3.2221 \(\int \frac{f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=209 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 c (b+2 c x) (-5 b e g+4 c d g+6 c e f)}{15 e (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (-5 b e g+4 c d g+6 c e f)}{15 e^2 (d+e x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(8*c*(6*c*e*f + 4*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^4*Sqrt[d*(c*
d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x
)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(6*c*e*f + 4*c*d*g - 5*b*e*g
))/(15*e^2*(2*c*d - b*e)^2*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

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Rubi [A]  time = 0.64042, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 c (b+2 c x) (-5 b e g+4 c d g+6 c e f)}{15 e (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{2 (-5 b e g+4 c d g+6 c e f)}{15 e^2 (d+e x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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)^(3/2)),x]

[Out]

(8*c*(6*c*e*f + 4*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^4*Sqrt[d*(c*
d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x
)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(6*c*e*f + 4*c*d*g - 5*b*e*g
))/(15*e^2*(2*c*d - b*e)^2*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])

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Rubi in Sympy [A]  time = 67.9914, size = 197, normalized size = 0.94 \[ - \frac{4 c \left (2 b + 4 c x\right ) \left (5 b e g - 4 c d g - 6 c e f\right )}{15 e \left (b e - 2 c d\right )^{4} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \left (5 b e g - 4 c d g - 6 c e f\right )}{15 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{2 \left (d g - e f\right )}{5 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

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)**(3/2),x)

[Out]

-4*c*(2*b + 4*c*x)*(5*b*e*g - 4*c*d*g - 6*c*e*f)/(15*e*(b*e - 2*c*d)**4*sqrt(-b*
e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) + 2*(5*b*e*g - 4*c*d*g - 6*c*e*f)/(15*e*
*2*(d + e*x)*(b*e - 2*c*d)**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) -
2*(d*g - e*f)/(5*e**2*(d + e*x)**2*(b*e - 2*c*d)*sqrt(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d)))

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Mathematica [A]  time = 0.914611, size = 179, normalized size = 0.86 \[ \frac{2 (d+e x)^2 (b e-c d+c e x)^2 \left (-\frac{15 c^2 (-b e g+c d g+c e f)}{b e-c d+c e x}-\frac{c (-25 b e g+17 c d g+33 c e f)}{d+e x}+\frac{(b e-2 c d) (-5 b e g+c d g+9 c e f)}{(d+e x)^2}+\frac{3 (b e-2 c d)^2 (d g-e f)}{(d+e x)^3}\right )}{15 e^2 (b e-2 c d)^4 ((d+e x) (c (d-e x)-b e))^{3/2}} \]

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)^(3/2)),x]

[Out]

(2*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^2*((3*(-2*c*d + b*e)^2*(-(e*f) + d*g))/(d
+ e*x)^3 + ((-2*c*d + b*e)*(9*c*e*f + c*d*g - 5*b*e*g))/(d + e*x)^2 - (c*(33*c*e
*f + 17*c*d*g - 25*b*e*g))/(d + e*x) - (15*c^2*(c*e*f + c*d*g - b*e*g))/(-(c*d)
+ b*e + c*e*x)))/(15*e^2*(-2*c*d + b*e)^4*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/
2))

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Maple [A]  time = 0.017, size = 382, normalized size = 1.8 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -40\,b{c}^{2}{e}^{4}g{x}^{3}+32\,{c}^{3}d{e}^{3}g{x}^{3}+48\,{c}^{3}{e}^{4}f{x}^{3}-20\,{b}^{2}c{e}^{4}g{x}^{2}-64\,b{c}^{2}d{e}^{3}g{x}^{2}+24\,b{c}^{2}{e}^{4}f{x}^{2}+64\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}+96\,{c}^{3}d{e}^{3}f{x}^{2}+5\,{b}^{3}{e}^{4}gx-64\,{b}^{2}cd{e}^{3}gx-6\,{b}^{2}c{e}^{4}fx+28\,b{c}^{2}{d}^{2}{e}^{2}gx+72\,b{c}^{2}d{e}^{3}fx+16\,{c}^{3}{d}^{3}egx+24\,{c}^{3}{d}^{2}{e}^{2}fx+2\,{b}^{3}d{e}^{3}g+3\,{b}^{3}{e}^{4}f-26\,{b}^{2}c{d}^{2}{e}^{2}g-24\,{b}^{2}cd{e}^{3}f+16\,b{c}^{2}{d}^{3}eg+84\,b{c}^{2}{d}^{2}{e}^{2}f+8\,{c}^{3}{d}^{4}g-48\,{c}^{3}{d}^{3}ef \right ) }{ \left ( 15\,ex+15\,d \right ){e}^{2} \left ({b}^{4}{e}^{4}-8\,{b}^{3}cd{e}^{3}+24\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}-32\,b{c}^{3}{d}^{3}e+16\,{c}^{4}{d}^{4} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

-2/15*(c*e*x+b*e-c*d)*(-40*b*c^2*e^4*g*x^3+32*c^3*d*e^3*g*x^3+48*c^3*e^4*f*x^3-2
0*b^2*c*e^4*g*x^2-64*b*c^2*d*e^3*g*x^2+24*b*c^2*e^4*f*x^2+64*c^3*d^2*e^2*g*x^2+9
6*c^3*d*e^3*f*x^2+5*b^3*e^4*g*x-64*b^2*c*d*e^3*g*x-6*b^2*c*e^4*f*x+28*b*c^2*d^2*
e^2*g*x+72*b*c^2*d*e^3*f*x+16*c^3*d^3*e*g*x+24*c^3*d^2*e^2*f*x+2*b^3*d*e^3*g+3*b
^3*e^4*f-26*b^2*c*d^2*e^2*g-24*b^2*c*d*e^3*f+16*b*c^2*d^3*e*g+84*b*c^2*d^2*e^2*f
+8*c^3*d^4*g-48*c^3*d^3*e*f)/(e*x+d)/e^2/(b^4*e^4-8*b^3*c*d*e^3+24*b^2*c^2*d^2*e
^2-32*b*c^3*d^3*e+16*c^4*d^4)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)

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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)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.37738, size = 876, normalized size = 4.19 \[ \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (8 \,{\left (6 \, c^{3} e^{4} f +{\left (4 \, c^{3} d e^{3} - 5 \, b c^{2} e^{4}\right )} g\right )} x^{3} + 4 \,{\left (6 \,{\left (4 \, c^{3} d e^{3} + b c^{2} e^{4}\right )} f +{\left (16 \, c^{3} d^{2} e^{2} - 16 \, b c^{2} d e^{3} - 5 \, b^{2} c e^{4}\right )} g\right )} x^{2} - 3 \,{\left (16 \, c^{3} d^{3} e - 28 \, b c^{2} d^{2} e^{2} + 8 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} f + 2 \,{\left (4 \, c^{3} d^{4} + 8 \, b c^{2} d^{3} e - 13 \, b^{2} c d^{2} e^{2} + b^{3} d e^{3}\right )} g +{\left (6 \,{\left (4 \, c^{3} d^{2} e^{2} + 12 \, b c^{2} d e^{3} - b^{2} c e^{4}\right )} f +{\left (16 \, c^{3} d^{3} e + 28 \, b c^{2} d^{2} e^{2} - 64 \, b^{2} c d e^{3} + 5 \, b^{3} e^{4}\right )} g\right )} x\right )}}{15 \,{\left (16 \, c^{5} d^{8} e^{2} - 48 \, b c^{4} d^{7} e^{3} + 56 \, b^{2} c^{3} d^{6} e^{4} - 32 \, b^{3} c^{2} d^{5} e^{5} + 9 \, b^{4} c d^{4} e^{6} - b^{5} d^{3} e^{7} -{\left (16 \, c^{5} d^{4} e^{6} - 32 \, b c^{4} d^{3} e^{7} + 24 \, b^{2} c^{3} d^{2} e^{8} - 8 \, b^{3} c^{2} d e^{9} + b^{4} c e^{10}\right )} x^{4} -{\left (32 \, c^{5} d^{5} e^{5} - 48 \, b c^{4} d^{4} e^{6} + 16 \, b^{2} c^{3} d^{3} e^{7} + 8 \, b^{3} c^{2} d^{2} e^{8} - 6 \, b^{4} c d e^{9} + b^{5} e^{10}\right )} x^{3} - 3 \,{\left (16 \, b c^{4} d^{5} e^{5} - 32 \, b^{2} c^{3} d^{4} e^{6} + 24 \, b^{3} c^{2} d^{3} e^{7} - 8 \, b^{4} c d^{2} e^{8} + b^{5} d e^{9}\right )} x^{2} +{\left (32 \, c^{5} d^{7} e^{3} - 112 \, b c^{4} d^{6} e^{4} + 144 \, b^{2} c^{3} d^{5} e^{5} - 88 \, b^{3} c^{2} d^{4} e^{6} + 26 \, b^{4} c d^{3} e^{7} - 3 \, b^{5} d^{2} e^{8}\right )} x\right )}} \]

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)^(3/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(6*c^3*e^4*f + (4*c^3*d*e^3 -
5*b*c^2*e^4)*g)*x^3 + 4*(6*(4*c^3*d*e^3 + b*c^2*e^4)*f + (16*c^3*d^2*e^2 - 16*b
*c^2*d*e^3 - 5*b^2*c*e^4)*g)*x^2 - 3*(16*c^3*d^3*e - 28*b*c^2*d^2*e^2 + 8*b^2*c*
d*e^3 - b^3*e^4)*f + 2*(4*c^3*d^4 + 8*b*c^2*d^3*e - 13*b^2*c*d^2*e^2 + b^3*d*e^3
)*g + (6*(4*c^3*d^2*e^2 + 12*b*c^2*d*e^3 - b^2*c*e^4)*f + (16*c^3*d^3*e + 28*b*c
^2*d^2*e^2 - 64*b^2*c*d*e^3 + 5*b^3*e^4)*g)*x)/(16*c^5*d^8*e^2 - 48*b*c^4*d^7*e^
3 + 56*b^2*c^3*d^6*e^4 - 32*b^3*c^2*d^5*e^5 + 9*b^4*c*d^4*e^6 - b^5*d^3*e^7 - (1
6*c^5*d^4*e^6 - 32*b*c^4*d^3*e^7 + 24*b^2*c^3*d^2*e^8 - 8*b^3*c^2*d*e^9 + b^4*c*
e^10)*x^4 - (32*c^5*d^5*e^5 - 48*b*c^4*d^4*e^6 + 16*b^2*c^3*d^3*e^7 + 8*b^3*c^2*
d^2*e^8 - 6*b^4*c*d*e^9 + b^5*e^10)*x^3 - 3*(16*b*c^4*d^5*e^5 - 32*b^2*c^3*d^4*e
^6 + 24*b^3*c^2*d^3*e^7 - 8*b^4*c*d^2*e^8 + b^5*d*e^9)*x^2 + (32*c^5*d^7*e^3 - 1
12*b*c^4*d^6*e^4 + 144*b^2*c^3*d^5*e^5 - 88*b^3*c^2*d^4*e^6 + 26*b^4*c*d^3*e^7 -
3*b^5*d^2*e^8)*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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)^(3/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError