3.2180 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{315 e^2 (d+e x)^3 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{105 e^2 (d+e x)^4 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{21 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (d+e x)^6 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*e^2*(2*c*d - b*e
)*(d + e*x)^6) - (2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(3/2))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^5) - (8*c*(2*c*e*f + 4*c*d*g -
3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105*e^2*(2*c*d - b*e)^3*(
d + e*x)^4) - (16*c^2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(3/2))/(315*e^2*(2*c*d - b*e)^4*(d + e*x)^3)

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Rubi [A]  time = 1.00412, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{16 c^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{315 e^2 (d+e x)^3 (2 c d-b e)^4}-\frac{8 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{105 e^2 (d+e x)^4 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g+4 c d g+2 c e f)}{21 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 e^2 (d+e x)^6 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^6,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*e^2*(2*c*d - b*e
)*(d + e*x)^6) - (2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(3/2))/(21*e^2*(2*c*d - b*e)^2*(d + e*x)^5) - (8*c*(2*c*e*f + 4*c*d*g -
3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(105*e^2*(2*c*d - b*e)^3*(
d + e*x)^4) - (16*c^2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2)^(3/2))/(315*e^2*(2*c*d - b*e)^4*(d + e*x)^3)

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Rubi in Sympy [A]  time = 107.789, size = 274, normalized size = 0.96 \[ \frac{16 c^{2} \left (3 b e g - 4 c d g - 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{315 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )^{4}} - \frac{8 c \left (3 b e g - 4 c d g - 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{105 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (3 b e g - 4 c d g - 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{21 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{9 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**6,x)

[Out]

16*c**2*(3*b*e*g - 4*c*d*g - 2*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))
**(3/2)/(315*e**2*(d + e*x)**3*(b*e - 2*c*d)**4) - 8*c*(3*b*e*g - 4*c*d*g - 2*c*
e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(105*e**2*(d + e*x)**4*(b
*e - 2*c*d)**3) + 2*(3*b*e*g - 4*c*d*g - 2*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(
-b*e + c*d))**(3/2)/(21*e**2*(d + e*x)**5*(b*e - 2*c*d)**2) - 2*(d*g - e*f)*(-b*
e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(9*e**2*(d + e*x)**6*(b*e - 2*c*d)
)

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Mathematica [A]  time = 0.385364, size = 193, normalized size = 0.68 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{8 c^3 (d+e x)^4 (-3 b e g+4 c d g+2 c e f)}{(b e-2 c d)^4}+\frac{4 c^2 (d+e x)^3 (-3 b e g+4 c d g+2 c e f)}{(2 c d-b e)^3}+\frac{3 c (d+e x)^2 (-3 b e g+4 c d g+2 c e f)}{(b e-2 c d)^2}-\frac{5 (d+e x) (9 b e g-19 c d g+c e f)}{b e-2 c d}+35 (d g-e f)\right )}{315 e^2 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^6,x]

[Out]

(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(35*(-(e*f) + d*g) - (5*(c*e*f - 19*c*
d*g + 9*b*e*g)*(d + e*x))/(-2*c*d + b*e) + (3*c*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d
+ e*x)^2)/(-2*c*d + b*e)^2 + (4*c^2*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d + e*x)^3)/
(2*c*d - b*e)^3 + (8*c^3*(2*c*e*f + 4*c*d*g - 3*b*e*g)*(d + e*x)^4)/(-2*c*d + b*
e)^4))/(315*e^2*(d + e*x)^5)

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Maple [A]  time = 0.017, size = 382, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 24\,b{c}^{2}{e}^{4}g{x}^{3}-32\,{c}^{3}d{e}^{3}g{x}^{3}-16\,{c}^{3}{e}^{4}f{x}^{3}-36\,{b}^{2}c{e}^{4}g{x}^{2}+192\,b{c}^{2}d{e}^{3}g{x}^{2}+24\,b{c}^{2}{e}^{4}f{x}^{2}-192\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-96\,{c}^{3}d{e}^{3}f{x}^{2}+45\,{b}^{3}{e}^{4}gx-312\,{b}^{2}cd{e}^{3}gx-30\,{b}^{2}c{e}^{4}fx+732\,b{c}^{2}{d}^{2}{e}^{2}gx+168\,b{c}^{2}d{e}^{3}fx-528\,{c}^{3}{d}^{3}egx-264\,{c}^{3}{d}^{2}{e}^{2}fx+10\,{b}^{3}d{e}^{3}g+35\,{b}^{3}{e}^{4}f-66\,{b}^{2}c{d}^{2}{e}^{2}g-240\,{b}^{2}cd{e}^{3}f+144\,b{c}^{2}{d}^{3}eg+564\,b{c}^{2}{d}^{2}{e}^{2}f-88\,{c}^{3}{d}^{4}g-464\,{c}^{3}{d}^{3}ef \right ) }{315\, \left ( ex+d \right ) ^{5}{e}^{2} \left ({b}^{4}{e}^{4}-8\,{b}^{3}cd{e}^{3}+24\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}-32\,b{c}^{3}{d}^{3}e+16\,{c}^{4}{d}^{4} \right ) }\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^6,x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(24*b*c^2*e^4*g*x^3-32*c^3*d*e^3*g*x^3-16*c^3*e^4*f*x^3-3
6*b^2*c*e^4*g*x^2+192*b*c^2*d*e^3*g*x^2+24*b*c^2*e^4*f*x^2-192*c^3*d^2*e^2*g*x^2
-96*c^3*d*e^3*f*x^2+45*b^3*e^4*g*x-312*b^2*c*d*e^3*g*x-30*b^2*c*e^4*f*x+732*b*c^
2*d^2*e^2*g*x+168*b*c^2*d*e^3*f*x-528*c^3*d^3*e*g*x-264*c^3*d^2*e^2*f*x+10*b^3*d
*e^3*g+35*b^3*e^4*f-66*b^2*c*d^2*e^2*g-240*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+564*b
*c^2*d^2*e^2*f-88*c^3*d^4*g-464*c^3*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1
/2)/(e*x+d)^5/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)

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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(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.4535, size = 1103, normalized size = 3.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^6,x, algorithm="fricas")

[Out]

2/315*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(2*c^4*e^5*f + (4*c^4*d*e^4
- 3*b*c^3*e^5)*g)*x^4 + 4*(2*(10*c^4*d*e^4 - b*c^3*e^5)*f + (40*c^4*d^2*e^3 - 34
*b*c^3*d*e^4 + 3*b^2*c^2*e^5)*g)*x^3 + 3*(2*(28*c^4*d^2*e^3 - 8*b*c^3*d*e^4 + b^
2*c^2*e^5)*f + (112*c^4*d^3*e^2 - 116*b*c^3*d^2*e^3 + 28*b^2*c^2*d*e^4 - 3*b^3*c
*e^5)*g)*x^2 - (464*c^4*d^4*e - 1028*b*c^3*d^3*e^2 + 804*b^2*c^2*d^2*e^3 - 275*b
^3*c*d*e^4 + 35*b^4*e^5)*f - 2*(44*c^4*d^5 - 116*b*c^3*d^4*e + 105*b^2*c^2*d^3*e
^2 - 38*b^3*c*d^2*e^3 + 5*b^4*d*e^4)*g + ((200*c^4*d^3*e^2 - 132*b*c^3*d^2*e^3 +
42*b^2*c^2*d*e^4 - 5*b^3*c*e^5)*f - (440*c^4*d^4*e - 1116*b*c^3*d^3*e^2 + 978*b
^2*c^2*d^2*e^3 - 347*b^3*c*d*e^4 + 45*b^4*e^5)*g)*x)/(16*c^4*d^9*e^2 - 32*b*c^3*
d^8*e^3 + 24*b^2*c^2*d^7*e^4 - 8*b^3*c*d^6*e^5 + b^4*d^5*e^6 + (16*c^4*d^4*e^7 -
32*b*c^3*d^3*e^8 + 24*b^2*c^2*d^2*e^9 - 8*b^3*c*d*e^10 + b^4*e^11)*x^5 + 5*(16*
c^4*d^5*e^6 - 32*b*c^3*d^4*e^7 + 24*b^2*c^2*d^3*e^8 - 8*b^3*c*d^2*e^9 + b^4*d*e^
10)*x^4 + 10*(16*c^4*d^6*e^5 - 32*b*c^3*d^5*e^6 + 24*b^2*c^2*d^4*e^7 - 8*b^3*c*d
^3*e^8 + b^4*d^2*e^9)*x^3 + 10*(16*c^4*d^7*e^4 - 32*b*c^3*d^6*e^5 + 24*b^2*c^2*d
^5*e^6 - 8*b^3*c*d^4*e^7 + b^4*d^3*e^8)*x^2 + 5*(16*c^4*d^8*e^3 - 32*b*c^3*d^7*e
^4 + 24*b^2*c^2*d^6*e^5 - 8*b^3*c*d^5*e^6 + b^4*d^4*e^7)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**6,x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**6, x)

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GIAC/XCAS [A]  time = 1.10622, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x