3.2252 \(\int \sqrt{d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=343 \[ -\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{195 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2} \]

[Out]

(-32*(2*c*d - b*e)^3*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(7/2))/(45045*c^5*e^2*(d + e*x)^(7/2)) - (16*(2*c*d - b*e)^2*(15*c*e*f +
c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6435*c^4*e^2*(d
+ e*x)^(5/2)) - (4*(2*c*d - b*e)*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b
*e^2*x - c*e^2*x^2)^(7/2))/(715*c^3*e^2*(d + e*x)^(3/2)) - (2*(15*c*e*f + c*d*g
- 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(195*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)^(7/2))/(15*c*e^2)

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Rubi [A]  time = 1.21356, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{32 (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{45045 c^5 e^2 (d+e x)^{7/2}}-\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{6435 c^4 e^2 (d+e x)^{5/2}}-\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{715 c^3 e^2 (d+e x)^{3/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-8 b e g+c d g+15 c e f)}{195 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{15 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

(-32*(2*c*d - b*e)^3*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(7/2))/(45045*c^5*e^2*(d + e*x)^(7/2)) - (16*(2*c*d - b*e)^2*(15*c*e*f +
c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6435*c^4*e^2*(d
+ e*x)^(5/2)) - (4*(2*c*d - b*e)*(15*c*e*f + c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b
*e^2*x - c*e^2*x^2)^(7/2))/(715*c^3*e^2*(d + e*x)^(3/2)) - (2*(15*c*e*f + c*d*g
- 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(195*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)^(7/2))/(15*c*e^2)

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Rubi in Sympy [A]  time = 118.881, size = 332, normalized size = 0.97 \[ - \frac{2 g \sqrt{d + e x} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{15 c e^{2}} + \frac{2 \left (8 b e g - c d g - 15 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{195 c^{2} e^{2} \sqrt{d + e x}} - \frac{4 \left (b e - 2 c d\right ) \left (8 b e g - c d g - 15 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{715 c^{3} e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{16 \left (b e - 2 c d\right )^{2} \left (8 b e g - c d g - 15 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{6435 c^{4} e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{32 \left (b e - 2 c d\right )^{3} \left (8 b e g - c d g - 15 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{45045 c^{5} e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(5/2),x)

[Out]

-2*g*sqrt(d + e*x)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(15*c*e**2)
+ 2*(8*b*e*g - c*d*g - 15*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7
/2)/(195*c**2*e**2*sqrt(d + e*x)) - 4*(b*e - 2*c*d)*(8*b*e*g - c*d*g - 15*c*e*f)
*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(715*c**3*e**2*(d + e*x)**(3/
2)) + 16*(b*e - 2*c*d)**2*(8*b*e*g - c*d*g - 15*c*e*f)*(-b*e**2*x - c*e**2*x**2
+ d*(-b*e + c*d))**(7/2)/(6435*c**4*e**2*(d + e*x)**(5/2)) - 32*(b*e - 2*c*d)**3
*(8*b*e*g - c*d*g - 15*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/
(45045*c**5*e**2*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.619899, size = 264, normalized size = 0.77 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (128 b^4 e^4 g-16 b^3 c e^3 (77 d g+15 e f+28 e g x)+24 b^2 c^2 e^2 \left (187 d^2 g+d e (95 f+161 g x)+7 e^2 x (5 f+6 g x)\right )-2 b c^3 e \left (3611 d^3 g+d^2 e (4065 f+5922 g x)+21 d e^2 x (170 f+183 g x)+21 e^3 x^2 (45 f+44 g x)\right )+c^4 \left (3838 d^4 g+d^3 e (12525 f+13433 g x)+147 d^2 e^2 x (145 f+129 g x)+21 d e^3 x^2 (675 f+583 g x)+231 e^4 x^3 (15 f+13 g x)\right )\right )}{45045 c^5 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^4*e^4*
g - 16*b^3*c*e^3*(15*e*f + 77*d*g + 28*e*g*x) + 24*b^2*c^2*e^2*(187*d^2*g + 7*e^
2*x*(5*f + 6*g*x) + d*e*(95*f + 161*g*x)) - 2*b*c^3*e*(3611*d^3*g + 21*e^3*x^2*(
45*f + 44*g*x) + 21*d*e^2*x*(170*f + 183*g*x) + d^2*e*(4065*f + 5922*g*x)) + c^4
*(3838*d^4*g + 231*e^4*x^3*(15*f + 13*g*x) + 147*d^2*e^2*x*(145*f + 129*g*x) + 2
1*d*e^3*x^2*(675*f + 583*g*x) + d^3*e*(12525*f + 13433*g*x))))/(45045*c^5*e^2*Sq
rt[d + e*x])

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Maple [A]  time = 0.014, size = 367, normalized size = 1.1 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3003\,g{e}^{4}{x}^{4}{c}^{4}-1848\,b{c}^{3}{e}^{4}g{x}^{3}+12243\,{c}^{4}d{e}^{3}g{x}^{3}+3465\,{c}^{4}{e}^{4}f{x}^{3}+1008\,{b}^{2}{c}^{2}{e}^{4}g{x}^{2}-7686\,b{c}^{3}d{e}^{3}g{x}^{2}-1890\,b{c}^{3}{e}^{4}f{x}^{2}+18963\,{c}^{4}{d}^{2}{e}^{2}g{x}^{2}+14175\,{c}^{4}d{e}^{3}f{x}^{2}-448\,{b}^{3}c{e}^{4}gx+3864\,{b}^{2}{c}^{2}d{e}^{3}gx+840\,{b}^{2}{c}^{2}{e}^{4}fx-11844\,b{c}^{3}{d}^{2}{e}^{2}gx-7140\,b{c}^{3}d{e}^{3}fx+13433\,{c}^{4}{d}^{3}egx+21315\,{c}^{4}{d}^{2}{e}^{2}fx+128\,{b}^{4}{e}^{4}g-1232\,{b}^{3}cd{e}^{3}g-240\,{b}^{3}c{e}^{4}f+4488\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}g+2280\,{b}^{2}{c}^{2}d{e}^{3}f-7222\,b{c}^{3}{d}^{3}eg-8130\,b{c}^{3}{d}^{2}{e}^{2}f+3838\,{c}^{4}{d}^{4}g+12525\,f{d}^{3}{c}^{4}e \right ) }{45045\,{c}^{5}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

2/45045*(c*e*x+b*e-c*d)*(3003*c^4*e^4*g*x^4-1848*b*c^3*e^4*g*x^3+12243*c^4*d*e^3
*g*x^3+3465*c^4*e^4*f*x^3+1008*b^2*c^2*e^4*g*x^2-7686*b*c^3*d*e^3*g*x^2-1890*b*c
^3*e^4*f*x^2+18963*c^4*d^2*e^2*g*x^2+14175*c^4*d*e^3*f*x^2-448*b^3*c*e^4*g*x+386
4*b^2*c^2*d*e^3*g*x+840*b^2*c^2*e^4*f*x-11844*b*c^3*d^2*e^2*g*x-7140*b*c^3*d*e^3
*f*x+13433*c^4*d^3*e*g*x+21315*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1232*b^3*c*d*e^3*g-
240*b^3*c*e^4*f+4488*b^2*c^2*d^2*e^2*g+2280*b^2*c^2*d*e^3*f-7222*b*c^3*d^3*e*g-8
130*b*c^3*d^2*e^2*f+3838*c^4*d^4*g+12525*c^4*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(5/2)/c^5/e^2/(e*x+d)^(5/2)

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Maxima [A]  time = 0.769492, size = 1185, normalized size = 3.45 \[ \text{result too large to display} \]

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)^(5/2)*sqrt(e*x + d)*(g*x + f),x, algorithm="maxima")

[Out]

2/3003*(231*c^6*e^6*x^6 - 835*c^6*d^6 + 3047*b*c^5*d^5*e - 4283*b^2*c^4*d^4*e^2
+ 2933*b^3*c^3*d^3*e^3 - 1046*b^4*c^2*d^2*e^4 + 200*b^5*c*d*e^5 - 16*b^6*e^6 + 6
3*(4*c^6*d*e^5 + 9*b*c^5*e^6)*x^5 - 7*(103*c^6*d^2*e^4 - 193*b*c^5*d*e^5 - 53*b^
2*c^4*e^6)*x^4 - (824*c^6*d^3*e^3 + 206*b*c^5*d^2*e^4 - 1454*b^2*c^4*d*e^5 - 5*b
^3*c^3*e^6)*x^3 + 3*(271*c^6*d^4*e^2 - 954*b*c^5*d^3*e^3 + 664*b^2*c^4*d^2*e^4 +
21*b^3*c^3*d*e^5 - 2*b^4*c^2*e^6)*x^2 + (1084*c^6*d^5*e - 1897*b*c^5*d^4*e^2 +
466*b^2*c^4*d^3*e^3 + 431*b^3*c^3*d^2*e^4 - 92*b^4*c^2*d*e^5 + 8*b^5*c*e^6)*x)*s
qrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^4*e^2*x + c^4*d*e) + 2/45045*(3003*c^7*e^
7*x^7 - 3838*c^7*d^7 + 18736*b*c^6*d^6*e - 37668*b^2*c^5*d^5*e^2 + 40200*b^3*c^4
*d^4*e^3 - 24510*b^4*c^3*d^3*e^4 + 8568*b^5*c^2*d^2*e^5 - 1616*b^6*c*d*e^6 + 128
*b^7*e^7 + 231*(14*c^7*d*e^6 + 31*b*c^6*e^7)*x^6 - 63*(139*c^7*d^2*e^5 - 263*b*c
^6*d*e^6 - 71*b^2*c^5*e^7)*x^5 - 35*(278*c^7*d^3*e^4 + 54*b*c^6*d^2*e^5 - 474*b^
2*c^5*d*e^6 - b^3*c^4*e^7)*x^4 + 5*(1637*c^7*d^4*e^3 - 5930*b*c^6*d^3*e^4 + 4224
*b^2*c^5*d^2*e^5 + 77*b^3*c^4*d*e^6 - 8*b^4*c^3*e^7)*x^3 + 3*(3274*c^7*d^5*e^2 -
6125*b*c^6*d^4*e^3 + 2290*b^2*c^5*d^3*e^4 + 715*b^3*c^4*d^2*e^5 - 170*b^4*c^3*d
*e^6 + 16*b^5*c^2*e^7)*x^2 - (1919*c^7*d^6*e - 7449*b*c^6*d^5*e^2 + 11385*b^2*c^
5*d^4*e^3 - 8715*b^3*c^4*d^3*e^4 + 3540*b^4*c^3*d^2*e^5 - 744*b^5*c^2*d*e^6 + 64
*b^6*c*e^7)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^5*e^3*x + c^5*d*e^2)

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Fricas [A]  time = 0.316162, size = 1797, normalized size = 5.24 \[ \text{result too large to display} \]

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)^(5/2)*sqrt(e*x + d)*(g*x + f),x, algorithm="fricas")

[Out]

-2/45045*(3003*c^8*e^9*g*x^9 + 231*(15*c^8*e^9*f + 2*(7*c^8*d*e^8 + 22*b*c^7*e^9
)*g)*x^8 + 42*(15*(6*c^8*d*e^8 + 19*b*c^7*e^9)*f - (280*c^8*d^2*e^7 - 543*b*c^7*
d*e^8 - 277*b^2*c^6*e^9)*g)*x^7 - 14*(15*(68*c^8*d^2*e^7 - 131*b*c^7*d*e^8 - 67*
b^2*c^6*e^9)*f + (926*c^8*d^3*e^6 + 1041*b*c^7*d^2*e^7 - 2880*b^2*c^6*d*e^8 - 32
2*b^3*c^5*e^9)*g)*x^6 - (30*(538*c^8*d^3*e^6 + 621*b*c^7*d^2*e^7 - 1686*b^2*c^6*
d*e^8 - 188*b^3*c^5*e^9)*f - (16942*c^8*d^4*e^5 - 64706*b*c^7*d^3*e^6 + 31326*b^
2*c^6*d^2*e^7 + 21448*b^3*c^5*d*e^8 - 5*b^4*c^4*e^9)*g)*x^5 + (15*(1534*c^8*d^4*
e^5 - 5758*b*c^7*d^3*e^6 + 2766*b^2*c^6*d^2*e^7 + 1888*b^3*c^5*d*e^8 - b^4*c^4*e
^9)*f + 2*(9776*c^8*d^5*e^4 - 9015*b*c^7*d^4*e^5 - 20630*b^2*c^6*d^3*e^6 + 19910
*b^3*c^5*d^2*e^7 - 45*b^4*c^4*d*e^8 + 4*b^5*c^3*e^9)*g)*x^4 + 2*(15*(954*c^8*d^5
*e^4 - 851*b*c^7*d^4*e^5 - 2028*b^2*c^6*d^3*e^6 + 1936*b^3*c^5*d^2*e^7 - 12*b^4*
c^4*d*e^8 + b^5*c^3*e^9)*f - (5052*c^8*d^6*e^3 - 27553*b*c^7*d^5*e^4 + 40265*b^2
*c^6*d^4*e^5 - 18160*b^3*c^5*d^3*e^6 + 485*b^4*c^4*d^2*e^7 - 97*b^5*c^3*d*e^8 +
8*b^6*c^2*e^9)*g)*x^3 - 2*(15*(824*c^8*d^6*e^3 - 3903*b*c^7*d^5*e^4 + 5517*b^2*c
^6*d^4*e^5 - 2664*b^3*c^5*d^3*e^6 + 273*b^4*c^4*d^2*e^7 - 51*b^5*c^3*d*e^8 + 4*b
^6*c^2*e^9)*f + (6830*c^8*d^7*e^2 - 22507*b*c^7*d^6*e^3 + 27732*b^2*c^6*d^5*e^4
- 16770*b^3*c^5*d^4*e^5 + 6570*b^4*c^4*d^3*e^6 - 2235*b^5*c^3*d^2*e^7 + 412*b^6*
c^2*d*e^8 - 32*b^7*c*e^9)*g)*x^2 + 15*(835*c^8*d^8*e - 3882*b*c^7*d^7*e^2 + 7330
*b^2*c^6*d^6*e^3 - 7216*b^3*c^5*d^5*e^4 + 3979*b^4*c^4*d^4*e^5 - 1246*b^5*c^3*d^
3*e^6 + 216*b^6*c^2*d^2*e^7 - 16*b^7*c*d*e^8)*f + 2*(1919*c^8*d^9 - 11287*b*c^7*
d^8*e + 28202*b^2*c^6*d^7*e^2 - 38934*b^3*c^5*d^6*e^3 + 32355*b^4*c^4*d^5*e^4 -
16539*b^5*c^3*d^4*e^5 + 5092*b^6*c^2*d^3*e^6 - 872*b^7*c*d^2*e^7 + 64*b^8*d*e^8)
*g - (30*(542*c^8*d^7*e^2 - 1073*b*c^7*d^6*e^3 - 342*b^2*c^6*d^5*e^4 + 2124*b^3*
c^5*d^4*e^5 - 1728*b^4*c^4*d^3*e^6 + 573*b^5*c^3*d^2*e^7 - 104*b^6*c^2*d*e^8 + 8
*b^7*c*e^9)*f - (1919*c^8*d^8*e - 13206*b*c^7*d^7*e^2 + 37570*b^2*c^6*d^6*e^3 -
57768*b^3*c^5*d^5*e^4 + 52455*b^4*c^4*d^4*e^5 - 28794*b^5*c^3*d^3*e^6 + 9376*b^6
*c^2*d^2*e^7 - 1680*b^7*c*d*e^8 + 128*b^8*e^9)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x
+ c*d^2 - b*d*e)*sqrt(e*x + d)*c^5*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)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),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)^(5/2)*sqrt(e*x + d)*(g*x + f),x, algorithm="giac")

[Out]

Timed out