3.2262 \(\int \frac{(d+e x)^{5/2} (f+g x)}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=270 \[ -\frac{16 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{35 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \]

[Out]

(-16*(2*c*d - b*e)^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2])/(105*c^4*e^2*Sqrt[d + e*x]) - (8*(2*c*d - b*e)*(7*c*e*f + 5*c*d*g
- 6*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*c^3*e^2
) - (2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2])/(35*c^2*e^2) - (2*g*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2])/(7*c*e^2)

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Rubi [A]  time = 1.01739, antiderivative size = 270, 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 \[ -\frac{16 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt{d+e x}}-\frac{8 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{35 c^2 e^2}-\frac{2 g (d+e x)^{5/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(5/2)*(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*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2])/(105*c^4*e^2*Sqrt[d + e*x]) - (8*(2*c*d - b*e)*(7*c*e*f + 5*c*d*g
- 6*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*c^3*e^2
) - (2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2])/(35*c^2*e^2) - (2*g*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2])/(7*c*e^2)

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Rubi in Sympy [A]  time = 92.7013, size = 262, normalized size = 0.97 \[ - \frac{2 g \left (d + e x\right )^{\frac{5}{2}} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{7 c e^{2}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (6 b e g - 5 c d g - 7 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{35 c^{2} e^{2}} - \frac{8 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (6 b e g - 5 c d g - 7 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{105 c^{3} e^{2}} + \frac{16 \left (b e - 2 c d\right )^{2} \left (6 b e g - 5 c d g - 7 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{105 c^{4} e^{2} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)*(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)**(5/2)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(7*c*e**2)
+ 2*(d + e*x)**(3/2)*(6*b*e*g - 5*c*d*g - 7*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2
+ d*(-b*e + c*d))/(35*c**2*e**2) - 8*sqrt(d + e*x)*(b*e - 2*c*d)*(6*b*e*g - 5*c*
d*g - 7*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(105*c**3*e**2) +
16*(b*e - 2*c*d)**2*(6*b*e*g - 5*c*d*g - 7*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))/(105*c**4*e**2*sqrt(d + e*x))

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Mathematica [A]  time = 0.3023, size = 181, normalized size = 0.67 \[ \frac{2 \sqrt{d+e x} (b e-c d+c e x) \left (-48 b^3 e^3 g+8 b^2 c e^2 (32 d g+7 e f+3 e g x)-2 b c^2 e \left (219 d^2 g+2 d e (63 f+26 g x)+e^2 x (14 f+9 g x)\right )+c^3 \left (230 d^3 g+d^2 e (301 f+115 g x)+2 d e^2 x (49 f+30 g x)+3 e^3 x^2 (7 f+5 g x)\right )\right )}{105 c^4 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(-(c*d) + b*e + c*e*x)*(-48*b^3*e^3*g + 8*b^2*c*e^2*(7*e*f + 32
*d*g + 3*e*g*x) - 2*b*c^2*e*(219*d^2*g + e^2*x*(14*f + 9*g*x) + 2*d*e*(63*f + 26
*g*x)) + c^3*(230*d^3*g + 3*e^3*x^2*(7*f + 5*g*x) + 2*d*e^2*x*(49*f + 30*g*x) +
d^2*e*(301*f + 115*g*x))))/(105*c^4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.012, size = 235, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -15\,{e}^{3}g{x}^{3}{c}^{3}+18\,b{c}^{2}{e}^{3}g{x}^{2}-60\,{c}^{3}d{e}^{2}g{x}^{2}-21\,{c}^{3}{e}^{3}f{x}^{2}-24\,{b}^{2}c{e}^{3}gx+104\,b{c}^{2}d{e}^{2}gx+28\,b{c}^{2}{e}^{3}fx-115\,{c}^{3}{d}^{2}egx-98\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-256\,{b}^{2}cd{e}^{2}g-56\,{b}^{2}c{e}^{3}f+438\,b{c}^{2}{d}^{2}eg+252\,b{c}^{2}d{e}^{2}f-230\,{c}^{3}{d}^{3}g-301\,{c}^{3}{d}^{2}ef \right ) }{105\,{c}^{4}{e}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(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^3*e^3*g*x^3+18*b*c^2*e^3*g*x^2-60*c^3*d*e^2*g*x^2-
21*c^3*e^3*f*x^2-24*b^2*c*e^3*g*x+104*b*c^2*d*e^2*g*x+28*b*c^2*e^3*f*x-115*c^3*d
^2*e*g*x-98*c^3*d*e^2*f*x+48*b^3*e^3*g-256*b^2*c*d*e^2*g-56*b^2*c*e^3*f+438*b*c^
2*d^2*e*g+252*b*c^2*d*e^2*f-230*c^3*d^3*g-301*c^3*d^2*e*f)*(e*x+d)^(1/2)/c^4/e^2
/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

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Maxima [A]  time = 0.731187, size = 431, normalized size = 1.6 \[ \frac{2 \,{\left (3 \, c^{3} e^{3} x^{3} - 43 \, c^{3} d^{3} + 79 \, b c^{2} d^{2} e - 44 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} +{\left (11 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} +{\left (29 \, c^{3} d^{2} e - 18 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} f}{15 \, \sqrt{-c e x + c d - b e} c^{3} e} + \frac{2 \,{\left (15 \, c^{4} e^{4} x^{4} - 230 \, c^{4} d^{4} + 668 \, b c^{3} d^{3} e - 694 \, b^{2} c^{2} d^{2} e^{2} + 304 \, b^{3} c d e^{3} - 48 \, b^{4} e^{4} + 3 \,{\left (15 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} +{\left (55 \, c^{4} d^{2} e^{2} - 26 \, b c^{3} d e^{3} + 6 \, b^{2} c^{2} e^{4}\right )} x^{2} +{\left (115 \, c^{4} d^{3} e - 219 \, b c^{3} d^{2} e^{2} + 128 \, b^{2} c^{2} d e^{3} - 24 \, b^{3} c e^{4}\right )} x\right )} g}{105 \, \sqrt{-c e x + c d - b e} c^{4} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="maxima")

[Out]

2/15*(3*c^3*e^3*x^3 - 43*c^3*d^3 + 79*b*c^2*d^2*e - 44*b^2*c*d*e^2 + 8*b^3*e^3 +
(11*c^3*d*e^2 - b*c^2*e^3)*x^2 + (29*c^3*d^2*e - 18*b*c^2*d*e^2 + 4*b^2*c*e^3)*
x)*f/(sqrt(-c*e*x + c*d - b*e)*c^3*e) + 2/105*(15*c^4*e^4*x^4 - 230*c^4*d^4 + 66
8*b*c^3*d^3*e - 694*b^2*c^2*d^2*e^2 + 304*b^3*c*d*e^3 - 48*b^4*e^4 + 3*(15*c^4*d
*e^3 - b*c^3*e^4)*x^3 + (55*c^4*d^2*e^2 - 26*b*c^3*d*e^3 + 6*b^2*c^2*e^4)*x^2 +
(115*c^4*d^3*e - 219*b*c^3*d^2*e^2 + 128*b^2*c^2*d*e^3 - 24*b^3*c*e^4)*x)*g/(sqr
t(-c*e*x + c*d - b*e)*c^4*e^2)

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Fricas [A]  time = 0.275651, size = 617, normalized size = 2.29 \[ \frac{2 \,{\left (15 \, c^{4} e^{5} g x^{5} + 3 \,{\left (7 \, c^{4} e^{5} f +{\left (20 \, c^{4} d e^{4} - b c^{3} e^{5}\right )} g\right )} x^{4} +{\left (7 \,{\left (14 \, c^{4} d e^{4} - b c^{3} e^{5}\right )} f +{\left (100 \, c^{4} d^{2} e^{3} - 29 \, b c^{3} d e^{4} + 6 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} +{\left (7 \,{\left (40 \, c^{4} d^{2} e^{3} - 19 \, b c^{3} d e^{4} + 4 \, b^{2} c^{2} e^{5}\right )} f +{\left (170 \, c^{4} d^{3} e^{2} - 245 \, b c^{3} d^{2} e^{3} + 134 \, b^{2} c^{2} d e^{4} - 24 \, b^{3} c e^{5}\right )} g\right )} x^{2} - 7 \,{\left (43 \, c^{4} d^{4} e - 79 \, b c^{3} d^{3} e^{2} + 44 \, b^{2} c^{2} d^{2} e^{3} - 8 \, b^{3} c d e^{4}\right )} f - 2 \,{\left (115 \, c^{4} d^{5} - 334 \, b c^{3} d^{4} e + 347 \, b^{2} c^{2} d^{3} e^{2} - 152 \, b^{3} c d^{2} e^{3} + 24 \, b^{4} d e^{4}\right )} g -{\left (7 \,{\left (14 \, c^{4} d^{3} e^{2} - 61 \, b c^{3} d^{2} e^{3} + 40 \, b^{2} c^{2} d e^{4} - 8 \, b^{3} c e^{5}\right )} f +{\left (115 \, c^{4} d^{4} e - 449 \, b c^{3} d^{3} e^{2} + 566 \, b^{2} c^{2} d^{2} e^{3} - 280 \, b^{3} c d e^{4} + 48 \, b^{4} e^{5}\right )} g\right )} x\right )}}{105 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{4} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="fricas")

[Out]

2/105*(15*c^4*e^5*g*x^5 + 3*(7*c^4*e^5*f + (20*c^4*d*e^4 - b*c^3*e^5)*g)*x^4 + (
7*(14*c^4*d*e^4 - b*c^3*e^5)*f + (100*c^4*d^2*e^3 - 29*b*c^3*d*e^4 + 6*b^2*c^2*e
^5)*g)*x^3 + (7*(40*c^4*d^2*e^3 - 19*b*c^3*d*e^4 + 4*b^2*c^2*e^5)*f + (170*c^4*d
^3*e^2 - 245*b*c^3*d^2*e^3 + 134*b^2*c^2*d*e^4 - 24*b^3*c*e^5)*g)*x^2 - 7*(43*c^
4*d^4*e - 79*b*c^3*d^3*e^2 + 44*b^2*c^2*d^2*e^3 - 8*b^3*c*d*e^4)*f - 2*(115*c^4*
d^5 - 334*b*c^3*d^4*e + 347*b^2*c^2*d^3*e^2 - 152*b^3*c*d^2*e^3 + 24*b^4*d*e^4)*
g - (7*(14*c^4*d^3*e^2 - 61*b*c^3*d^2*e^3 + 40*b^2*c^2*d*e^4 - 8*b^3*c*e^5)*f +
(115*c^4*d^4*e - 449*b*c^3*d^3*e^2 + 566*b^2*c^2*d^2*e^3 - 280*b^3*c*d*e^4 + 48*
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^4*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((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(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((e*x + d)^(5/2)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="giac")

[Out]

Timed out