3.2233 \(\int \sqrt{d+e x} (f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\)

Optimal. Leaf size=191 \[ -\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{105 c^3 e^2 (d+e x)^{3/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{35 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2} \]

[Out]

(-4*(2*c*d - b*e)*(7*c*e*f + c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(3/2))/(105*c^3*e^2*(d + e*x)^(3/2)) - (2*(7*c*e*f + c*d*g - 4*b*e*g)*(d*(c*
d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(35*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d
+ e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(7*c*e^2)

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Rubi [A]  time = 0.700354, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{105 c^3 e^2 (d+e x)^{3/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-4 b e g+c d g+7 c e f)}{35 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(2*c*d - b*e)*(7*c*e*f + c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(3/2))/(105*c^3*e^2*(d + e*x)^(3/2)) - (2*(7*c*e*f + c*d*g - 4*b*e*g)*(d*(c*
d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(35*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d
+ e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(7*c*e^2)

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Rubi in Sympy [A]  time = 66.044, size = 182, normalized size = 0.95 \[ - \frac{2 g \sqrt{d + e x} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{7 c e^{2}} + \frac{2 \left (4 b e g - c d g - 7 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{35 c^{2} e^{2} \sqrt{d + e x}} - \frac{4 \left (b e - 2 c d\right ) \left (4 b e g - c d g - 7 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{105 c^{3} e^{2} \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*g*sqrt(d + e*x)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(7*c*e**2)
+ 2*(4*b*e*g - c*d*g - 7*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2
)/(35*c**2*e**2*sqrt(d + e*x)) - 4*(b*e - 2*c*d)*(4*b*e*g - c*d*g - 7*c*e*f)*(-b
*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(105*c**3*e**2*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.155377, size = 119, normalized size = 0.62 \[ \frac{2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (15 d g+7 e f+6 e g x)+c^2 \left (22 d^2 g+d e (49 f+33 g x)+3 e^2 x (7 f+5 g x)\right )\right )}{105 c^3 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(8*b^2*e^2*g -
2*b*c*e*(7*e*f + 15*d*g + 6*e*g*x) + c^2*(22*d^2*g + 3*e^2*x*(7*f + 5*g*x) + d*e
*(49*f + 33*g*x))))/(105*c^3*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.007, size = 139, normalized size = 0.7 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 15\,g{x}^{2}{c}^{2}{e}^{2}-12\,bc{e}^{2}gx+33\,{c}^{2}degx+21\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-30\,bcdeg-14\,bc{e}^{2}f+22\,{c}^{2}{d}^{2}g+49\,{c}^{2}def \right ) }{105\,{c}^{3}{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(c*e*x+b*e-c*d)*(15*c^2*e^2*g*x^2-12*b*c*e^2*g*x+33*c^2*d*e*g*x+21*c^2*e^2
*f*x+8*b^2*e^2*g-30*b*c*d*e*g-14*b*c*e^2*f+22*c^2*d^2*g+49*c^2*d*e*f)*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2)/c^3/e^2/(e*x+d)^(1/2)

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Maxima [A]  time = 0.732271, size = 319, normalized size = 1.67 \[ \frac{2 \,{\left (3 \, c^{2} e^{2} x^{2} - 7 \, c^{2} d^{2} + 9 \, b c d e - 2 \, b^{2} e^{2} +{\left (4 \, c^{2} d e + b c e^{2}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{15 \,{\left (c^{2} e^{2} x + c^{2} d e\right )}} + \frac{2 \,{\left (15 \, c^{3} e^{3} x^{3} - 22 \, c^{3} d^{3} + 52 \, b c^{2} d^{2} e - 38 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \,{\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} -{\left (11 \, c^{3} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{105 \,{\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \]

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)*sqrt(e*x + d)*(g*x + f),x, algorithm="maxima")

[Out]

2/15*(3*c^2*e^2*x^2 - 7*c^2*d^2 + 9*b*c*d*e - 2*b^2*e^2 + (4*c^2*d*e + b*c*e^2)*
x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^2*e^2*x + c^2*d*e) + 2/105*(15*c^3*e^
3*x^3 - 22*c^3*d^3 + 52*b*c^2*d^2*e - 38*b^2*c*d*e^2 + 8*b^3*e^3 + 3*(6*c^3*d*e^
2 + b*c^2*e^3)*x^2 - (11*c^3*d^2*e - 15*b*c^2*d*e^2 + 4*b^2*c*e^3)*x)*sqrt(-c*e*
x + c*d - b*e)*(e*x + d)*g/(c^3*e^3*x + c^3*d*e^2)

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Fricas [A]  time = 0.28676, size = 614, normalized size = 3.21 \[ -\frac{2 \,{\left (15 \, c^{4} e^{5} g x^{5} + 3 \,{\left (7 \, c^{4} e^{5} f + 6 \,{\left (c^{4} d e^{4} + b c^{3} e^{5}\right )} g\right )} x^{4} +{\left (28 \,{\left (c^{4} d e^{4} + b c^{3} e^{5}\right )} f -{\left (26 \, c^{4} d^{2} e^{3} - 48 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}\right )} g\right )} x^{3} -{\left (7 \,{\left (10 \, c^{4} d^{2} e^{3} - 16 \, b c^{3} d e^{4} + b^{2} c^{2} e^{5}\right )} f + 4 \,{\left (10 \, c^{4} d^{3} e^{2} - 14 \, b c^{3} d^{2} e^{3} + 5 \, b^{2} c^{2} d e^{4} - b^{3} c e^{5}\right )} g\right )} x^{2} + 7 \,{\left (7 \, c^{4} d^{4} e - 16 \, b c^{3} d^{3} e^{2} + 11 \, b^{2} c^{2} d^{2} e^{3} - 2 \, b^{3} c d e^{4}\right )} f + 2 \,{\left (11 \, c^{4} d^{5} - 37 \, b c^{3} d^{4} e + 45 \, b^{2} c^{2} d^{3} e^{2} - 23 \, b^{3} c d^{2} e^{3} + 4 \, b^{4} d e^{4}\right )} g -{\left (14 \,{\left (2 \, c^{4} d^{3} e^{2} + 2 \, b c^{3} d^{2} e^{3} - 5 \, b^{2} c^{2} d e^{4} + b^{3} c e^{5}\right )} f -{\left (11 \, c^{4} d^{4} e - 48 \, b c^{3} d^{3} e^{2} + 71 \, b^{2} c^{2} d^{2} e^{3} - 42 \, b^{3} c d e^{4} + 8 \, b^{4} e^{5}\right )} g\right )} x\right )}}{105 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{3} e^{2}} \]

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)*sqrt(e*x + d)*(g*x + f),x, algorithm="fricas")

[Out]

-2/105*(15*c^4*e^5*g*x^5 + 3*(7*c^4*e^5*f + 6*(c^4*d*e^4 + b*c^3*e^5)*g)*x^4 + (
28*(c^4*d*e^4 + b*c^3*e^5)*f - (26*c^4*d^2*e^3 - 48*b*c^3*d*e^4 + b^2*c^2*e^5)*g
)*x^3 - (7*(10*c^4*d^2*e^3 - 16*b*c^3*d*e^4 + b^2*c^2*e^5)*f + 4*(10*c^4*d^3*e^2
- 14*b*c^3*d^2*e^3 + 5*b^2*c^2*d*e^4 - b^3*c*e^5)*g)*x^2 + 7*(7*c^4*d^4*e - 16*
b*c^3*d^3*e^2 + 11*b^2*c^2*d^2*e^3 - 2*b^3*c*d*e^4)*f + 2*(11*c^4*d^5 - 37*b*c^3
*d^4*e + 45*b^2*c^2*d^3*e^2 - 23*b^3*c*d^2*e^3 + 4*b^4*d*e^4)*g - (14*(2*c^4*d^3
*e^2 + 2*b*c^3*d^2*e^3 - 5*b^2*c^2*d*e^4 + b^3*c*e^5)*f - (11*c^4*d^4*e - 48*b*c
^3*d^3*e^2 + 71*b^2*c^2*d^2*e^3 - 42*b^3*c*d*e^4 + 8*b^4*e^5)*g)*x)/(sqrt(-c*e^2
*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c^3*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(1/2)*(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))*sqrt(d + e*x)*(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)*sqrt(e*x + d)*(g*x + f),x, algorithm="giac")

[Out]

Exception raised: AttributeError