3.2232 \(\int (d+e x)^{3/2} (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{315 c^4 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{105 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2} \]

[Out]

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

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Rubi [A]  time = 0.951673, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{315 c^4 e^2 (d+e x)^{3/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{105 c^3 e^2 \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-2 b e g+c d g+3 c e f)}{21 c^2 e^2}-\frac{2 g (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)*(f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 92.2088, size = 257, normalized size = 0.96 \[ - \frac{2 g \left (d + e x\right )^{\frac{3}{2}} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{9 c e^{2}} + \frac{2 \sqrt{d + e x} \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{21 c^{2} e^{2}} - \frac{8 \left (b e - 2 c d\right ) \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{105 c^{3} e^{2} \sqrt{d + e x}} + \frac{16 \left (b e - 2 c d\right )^{2} \left (2 b e g - c d g - 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{315 c^{4} e^{2} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.33084, size = 167, normalized size = 0.63 \[ -\frac{2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (-16 b^3 e^3 g+24 b^2 c e^2 (4 d g+e (f+g x))-6 b c^2 e \left (31 d^2 g+d e (22 f+20 g x)+e^2 x (6 f+5 g x)\right )+c^3 \left (106 d^3 g+3 d^2 e (71 f+53 g x)+6 d e^2 x (27 f+20 g x)+5 e^3 x^2 (9 f+7 g x)\right )\right )}{315 c^4 e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*(f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]

[Out]

(-2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*(-16*b^3*e^3*g + 24*b^2*c*e^2*(4*d*
g + e*(f + g*x)) - 6*b*c^2*e*(31*d^2*g + e^2*x*(6*f + 5*g*x) + d*e*(22*f + 20*g*
x)) + c^3*(106*d^3*g + 5*e^3*x^2*(9*f + 7*g*x) + 6*d*e^2*x*(27*f + 20*g*x) + 3*d
^2*e*(71*f + 53*g*x))))/(315*c^4*e^2*(d + e*x)^(3/2))

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Maple [A]  time = 0.012, size = 235, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -35\,{e}^{3}g{x}^{3}{c}^{3}+30\,b{c}^{2}{e}^{3}g{x}^{2}-120\,{c}^{3}d{e}^{2}g{x}^{2}-45\,{c}^{3}{e}^{3}f{x}^{2}-24\,{b}^{2}c{e}^{3}gx+120\,b{c}^{2}d{e}^{2}gx+36\,b{c}^{2}{e}^{3}fx-159\,{c}^{3}{d}^{2}egx-162\,{c}^{3}d{e}^{2}fx+16\,{b}^{3}{e}^{3}g-96\,{b}^{2}cd{e}^{2}g-24\,{b}^{2}c{e}^{3}f+186\,b{c}^{2}{d}^{2}eg+132\,b{c}^{2}d{e}^{2}f-106\,{c}^{3}{d}^{3}g-213\,f{d}^{2}{c}^{3}e \right ) }{315\,{c}^{4}{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/315*(c*e*x+b*e-c*d)*(-35*c^3*e^3*g*x^3+30*b*c^2*e^3*g*x^2-120*c^3*d*e^2*g*x^2
-45*c^3*e^3*f*x^2-24*b^2*c*e^3*g*x+120*b*c^2*d*e^2*g*x+36*b*c^2*e^3*f*x-159*c^3*
d^2*e*g*x-162*c^3*d*e^2*f*x+16*b^3*e^3*g-96*b^2*c*d*e^2*g-24*b^2*c*e^3*f+186*b*c
^2*d^2*e*g+132*b*c^2*d*e^2*f-106*c^3*d^3*g-213*c^3*d^2*e*f)*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)/c^4/e^2/(e*x+d)^(1/2)

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Maxima [A]  time = 0.740073, size = 478, normalized size = 1.79 \[ \frac{2 \,{\left (15 \, c^{3} e^{3} x^{3} - 71 \, c^{3} d^{3} + 115 \, b c^{2} d^{2} e - 52 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \,{\left (13 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} +{\left (17 \, c^{3} d^{2} e + 22 \, b c^{2} d e^{2} - 4 \, b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{105 \,{\left (c^{3} e^{2} x + c^{3} d e\right )}} + \frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} - 106 \, c^{4} d^{4} + 292 \, b c^{3} d^{3} e - 282 \, b^{2} c^{2} d^{2} e^{2} + 112 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \,{\left (17 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \,{\left (13 \, c^{4} d^{2} e^{2} + 10 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} -{\left (53 \, c^{4} d^{3} e - 93 \, b c^{3} d^{2} e^{2} + 48 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{315 \,{\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \]

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

[Out]

2/105*(15*c^3*e^3*x^3 - 71*c^3*d^3 + 115*b*c^2*d^2*e - 52*b^2*c*d*e^2 + 8*b^3*e^
3 + 3*(13*c^3*d*e^2 + b*c^2*e^3)*x^2 + (17*c^3*d^2*e + 22*b*c^2*d*e^2 - 4*b^2*c*
e^3)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^3*e^2*x + c^3*d*e) + 2/315*(35*c
^4*e^4*x^4 - 106*c^4*d^4 + 292*b*c^3*d^3*e - 282*b^2*c^2*d^2*e^2 + 112*b^3*c*d*e
^3 - 16*b^4*e^4 + 5*(17*c^4*d*e^3 + b*c^3*e^4)*x^3 + 3*(13*c^4*d^2*e^2 + 10*b*c^
3*d*e^3 - 2*b^2*c^2*e^4)*x^2 - (53*c^4*d^3*e - 93*b*c^3*d^2*e^2 + 48*b^2*c^2*d*e
^3 - 8*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^4*e^3*x + c^4*d*e^2
)

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Fricas [A]  time = 0.296774, size = 856, normalized size = 3.21 \[ -\frac{2 \,{\left (35 \, c^{5} e^{6} g x^{6} + 5 \,{\left (9 \, c^{5} e^{6} f +{\left (17 \, c^{5} d e^{5} + 8 \, b c^{4} e^{6}\right )} g\right )} x^{5} +{\left (9 \,{\left (13 \, c^{5} d e^{5} + 6 \, b c^{4} e^{6}\right )} f +{\left (4 \, c^{5} d^{2} e^{4} + 150 \, b c^{4} d e^{5} - b^{2} c^{3} e^{6}\right )} g\right )} x^{4} +{\left (3 \,{\left (2 \, c^{5} d^{2} e^{4} + 76 \, b c^{4} d e^{5} - b^{2} c^{3} e^{6}\right )} f -{\left (138 \, c^{5} d^{3} e^{3} - 212 \, b c^{4} d^{2} e^{4} + 13 \, b^{2} c^{3} d e^{5} - 2 \, b^{3} c^{2} e^{6}\right )} g\right )} x^{3} -{\left (3 \,{\left (110 \, c^{5} d^{3} e^{3} - 168 \, b c^{4} d^{2} e^{4} + 27 \, b^{2} c^{3} d e^{5} - 4 \, b^{3} c^{2} e^{6}\right )} f +{\left (145 \, c^{5} d^{4} e^{2} - 248 \, b c^{4} d^{3} e^{3} + 153 \, b^{2} c^{3} d^{2} e^{4} - 58 \, b^{3} c^{2} d e^{5} + 8 \, b^{4} c e^{6}\right )} g\right )} x^{2} + 3 \,{\left (71 \, c^{5} d^{5} e - 186 \, b c^{4} d^{4} e^{2} + 167 \, b^{2} c^{3} d^{3} e^{3} - 60 \, b^{3} c^{2} d^{2} e^{4} + 8 \, b^{4} c d e^{5}\right )} f + 2 \,{\left (53 \, c^{5} d^{6} - 199 \, b c^{4} d^{5} e + 287 \, b^{2} c^{3} d^{4} e^{2} - 197 \, b^{3} c^{2} d^{3} e^{3} + 64 \, b^{4} c d^{2} e^{4} - 8 \, b^{5} d e^{5}\right )} g -{\left (3 \,{\left (17 \, c^{5} d^{4} e^{2} + 76 \, b c^{4} d^{3} e^{3} - 141 \, b^{2} c^{3} d^{2} e^{4} + 56 \, b^{3} c^{2} d e^{5} - 8 \, b^{4} c e^{6}\right )} f -{\left (53 \, c^{5} d^{5} e - 252 \, b c^{4} d^{4} e^{2} + 433 \, b^{2} c^{3} d^{3} e^{3} - 338 \, b^{3} c^{2} d^{2} e^{4} + 120 \, b^{4} c d e^{5} - 16 \, b^{5} e^{6}\right )} g\right )} x\right )}}{315 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{4} e^{2}} \]

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

[Out]

-2/315*(35*c^5*e^6*g*x^6 + 5*(9*c^5*e^6*f + (17*c^5*d*e^5 + 8*b*c^4*e^6)*g)*x^5
+ (9*(13*c^5*d*e^5 + 6*b*c^4*e^6)*f + (4*c^5*d^2*e^4 + 150*b*c^4*d*e^5 - b^2*c^3
*e^6)*g)*x^4 + (3*(2*c^5*d^2*e^4 + 76*b*c^4*d*e^5 - b^2*c^3*e^6)*f - (138*c^5*d^
3*e^3 - 212*b*c^4*d^2*e^4 + 13*b^2*c^3*d*e^5 - 2*b^3*c^2*e^6)*g)*x^3 - (3*(110*c
^5*d^3*e^3 - 168*b*c^4*d^2*e^4 + 27*b^2*c^3*d*e^5 - 4*b^3*c^2*e^6)*f + (145*c^5*
d^4*e^2 - 248*b*c^4*d^3*e^3 + 153*b^2*c^3*d^2*e^4 - 58*b^3*c^2*d*e^5 + 8*b^4*c*e
^6)*g)*x^2 + 3*(71*c^5*d^5*e - 186*b*c^4*d^4*e^2 + 167*b^2*c^3*d^3*e^3 - 60*b^3*
c^2*d^2*e^4 + 8*b^4*c*d*e^5)*f + 2*(53*c^5*d^6 - 199*b*c^4*d^5*e + 287*b^2*c^3*d
^4*e^2 - 197*b^3*c^2*d^3*e^3 + 64*b^4*c*d^2*e^4 - 8*b^5*d*e^5)*g - (3*(17*c^5*d^
4*e^2 + 76*b*c^4*d^3*e^3 - 141*b^2*c^3*d^2*e^4 + 56*b^3*c^2*d*e^5 - 8*b^4*c*e^6)
*f - (53*c^5*d^5*e - 252*b*c^4*d^4*e^2 + 433*b^2*c^3*d^3*e^3 - 338*b^3*c^2*d^2*e
^4 + 120*b^4*c*d*e^5 - 16*b^5*e^6)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*
d*e)*sqrt(e*x + d)*c^4*e^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**(3/2)*(f + g*x), x)

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

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

[Out]

Exception raised: AttributeError