x^2-3216*
b^3*c^3*d^2*e^5*g*x^2-384*b^3*c^3*d*e^6*f*x^2-288*b^2*c^4*d^3*e^4*g*x^2+4032*b^2
*c^4*d^2*e^5*f*x^2+4704*b*c^5*d^4*e^3*g*x^2+3072*b*c^5*d^3*e^4*f*x^2-2112*c^6*d^
5*e^2*g*x^2-4224*c^6*d^4*e^3*f*x^2+9*b^6*e^7*g*x-132*b^5*c*d*e^6*g*x-12*b^5*c*e^
7*f*x+876*b^4*c^2*d^2*e^5*g*x+168*b^4*c^2*d*e^6*f*x-4128*b^3*c^3*d^3*e^4*g*x-105
6*b^3*c^3*d^2*e^5*f*x+4848*b^2*c^4*d^4*e^3*g*x+4800*b^2*c^4*d^3*e^4*f*x-1344*b*c
^5*d^5*e^2*g*x-3264*b*c^5*d^4*e^3*f*x-192*c^6*d^6*e*g*x-384*c^6*d^5*e^2*f*x+2*b^
6*d*e^6*g+7*b^6*e^7*f-30*b^5*c*d^2*e^5*g-96*b^5*c*d*e^6*f+204*b^4*c^2*d^3*e^4*g+
564*b^4*c^2*d^2*e^5*f-976*b^3*c^3*d^4*e^3*g-1856*b^3*c^3*d^3*e^4*f+1344*b^2*c^4*
d^5*e^2*g+3984*b^2*c^4*d^4*e^3*f-480*b*c^5*d^6*e*g-3840*b*c^5*d^5*e^2*f-64*c^6*d
^7*g+1216*c^6*d^6*e*f)/(e*x+d)^2/(b^7*e^7-14*b^6*c*d*e^6+84*b^5*c^2*d^2*e^5-280*
b^4*c^3*d^3*e^4+560*b^3*c^4*d^4*e^3-672*b^2*c^5*d^5*e^2+448*b*c^6*d^6*e-128*c^7*
d^7)/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)^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 90.6098, size = 2485, normalized size = 6.94 \[ \text{result too large to display} \]
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)^3),x, algorithm="fricas")
[Out]
-2/63*(256*(4*c^6*e^7*f + (2*c^6*d*e^6 - 3*b*c^5*e^7)*g)*x^6 + 384*(4*(2*c^6*d*e
^6 + b*c^5*e^7)*f + (4*c^6*d^2*e^5 - 4*b*c^5*d*e^6 - 3*b^2*c^4*e^7)*g)*x^5 + 96*
(4*(4*c^6*d^2*e^5 + 16*b*c^5*d*e^6 + b^2*c^4*e^7)*f + (8*c^6*d^3*e^4 + 20*b*c^5*
d^2*e^5 - 46*b^2*c^4*d*e^6 - 3*b^3*c^3*e^7)*g)*x^4 - 16*(4*(56*c^6*d^3*e^4 - 132
*b*c^5*d^2*e^5 - 30*b^2*c^4*d*e^6 + b^3*c^3*e^7)*f + (112*c^6*d^4*e^3 - 432*b*c^
5*d^3*e^4 + 336*b^2*c^4*d^2*e^5 + 92*b^3*c^3*d*e^6 - 3*b^4*c^2*e^7)*g)*x^3 - 6*(
4*(176*c^6*d^4*e^3 - 128*b*c^5*d^3*e^4 - 168*b^2*c^4*d^2*e^5 + 16*b^3*c^3*d*e^6
- b^4*c^2*e^7)*f + (352*c^6*d^5*e^2 - 784*b*c^5*d^4*e^3 + 48*b^2*c^4*d^3*e^4 + 5
36*b^3*c^3*d^2*e^5 - 50*b^4*c^2*d*e^6 + 3*b^5*c*e^7)*g)*x^2 + (1216*c^6*d^6*e -
3840*b*c^5*d^5*e^2 + 3984*b^2*c^4*d^4*e^3 - 1856*b^3*c^3*d^3*e^4 + 564*b^4*c^2*d
^2*e^5 - 96*b^5*c*d*e^6 + 7*b^6*e^7)*f - 2*(32*c^6*d^7 + 240*b*c^5*d^6*e - 672*b
^2*c^4*d^5*e^2 + 488*b^3*c^3*d^4*e^3 - 102*b^4*c^2*d^3*e^4 + 15*b^5*c*d^2*e^5 -
b^6*d*e^6)*g - 3*(4*(32*c^6*d^5*e^2 + 272*b*c^5*d^4*e^3 - 400*b^2*c^4*d^3*e^4 +
88*b^3*c^3*d^2*e^5 - 14*b^4*c^2*d*e^6 + b^5*c*e^7)*f + (64*c^6*d^6*e + 448*b*c^5
*d^5*e^2 - 1616*b^2*c^4*d^4*e^3 + 1376*b^3*c^3*d^3*e^4 - 292*b^4*c^2*d^2*e^5 + 4
4*b^5*c*d*e^6 - 3*b^6*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)/(128
*c^9*d^14*e^2 - 704*b*c^8*d^13*e^3 + 1696*b^2*c^7*d^12*e^4 - 2352*b^3*c^6*d^11*e
^5 + 2072*b^4*c^5*d^10*e^6 - 1204*b^5*c^4*d^9*e^7 + 462*b^6*c^3*d^8*e^8 - 113*b^
7*c^2*d^7*e^9 + 16*b^8*c*d^6*e^10 - b^9*d^5*e^11 + (128*c^9*d^7*e^9 - 448*b*c^8*
d^6*e^10 + 672*b^2*c^7*d^5*e^11 - 560*b^3*c^6*d^4*e^12 + 280*b^4*c^5*d^3*e^13 -
84*b^5*c^4*d^2*e^14 + 14*b^6*c^3*d*e^15 - b^7*c^2*e^16)*x^7 + (384*c^9*d^8*e^8 -
1088*b*c^8*d^7*e^9 + 1120*b^2*c^7*d^6*e^10 - 336*b^3*c^6*d^5*e^11 - 280*b^4*c^5
*d^4*e^12 + 308*b^5*c^4*d^3*e^13 - 126*b^6*c^3*d^2*e^14 + 25*b^7*c^2*d*e^15 - 2*
b^8*c*e^16)*x^6 + (128*c^9*d^9*e^7 + 576*b*c^8*d^8*e^8 - 2784*b^2*c^7*d^7*e^9 +
4368*b^3*c^6*d^6*e^10 - 3528*b^4*c^5*d^5*e^11 + 1596*b^5*c^4*d^4*e^12 - 378*b^6*
c^3*d^3*e^13 + 27*b^7*c^2*d^2*e^14 + 6*b^8*c*d*e^15 - b^9*e^16)*x^5 - 5*(128*c^9
*d^10*e^6 - 704*b*c^8*d^9*e^7 + 1440*b^2*c^7*d^8*e^8 - 1456*b^3*c^6*d^7*e^9 + 72
8*b^4*c^5*d^6*e^10 - 84*b^5*c^4*d^5*e^11 - 98*b^6*c^3*d^4*e^12 + 55*b^7*c^2*d^3*
e^13 - 12*b^8*c*d^2*e^14 + b^9*d*e^15)*x^4 - 5*(128*c^9*d^11*e^5 - 448*b*c^8*d^1
0*e^6 + 416*b^2*c^7*d^9*e^7 + 336*b^3*c^6*d^8*e^8 - 1064*b^4*c^5*d^7*e^9 + 1036*
b^5*c^4*d^6*e^10 - 546*b^6*c^3*d^5*e^11 + 167*b^7*c^2*d^4*e^12 - 28*b^8*c*d^3*e^
13 + 2*b^9*d^2*e^14)*x^3 + (128*c^9*d^12*e^4 - 1728*b*c^8*d^11*e^5 + 6432*b^2*c^
7*d^10*e^6 - 11760*b^3*c^6*d^9*e^7 + 12600*b^4*c^5*d^8*e^8 - 8484*b^5*c^4*d^7*e^
9 + 3654*b^6*c^3*d^6*e^10 - 981*b^7*c^2*d^5*e^11 + 150*b^8*c*d^4*e^12 - 10*b^9*d
^3*e^13)*x^2 + (384*c^9*d^13*e^3 - 2368*b*c^8*d^12*e^4 + 6240*b^2*c^7*d^11*e^5 -
9296*b^3*c^6*d^10*e^6 + 8680*b^4*c^5*d^9*e^7 - 5292*b^5*c^4*d^8*e^8 + 2114*b^6*
c^3*d^7*e^9 - 535*b^7*c^2*d^6*e^10 + 78*b^8*c*d^5*e^11 - 5*b^9*d^4*e^12)*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)**3/(-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] time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]
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)^3),x, algorithm="giac")
[Out]
[undef, undef, undef, undef, undef, 1]