3.2228 \(\int \frac{f+g x}{(d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^3*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x
)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(8*c*e*f + 2*c*d*g - 5*b*
e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
])

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Rubi [A]  time = 0.508247, antiderivative size = 208, 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 \[ -\frac{2 (e f-d g)}{5 e^2 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{16 c (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (b+2 c x) (-5 b e g+2 c d g+8 c e f)}{15 e (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)/((d + e*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]

[Out]

(2*(8*c*e*f + 2*c*d*g - 5*b*e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^3*(d*(c*d - b*
e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x
)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (16*c*(8*c*e*f + 2*c*d*g - 5*b*
e*g)*(b + 2*c*x))/(15*e*(2*c*d - b*e)^5*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2
])

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Rubi in Sympy [A]  time = 55.5068, size = 197, normalized size = 0.95 \[ \frac{8 c \left (2 b + 4 c x\right ) \left (5 b e g - 2 c d g - 8 c e f\right )}{15 e \left (b e - 2 c d\right )^{5} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \left (b + 2 c x\right ) \left (5 b e g - 2 c d g - 8 c e f\right )}{15 e \left (b e - 2 c d\right )^{3} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 \left (d g - e f\right )}{5 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

8*c*(2*b + 4*c*x)*(5*b*e*g - 2*c*d*g - 8*c*e*f)/(15*e*(b*e - 2*c*d)**5*sqrt(-b*e
**2*x - c*e**2*x**2 + d*(-b*e + c*d))) + 2*(b + 2*c*x)*(5*b*e*g - 2*c*d*g - 8*c*
e*f)/(15*e*(b*e - 2*c*d)**3*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)) -
2*(d*g - e*f)/(5*e**2*(d + e*x)*(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 = 1.18286, size = 225, normalized size = 1.08 \[ \frac{2 (d+e x)^3 (c (d-e x)-b e)^3 \left (\frac{5 c^2 (-8 b e g+5 c d g+11 c e f)}{b e-c d+c e x}+\frac{5 c^2 (b e-2 c d) (-b e g+c d g+c e f)}{(b e-c d+c e x)^2}+\frac{c (-40 b e g+7 c d g+73 c e f)}{d+e x}-\frac{(2 c d-b e) (5 b e g+4 c d g-14 c e f)}{(d+e x)^2}+\frac{3 (b e-2 c d)^2 (e f-d g)}{(d+e x)^3}\right )}{15 e^2 (b e-2 c d)^5 ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)/((d + e*x)*(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*((3*(-2*c*d + b*e)^2*(e*f - d*g))/(d + e
*x)^3 - ((2*c*d - b*e)*(-14*c*e*f + 4*c*d*g + 5*b*e*g))/(d + e*x)^2 + (c*(73*c*e
*f + 7*c*d*g - 40*b*e*g))/(d + e*x) + (5*c^2*(-2*c*d + b*e)*(c*e*f + c*d*g - b*e
*g))/(-(c*d) + b*e + c*e*x)^2 + (5*c^2*(11*c*e*f + 5*c*d*g - 8*b*e*g))/(-(c*d) +
b*e + c*e*x)))/(15*e^2*(-2*c*d + b*e)^5*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2
))

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Maple [B]  time = 0.02, size = 557, normalized size = 2.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -80\,b{c}^{3}{e}^{5}g{x}^{4}+32\,{c}^{4}d{e}^{4}g{x}^{4}+128\,{c}^{4}{e}^{5}f{x}^{4}-120\,{b}^{2}{c}^{2}{e}^{5}g{x}^{3}-32\,b{c}^{3}d{e}^{4}g{x}^{3}+192\,b{c}^{3}{e}^{5}f{x}^{3}+32\,{c}^{4}{d}^{2}{e}^{3}g{x}^{3}+128\,{c}^{4}d{e}^{4}f{x}^{3}-30\,{b}^{3}c{e}^{5}g{x}^{2}-228\,{b}^{2}{c}^{2}d{e}^{4}g{x}^{2}+48\,{b}^{2}{c}^{2}{e}^{5}f{x}^{2}+216\,b{c}^{3}{d}^{2}{e}^{3}g{x}^{2}+384\,b{c}^{3}d{e}^{4}f{x}^{2}-48\,{c}^{4}{d}^{3}{e}^{2}g{x}^{2}-192\,{c}^{4}{d}^{2}{e}^{3}f{x}^{2}+5\,{b}^{4}{e}^{5}gx-92\,{b}^{3}cd{e}^{4}gx-8\,{b}^{3}c{e}^{5}fx-24\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}gx+144\,{b}^{2}{c}^{2}d{e}^{4}fx+144\,b{c}^{3}{d}^{3}{e}^{2}gx+96\,b{c}^{3}{d}^{2}{e}^{3}fx-48\,{c}^{4}{d}^{4}egx-192\,{c}^{4}{d}^{3}{e}^{2}fx+2\,{b}^{4}d{e}^{4}g+3\,{b}^{4}{e}^{5}f-38\,{b}^{3}c{d}^{2}{e}^{3}g-32\,{b}^{3}cd{e}^{4}f+12\,{b}^{2}{c}^{2}{d}^{3}{e}^{2}g+168\,{b}^{2}{c}^{2}{d}^{2}{e}^{3}f+72\,b{c}^{3}{d}^{4}eg-192\,b{c}^{3}{d}^{3}{e}^{2}f-48\,{c}^{4}{d}^{5}g+48\,{c}^{4}{d}^{4}ef \right ) }{ \left ( 15\,{b}^{5}{e}^{5}-150\,{b}^{4}cd{e}^{4}+600\,{b}^{3}{c}^{2}{d}^{2}{e}^{3}-1200\,{b}^{2}{c}^{3}{d}^{3}{e}^{2}+1200\,b{c}^{4}{d}^{4}e-480\,{c}^{5}{d}^{5} \right ){e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)/(e*x+d)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/15*(c*e*x+b*e-c*d)*(-80*b*c^3*e^5*g*x^4+32*c^4*d*e^4*g*x^4+128*c^4*e^5*f*x^4-
120*b^2*c^2*e^5*g*x^3-32*b*c^3*d*e^4*g*x^3+192*b*c^3*e^5*f*x^3+32*c^4*d^2*e^3*g*
x^3+128*c^4*d*e^4*f*x^3-30*b^3*c*e^5*g*x^2-228*b^2*c^2*d*e^4*g*x^2+48*b^2*c^2*e^
5*f*x^2+216*b*c^3*d^2*e^3*g*x^2+384*b*c^3*d*e^4*f*x^2-48*c^4*d^3*e^2*g*x^2-192*c
^4*d^2*e^3*f*x^2+5*b^4*e^5*g*x-92*b^3*c*d*e^4*g*x-8*b^3*c*e^5*f*x-24*b^2*c^2*d^2
*e^3*g*x+144*b^2*c^2*d*e^4*f*x+144*b*c^3*d^3*e^2*g*x+96*b*c^3*d^2*e^3*f*x-48*c^4
*d^4*e*g*x-192*c^4*d^3*e^2*f*x+2*b^4*d*e^4*g+3*b^4*e^5*f-38*b^3*c*d^2*e^3*g-32*b
^3*c*d*e^4*f+12*b^2*c^2*d^3*e^2*g+168*b^2*c^2*d^2*e^3*f+72*b*c^3*d^4*e*g-192*b*c
^3*d^3*e^2*f-48*c^4*d^5*g+48*c^4*d^4*e*f)/(b^5*e^5-10*b^4*c*d*e^4+40*b^3*c^2*d^2
*e^3-80*b^2*c^3*d^3*e^2+80*b*c^4*d^4*e-32*c^5*d^5)/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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 9.43221, size = 1388, normalized size = 6.67 \[ \text{result too large to display} \]

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)),x, algorithm="fricas")

[Out]

-2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(16*(8*c^4*e^5*f + (2*c^4*d*e^4
- 5*b*c^3*e^5)*g)*x^4 + 8*(8*(2*c^4*d*e^4 + 3*b*c^3*e^5)*f + (4*c^4*d^2*e^3 - 4
*b*c^3*d*e^4 - 15*b^2*c^2*e^5)*g)*x^3 - 6*(8*(4*c^4*d^2*e^3 - 8*b*c^3*d*e^4 - b^
2*c^2*e^5)*f + (8*c^4*d^3*e^2 - 36*b*c^3*d^2*e^3 + 38*b^2*c^2*d*e^4 + 5*b^3*c*e^
5)*g)*x^2 + (48*c^4*d^4*e - 192*b*c^3*d^3*e^2 + 168*b^2*c^2*d^2*e^3 - 32*b^3*c*d
*e^4 + 3*b^4*e^5)*f - 2*(24*c^4*d^5 - 36*b*c^3*d^4*e - 6*b^2*c^2*d^3*e^2 + 19*b^
3*c*d^2*e^3 - b^4*d*e^4)*g - (8*(24*c^4*d^3*e^2 - 12*b*c^3*d^2*e^3 - 18*b^2*c^2*
d*e^4 + b^3*c*e^5)*f + (48*c^4*d^4*e - 144*b*c^3*d^3*e^2 + 24*b^2*c^2*d^2*e^3 +
92*b^3*c*d*e^4 - 5*b^4*e^5)*g)*x)/(32*c^7*d^10*e^2 - 144*b*c^6*d^9*e^3 + 272*b^2
*c^5*d^8*e^4 - 280*b^3*c^4*d^7*e^5 + 170*b^4*c^3*d^6*e^6 - 61*b^5*c^2*d^5*e^7 +
12*b^6*c*d^4*e^8 - b^7*d^3*e^9 + (32*c^7*d^5*e^7 - 80*b*c^6*d^4*e^8 + 80*b^2*c^5
*d^3*e^9 - 40*b^3*c^4*d^2*e^10 + 10*b^4*c^3*d*e^11 - b^5*c^2*e^12)*x^5 + (32*c^7
*d^6*e^6 - 16*b*c^6*d^5*e^7 - 80*b^2*c^5*d^4*e^8 + 120*b^3*c^4*d^3*e^9 - 70*b^4*
c^3*d^2*e^10 + 19*b^5*c^2*d*e^11 - 2*b^6*c*e^12)*x^4 - (64*c^7*d^7*e^5 - 288*b*c
^6*d^6*e^6 + 448*b^2*c^5*d^5*e^7 - 320*b^3*c^4*d^4*e^8 + 100*b^4*c^3*d^3*e^9 - 2
*b^5*c^2*d^2*e^10 - 6*b^6*c*d*e^11 + b^7*e^12)*x^3 - (64*c^7*d^8*e^4 - 160*b*c^6
*d^7*e^5 + 64*b^2*c^5*d^6*e^6 + 160*b^3*c^4*d^5*e^7 - 220*b^4*c^3*d^4*e^8 + 118*
b^5*c^2*d^3*e^9 - 30*b^6*c*d^2*e^10 + 3*b^7*d*e^11)*x^2 + (32*c^7*d^9*e^3 - 208*
b*c^6*d^8*e^4 + 496*b^2*c^5*d^7*e^5 - 600*b^3*c^4*d^6*e^6 + 410*b^4*c^3*d^5*e^7
- 161*b^5*c^2*d^4*e^8 + 34*b^6*c*d^3*e^9 - 3*b^7*d^2*e^10)*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)/(-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)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]