3.2243 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=193 \[ -\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt{d+e x}} \]

[Out]

(-4*(2*c*d - b*e)*(9*c*e*f - c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(5/2))/(315*c^3*e^2*(d + e*x)^(5/2)) - (2*(9*c*e*f - c*d*g - 4*b*e*g)*(d*(c*
d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(63*c^2*e^2*(d + e*x)^(3/2)) - (2*g*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*c*e^2*Sqrt[d + e*x])

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Rubi [A]  time = 0.715388, antiderivative size = 193, 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 \[ -\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{315 c^3 e^2 (d+e x)^{5/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-4 b e g-c d g+9 c e f)}{63 c^2 e^2 (d+e x)^{3/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(2*c*d - b*e)*(9*c*e*f - c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2)^(5/2))/(315*c^3*e^2*(d + e*x)^(5/2)) - (2*(9*c*e*f - c*d*g - 4*b*e*g)*(d*(c*
d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(63*c^2*e^2*(d + e*x)^(3/2)) - (2*g*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(9*c*e^2*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 63.2022, size = 182, normalized size = 0.94 \[ - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{9 c e^{2} \sqrt{d + e x}} + \frac{2 \left (4 b e g + c d g - 9 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{63 c^{2} e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 \left (b e - 2 c d\right ) \left (4 b e g + c d g - 9 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{315 c^{3} e^{2} \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(3/2)/(e*x+d)**(1/2),x)

[Out]

-2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(9*c*e**2*sqrt(d + e*x))
+ 2*(4*b*e*g + c*d*g - 9*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2
)/(63*c**2*e**2*(d + e*x)**(3/2)) - 4*(b*e - 2*c*d)*(4*b*e*g + c*d*g - 9*c*e*f)*
(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(315*c**3*e**2*(d + e*x)**(5/2
))

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Mathematica [A]  time = 0.191095, size = 121, normalized size = 0.63 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (17 d g+9 e f+10 e g x)+c^2 \left (26 d^2 g+d e (81 f+65 g x)+5 e^2 x (9 f+7 g x)\right )\right )}{315 c^3 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(8*b^2*e^2*g
- 2*b*c*e*(9*e*f + 17*d*g + 10*e*g*x) + c^2*(26*d^2*g + 5*e^2*x*(9*f + 7*g*x) +
d*e*(81*f + 65*g*x))))/(315*c^3*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 139, normalized size = 0.7 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 35\,g{x}^{2}{c}^{2}{e}^{2}-20\,bc{e}^{2}gx+65\,{c}^{2}degx+45\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-34\,bcdeg-18\,bc{e}^{2}f+26\,{c}^{2}{d}^{2}g+81\,{c}^{2}def \right ) }{315\,{c}^{3}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^(1/2),x)

[Out]

2/315*(c*e*x+b*e-c*d)*(35*c^2*e^2*g*x^2-20*b*c*e^2*g*x+65*c^2*d*e*g*x+45*c^2*e^2
*f*x+8*b^2*e^2*g-34*b*c*d*e*g-18*b*c*e^2*f+26*c^2*d^2*g+81*c^2*d*e*f)*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^3/e^2/(e*x+d)^(3/2)

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Maxima [A]  time = 0.738112, size = 432, normalized size = 2.24 \[ -\frac{2 \,{\left (5 \, c^{3} e^{3} x^{3} + 9 \, c^{3} d^{3} - 20 \, b c^{2} d^{2} e + 13 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} -{\left (c^{3} d e^{2} - 8 \, b c^{2} e^{3}\right )} x^{2} -{\left (13 \, c^{3} d^{2} e - 12 \, b c^{2} d e^{2} - b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{35 \, c^{2} e} - \frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} + 26 \, c^{4} d^{4} - 86 \, b c^{3} d^{3} e + 102 \, b^{2} c^{2} d^{2} e^{2} - 50 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} - 5 \,{\left (c^{4} d e^{3} - 10 \, b c^{3} e^{4}\right )} x^{3} - 3 \,{\left (23 \, c^{4} d^{2} e^{2} - 22 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} +{\left (13 \, c^{4} d^{3} e - 30 \, b c^{3} d^{2} e^{2} + 21 \, b^{2} c^{2} d e^{3} - 4 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{315 \, c^{3} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/sqrt(e*x + d),x, algorithm="maxima")

[Out]

-2/35*(5*c^3*e^3*x^3 + 9*c^3*d^3 - 20*b*c^2*d^2*e + 13*b^2*c*d*e^2 - 2*b^3*e^3 -
(c^3*d*e^2 - 8*b*c^2*e^3)*x^2 - (13*c^3*d^2*e - 12*b*c^2*d*e^2 - b^2*c*e^3)*x)*
sqrt(-c*e*x + c*d - b*e)*f/(c^2*e) - 2/315*(35*c^4*e^4*x^4 + 26*c^4*d^4 - 86*b*c
^3*d^3*e + 102*b^2*c^2*d^2*e^2 - 50*b^3*c*d*e^3 + 8*b^4*e^4 - 5*(c^4*d*e^3 - 10*
b*c^3*e^4)*x^3 - 3*(23*c^4*d^2*e^2 - 22*b*c^3*d*e^3 - b^2*c^2*e^4)*x^2 + (13*c^4
*d^3*e - 30*b*c^3*d^2*e^2 + 21*b^2*c^2*d*e^3 - 4*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d
- b*e)*g/(c^3*e^2)

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Fricas [A]  time = 0.287242, size = 855, normalized size = 4.43 \[ \frac{2 \,{\left (35 \, c^{5} e^{6} g x^{6} + 5 \,{\left (9 \, c^{5} e^{6} f -{\left (c^{5} d e^{5} - 17 \, b c^{4} e^{6}\right )} g\right )} x^{5} -{\left (9 \,{\left (c^{5} d e^{5} - 13 \, b c^{4} e^{6}\right )} f +{\left (104 \, c^{5} d^{2} e^{4} - 96 \, b c^{4} d e^{5} - 53 \, b^{2} c^{3} e^{6}\right )} g\right )} x^{4} -{\left (9 \,{\left (18 \, c^{5} d^{2} e^{4} - 16 \, b c^{4} d e^{5} - 9 \, b^{2} c^{3} e^{6}\right )} f -{\left (18 \, c^{5} d^{3} e^{3} - 154 \, b c^{4} d^{2} e^{4} + 137 \, b^{2} c^{3} d e^{5} - b^{3} c^{2} e^{6}\right )} g\right )} x^{3} +{\left (9 \,{\left (10 \, c^{5} d^{3} e^{3} - 42 \, b c^{4} d^{2} e^{4} + 33 \, b^{2} c^{3} d e^{5} - b^{3} c^{2} e^{6}\right )} f +{\left (95 \, c^{5} d^{4} e^{2} - 208 \, b c^{4} d^{3} e^{3} + 135 \, b^{2} c^{3} d^{2} e^{4} - 26 \, b^{3} c^{2} d e^{5} + 4 \, b^{4} c e^{6}\right )} g\right )} x^{2} - 9 \,{\left (9 \, c^{5} d^{5} e - 29 \, b c^{4} d^{4} e^{2} + 33 \, b^{2} c^{3} d^{3} e^{3} - 15 \, b^{3} c^{2} d^{2} e^{4} + 2 \, b^{4} c d e^{5}\right )} f - 2 \,{\left (13 \, c^{5} d^{6} - 56 \, b c^{4} d^{5} e + 94 \, b^{2} c^{3} d^{4} e^{2} - 76 \, b^{3} c^{2} d^{3} e^{3} + 29 \, b^{4} c d^{2} e^{4} - 4 \, b^{5} d e^{5}\right )} g +{\left (9 \,{\left (13 \, c^{5} d^{4} e^{2} - 16 \, b c^{4} d^{3} e^{3} - 9 \, b^{2} c^{3} d^{2} e^{4} + 14 \, b^{3} c^{2} d e^{5} - 2 \, b^{4} c e^{6}\right )} f -{\left (13 \, c^{5} d^{5} e - 69 \, b c^{4} d^{4} e^{2} + 137 \, b^{2} c^{3} d^{3} e^{3} - 127 \, b^{3} c^{2} d^{2} e^{4} + 54 \, b^{4} c d e^{5} - 8 \, b^{5} e^{6}\right )} g\right )} x\right )}}{315 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{3} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/315*(35*c^5*e^6*g*x^6 + 5*(9*c^5*e^6*f - (c^5*d*e^5 - 17*b*c^4*e^6)*g)*x^5 - (
9*(c^5*d*e^5 - 13*b*c^4*e^6)*f + (104*c^5*d^2*e^4 - 96*b*c^4*d*e^5 - 53*b^2*c^3*
e^6)*g)*x^4 - (9*(18*c^5*d^2*e^4 - 16*b*c^4*d*e^5 - 9*b^2*c^3*e^6)*f - (18*c^5*d
^3*e^3 - 154*b*c^4*d^2*e^4 + 137*b^2*c^3*d*e^5 - b^3*c^2*e^6)*g)*x^3 + (9*(10*c^
5*d^3*e^3 - 42*b*c^4*d^2*e^4 + 33*b^2*c^3*d*e^5 - b^3*c^2*e^6)*f + (95*c^5*d^4*e
^2 - 208*b*c^4*d^3*e^3 + 135*b^2*c^3*d^2*e^4 - 26*b^3*c^2*d*e^5 + 4*b^4*c*e^6)*g
)*x^2 - 9*(9*c^5*d^5*e - 29*b*c^4*d^4*e^2 + 33*b^2*c^3*d^3*e^3 - 15*b^3*c^2*d^2*
e^4 + 2*b^4*c*d*e^5)*f - 2*(13*c^5*d^6 - 56*b*c^4*d^5*e + 94*b^2*c^3*d^4*e^2 - 7
6*b^3*c^2*d^3*e^3 + 29*b^4*c*d^2*e^4 - 4*b^5*d*e^5)*g + (9*(13*c^5*d^4*e^2 - 16*
b*c^4*d^3*e^3 - 9*b^2*c^3*d^2*e^4 + 14*b^3*c^2*d*e^5 - 2*b^4*c*e^6)*f - (13*c^5*
d^5*e - 69*b*c^4*d^4*e^2 + 137*b^2*c^3*d^3*e^3 - 127*b^3*c^2*d^2*e^4 + 54*b^4*c*
d*e^5 - 8*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^3*e^2)

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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)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/sqrt(e*x + d),x, algorithm="giac")

[Out]

Timed out