3.2183 \(\int (d+e x)^3 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=488 \[ \frac{9 (2 c d-b e)^7 (-11 b e g+6 c d g+16 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{13/2} e^2}+\frac{9 (b+2 c x) (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+6 c d g+16 c e f)}{16384 c^6 e}+\frac{3 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+6 c d g+16 c e f)}{2048 c^5 e}-\frac{3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{640 c^4 e^2}-\frac{3 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{448 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{112 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2} \]

[Out]

(9*(2*c*d - b*e)^5*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2])/(16384*c^6*e) + (3*(2*c*d - b*e)^3*(16*c*e*f + 6*c*d*g
- 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(2048*c^5*
e) - (3*(2*c*d - b*e)^2*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2)^(5/2))/(640*c^4*e^2) - (3*(2*c*d - b*e)*(16*c*e*f + 6*c*d*g - 11*b
*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(448*c^3*e^2) - ((1
6*c*e*f + 6*c*d*g - 11*b*e*g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^
(5/2))/(112*c^2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2
))/(8*c*e^2) + (9*(2*c*d - b*e)^7*(16*c*e*f + 6*c*d*g - 11*b*e*g)*ArcTan[(e*(b +
2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(13/2)
*e^2)

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Rubi [A]  time = 1.95514, antiderivative size = 488, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{9 (2 c d-b e)^7 (-11 b e g+6 c d g+16 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{13/2} e^2}+\frac{9 (b+2 c x) (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+6 c d g+16 c e f)}{16384 c^6 e}+\frac{3 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+6 c d g+16 c e f)}{2048 c^5 e}-\frac{3 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{640 c^4 e^2}-\frac{3 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{448 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+6 c d g+16 c e f)}{112 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{8 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

(9*(2*c*d - b*e)^5*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2])/(16384*c^6*e) + (3*(2*c*d - b*e)^3*(16*c*e*f + 6*c*d*g
- 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(2048*c^5*
e) - (3*(2*c*d - b*e)^2*(16*c*e*f + 6*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2)^(5/2))/(640*c^4*e^2) - (3*(2*c*d - b*e)*(16*c*e*f + 6*c*d*g - 11*b
*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(448*c^3*e^2) - ((1
6*c*e*f + 6*c*d*g - 11*b*e*g)*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^
(5/2))/(112*c^2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2
))/(8*c*e^2) + (9*(2*c*d - b*e)^7*(16*c*e*f + 6*c*d*g - 11*b*e*g)*ArcTan[(e*(b +
2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(13/2)
*e^2)

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Rubi in Sympy [A]  time = 163.819, size = 479, normalized size = 0.98 \[ - \frac{g \left (d + e x\right )^{3} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{8 c e^{2}} + \frac{\left (d + e x\right )^{2} \left (11 b e g - 6 c d g - 16 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{112 c^{2} e^{2}} - \frac{3 \left (d + e x\right ) \left (b e - 2 c d\right ) \left (11 b e g - 6 c d g - 16 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{448 c^{3} e^{2}} + \frac{3 \left (b e - 2 c d\right )^{2} \left (11 b e g - 6 c d g - 16 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{640 c^{4} e^{2}} + \frac{3 \left (b + 2 c x\right ) \left (b e - 2 c d\right )^{3} \left (11 b e g - 6 c d g - 16 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{2048 c^{5} e} + \frac{9 \left (b + 2 c x\right ) \left (b e - 2 c d\right )^{5} \left (11 b e g - 6 c d g - 16 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{16384 c^{6} e} + \frac{9 \left (b e - 2 c d\right )^{7} \left (11 b e g - 6 c d g - 16 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{32768 c^{\frac{13}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-g*(d + e*x)**3*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(8*c*e**2) + (
d + e*x)**2*(11*b*e*g - 6*c*d*g - 16*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e +
c*d))**(5/2)/(112*c**2*e**2) - 3*(d + e*x)*(b*e - 2*c*d)*(11*b*e*g - 6*c*d*g -
16*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(448*c**3*e**2) + 3*
(b*e - 2*c*d)**2*(11*b*e*g - 6*c*d*g - 16*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-
b*e + c*d))**(5/2)/(640*c**4*e**2) + 3*(b + 2*c*x)*(b*e - 2*c*d)**3*(11*b*e*g -
6*c*d*g - 16*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(2048*c**5
*e) + 9*(b + 2*c*x)*(b*e - 2*c*d)**5*(11*b*e*g - 6*c*d*g - 16*c*e*f)*sqrt(-b*e**
2*x - c*e**2*x**2 + d*(-b*e + c*d))/(16384*c**6*e) + 9*(b*e - 2*c*d)**7*(11*b*e*
g - 6*c*d*g - 16*c*e*f)*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x - c*e**2*
x**2 + d*(-b*e + c*d))))/(32768*c**(13/2)*e**2)

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Mathematica [C]  time = 5.11493, size = 739, normalized size = 1.51 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{2 \left (-3465 b^7 e^7 g+210 b^6 c e^6 (218 d g+24 e f+11 e g x)-84 b^5 c^2 e^5 \left (3057 d^2 g+d e (760 f+334 g x)+2 e^2 x (20 f+11 g x)\right )+24 b^4 c^3 e^4 \left (32924 d^3 g+3 d^2 e (4704 f+1963 g x)+8 d e^2 x (203 f+107 g x)+2 e^3 x^2 (56 f+33 g x)\right )-16 b^3 c^4 e^3 \left (89587 d^4 g+4 d^3 e (15072 f+5887 g x)+12 d^2 e^2 x (960 f+479 g x)+8 d e^3 x^2 (222 f+125 g x)+8 e^4 x^3 (18 f+11 g x)\right )+32 b^2 c^5 e^2 \left (47490 d^5 g+d^4 e (48712 f+17401 g x)+8 d^3 e^2 x (1748 f+809 g x)+12 d^2 e^3 x^2 (308 f+163 g x)+16 d e^4 x^3 (43 f+25 g x)+8 e^5 x^4 (8 f+5 g x)\right )+64 b c^6 e \left (-13647 d^6 g-2 d^5 e (9812 f+3263 g x)+6 d^4 e^2 x (123 g x-116 f)+8 d^3 e^3 x^2 (1574 f+1187 g x)+8 d^2 e^4 x^3 (1882 f+1483 g x)+48 d e^5 x^4 (164 f+135 g x)+80 e^6 x^5 (20 f+17 g x)\right )+128 c^7 \left (1664 d^7 g+d^6 e (2944 f+945 g x)-8 d^5 e^2 x (245 f+176 g x)-2 d^4 e^3 x^2 (2624 f+1925 g x)-16 d^3 e^4 x^3 (175 f+136 g x)+8 d^2 e^5 x^4 (208 f+175 g x)+320 d e^6 x^5 (7 f+6 g x)+80 e^7 x^6 (8 f+7 g x)\right )\right )}{35 c^6 e^2 (d+e x) (b e-c d+c e x)}+\frac{9 i (2 c d-b e)^7 (-11 b e g+6 c d g+16 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{c^{13/2} e^2 (d+e x)^{3/2} (c (d-e x)-b e)^{3/2}}\right )}{32768} \]

Antiderivative was successfully verified.

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

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((2*(-3465*b^7*e^7*g + 210*b^6*c*e^6*(
24*e*f + 218*d*g + 11*e*g*x) - 84*b^5*c^2*e^5*(3057*d^2*g + 2*e^2*x*(20*f + 11*g
*x) + d*e*(760*f + 334*g*x)) + 128*c^7*(1664*d^7*g + 320*d*e^6*x^5*(7*f + 6*g*x)
+ 80*e^7*x^6*(8*f + 7*g*x) - 16*d^3*e^4*x^3*(175*f + 136*g*x) + 8*d^2*e^5*x^4*(
208*f + 175*g*x) - 8*d^5*e^2*x*(245*f + 176*g*x) + d^6*e*(2944*f + 945*g*x) - 2*
d^4*e^3*x^2*(2624*f + 1925*g*x)) + 24*b^4*c^3*e^4*(32924*d^3*g + 2*e^3*x^2*(56*f
+ 33*g*x) + 8*d*e^2*x*(203*f + 107*g*x) + 3*d^2*e*(4704*f + 1963*g*x)) + 64*b*c
^6*e*(-13647*d^6*g + 80*e^6*x^5*(20*f + 17*g*x) + 6*d^4*e^2*x*(-116*f + 123*g*x)
+ 48*d*e^5*x^4*(164*f + 135*g*x) + 8*d^3*e^3*x^2*(1574*f + 1187*g*x) + 8*d^2*e^
4*x^3*(1882*f + 1483*g*x) - 2*d^5*e*(9812*f + 3263*g*x)) - 16*b^3*c^4*e^3*(89587
*d^4*g + 8*e^4*x^3*(18*f + 11*g*x) + 8*d*e^3*x^2*(222*f + 125*g*x) + 12*d^2*e^2*
x*(960*f + 479*g*x) + 4*d^3*e*(15072*f + 5887*g*x)) + 32*b^2*c^5*e^2*(47490*d^5*
g + 8*e^5*x^4*(8*f + 5*g*x) + 16*d*e^4*x^3*(43*f + 25*g*x) + 12*d^2*e^3*x^2*(308
*f + 163*g*x) + 8*d^3*e^2*x*(1748*f + 809*g*x) + d^4*e*(48712*f + 17401*g*x))))/
(35*c^6*e^2*(d + e*x)*(-(c*d) + b*e + c*e*x)) + ((9*I)*(2*c*d - b*e)^7*(16*c*e*f
+ 6*c*d*g - 11*b*e*g)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-
(b*e) + c*(d - e*x)]])/(c^(13/2)*e^2*(d + e*x)^(3/2)*(-(b*e) + c*(d - e*x))^(3/2
))))/32768

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Maple [B]  time = 0.044, size = 3576, normalized size = 7.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

99/16384*e^5*g*b^7/c^6*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)+99/8192*e^5*g*b^6/
c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+315/128*b^2*e/(c*e^2)^(1/2)*arctan(
(c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^6*g-3/64*b^3
/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*e^3*f-27/256*b^4/c^4*(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(3/2)*d*e^2*g+677/1120/c^2/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(5/2)*b*d^2*g+63/256*b^3/c^3*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*g+9/
64*b^3/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f+765/1024*b^3/c^2*e*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4*g-9/32*b^2/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(3/2)*d^2*f-9/512*b^5/c^4*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x
*f+43/112*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d*g-9/8*b*c/(c*e^2)^(1/
2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^7*
g-15/32*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*g+3/8*d^3*f*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+9/32*d^5*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*
b+33/640*e*g*b^3/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)+45/512*b^5/c^4*e^4*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-9/2048*b^7/c^5*e^7/(c*e^2)^(1/2)*arcta
n((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*f+3/28*b/c^2
*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e*f+1125/4096*b^5/c^4*e^3*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*g-261/4096*b^6/c^5*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2)*d*g-315/512*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*e^2*
g-45/128*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*e^3*f+27/128*c/e*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^6*g-9/16*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(5/2)/c/e*d^2*g+27/128*c^2/e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^8*g+9/128/c/e*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*b*d^4*g+45/64*d^3*f/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)*b^3-45/64*d^4*f/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2+189/64*d^5*f*e^
2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(1/2))*b^2-45/32*d^4*f*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b+33/1024*e^
3*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x-45/1024*b^7/c^5*e^6/(c*e^2)
^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*
d*g-189/64*b^3/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2))*d^5*e^2*g-261/2048*b^5/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(1/2)*x*d*g+1125/2048*b^4/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)
*x*d^2*g+45/256*b^4/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*f-9/16*b/
c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*f+63/128*b^2/c^2*e*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*g-27/128*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(3/2)*x*d*e^2*g+9/32*b^2/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*f
-63/64*b^5/c^3*e^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(1/2))*d^3*g-189/512*b^5/c^3*e^5/(c*e^2)^(1/2)*arctan((c*e^2)
^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2*f+2205/1024*b^4/c
^2*e^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2))*d^4*g+45/32*d^3*f/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b
^2+315/256*d^3*f/c^2*e^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^4-315/128*d^4*f/c*e^3/(c*e^2)^(1/2)*arctan((c*
e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^3-63/32*d^6*f*c
*e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2))*b+765/512*b^2/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*g+567/2
048*b^6/c^4*e^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(1/2))*d^2*g+63/1024*b^6/c^4*e^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1
/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d*f-315/256*b^3/c^2*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*e^2*g-45/64*b^3/c^2*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*x*d^2*e^3*f+9/64/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*
d^4*g+27/256/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^6*g-1/2*x*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(5/2)/c*d*f-3/40*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
5/2)*e*f-3/7*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c*d*g-117/128*b*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*g-69/224*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(5/2)*d*g+57/140/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d*f-15/64*b
^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*g-117/256*b^2/c*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(1/2)*d^5*g-23/35/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*
d^2*f-1/7*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c*e*f+3/16*d^3*f/c*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b+9/16*d^7*f*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/
2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-9/1024*b^6/c^5*e^5*(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-13/35/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(5/2)*d^3*g+9/16*d^5*f*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+11/112*e*g*b/c
^2*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)+99/32768*e^7*g*b^8/c^6/(c*e^2)^(1/
2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-33/4
48*e*g*b^2/c^3*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-1/8*e*g*x^3*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(5/2)/c+33/2048*e^3*g*b^5/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d
^2)^(3/2)-3/128*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*e^3*f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 11.3679, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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

[Out]

[-1/2293760*(4*(71680*c^7*e^7*g*x^7 + 5120*(16*c^7*e^7*f + (48*c^7*d*e^6 + 17*b*
c^6*e^7)*g)*x^6 + 1280*(16*(14*c^7*d*e^6 + 5*b*c^6*e^7)*f + (140*c^7*d^2*e^5 + 3
24*b*c^6*d*e^6 + b^2*c^5*e^7)*g)*x^5 + 128*(16*(104*c^7*d^2*e^5 + 246*b*c^6*d*e^
6 + b^2*c^5*e^7)*f - (2176*c^7*d^3*e^4 - 5932*b*c^6*d^2*e^5 - 100*b^2*c^5*d*e^6
+ 11*b^3*c^4*e^7)*g)*x^4 - 16*(16*(1400*c^7*d^3*e^4 - 3764*b*c^6*d^2*e^5 - 86*b^
2*c^5*d*e^6 + 9*b^3*c^4*e^7)*f + (30800*c^7*d^4*e^3 - 37984*b*c^6*d^3*e^4 - 3912
*b^2*c^5*d^2*e^5 + 1000*b^3*c^4*d*e^6 - 99*b^4*c^3*e^7)*g)*x^3 - 8*(16*(5248*c^7
*d^4*e^3 - 6296*b*c^6*d^3*e^4 - 924*b^2*c^5*d^2*e^5 + 222*b^3*c^4*d*e^6 - 21*b^4
*c^3*e^7)*f + (22528*c^7*d^5*e^2 - 5904*b*c^6*d^4*e^3 - 25888*b^2*c^5*d^3*e^4 +
11496*b^3*c^4*d^2*e^5 - 2568*b^4*c^3*d*e^6 + 231*b^5*c^2*e^7)*g)*x^2 + 16*(23552
*c^7*d^6*e - 78496*b*c^6*d^5*e^2 + 97424*b^2*c^5*d^4*e^3 - 60288*b^3*c^4*d^3*e^4
+ 21168*b^4*c^3*d^2*e^5 - 3990*b^5*c^2*d*e^6 + 315*b^6*c*e^7)*f + (212992*c^7*d
^7 - 873408*b*c^6*d^6*e + 1519680*b^2*c^5*d^5*e^2 - 1433392*b^3*c^4*d^4*e^3 + 79
0176*b^4*c^3*d^3*e^4 - 256788*b^5*c^2*d^2*e^5 + 45780*b^6*c*d*e^6 - 3465*b^7*e^7
)*g - 2*(16*(7840*c^7*d^5*e^2 + 1392*b*c^6*d^4*e^3 - 13984*b^2*c^5*d^3*e^4 + 576
0*b^3*c^4*d^2*e^5 - 1218*b^4*c^3*d*e^6 + 105*b^5*c^2*e^7)*f - (60480*c^7*d^6*e -
208832*b*c^6*d^5*e^2 + 278416*b^2*c^5*d^4*e^3 - 188384*b^3*c^4*d^3*e^4 + 70668*
b^4*c^3*d^2*e^5 - 14028*b^5*c^2*d*e^6 + 1155*b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 -
b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) - 315*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 +
672*b^2*c^6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*c^4*d^3*e^5 - 84*b^5*c^3*d^
2*e^6 + 14*b^6*c^2*d*e^7 - b^7*c*e^8)*f + (768*c^8*d^8 - 4096*b*c^7*d^7*e + 8960
*b^2*c^6*d^6*e^2 - 10752*b^3*c^5*d^5*e^3 + 7840*b^4*c^4*d^4*e^4 - 3584*b^5*c^3*d
^3*e^5 + 1008*b^6*c^2*d^2*e^6 - 160*b^7*c*d*e^7 + 11*b^8*e^8)*g)*log(4*sqrt(-c*e
^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c^2*e*x + b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e
^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*sqrt(-c)))/(sqrt(-c)*c^6*e^2), -1/114688
0*(2*(71680*c^7*e^7*g*x^7 + 5120*(16*c^7*e^7*f + (48*c^7*d*e^6 + 17*b*c^6*e^7)*g
)*x^6 + 1280*(16*(14*c^7*d*e^6 + 5*b*c^6*e^7)*f + (140*c^7*d^2*e^5 + 324*b*c^6*d
*e^6 + b^2*c^5*e^7)*g)*x^5 + 128*(16*(104*c^7*d^2*e^5 + 246*b*c^6*d*e^6 + b^2*c^
5*e^7)*f - (2176*c^7*d^3*e^4 - 5932*b*c^6*d^2*e^5 - 100*b^2*c^5*d*e^6 + 11*b^3*c
^4*e^7)*g)*x^4 - 16*(16*(1400*c^7*d^3*e^4 - 3764*b*c^6*d^2*e^5 - 86*b^2*c^5*d*e^
6 + 9*b^3*c^4*e^7)*f + (30800*c^7*d^4*e^3 - 37984*b*c^6*d^3*e^4 - 3912*b^2*c^5*d
^2*e^5 + 1000*b^3*c^4*d*e^6 - 99*b^4*c^3*e^7)*g)*x^3 - 8*(16*(5248*c^7*d^4*e^3 -
6296*b*c^6*d^3*e^4 - 924*b^2*c^5*d^2*e^5 + 222*b^3*c^4*d*e^6 - 21*b^4*c^3*e^7)*
f + (22528*c^7*d^5*e^2 - 5904*b*c^6*d^4*e^3 - 25888*b^2*c^5*d^3*e^4 + 11496*b^3*
c^4*d^2*e^5 - 2568*b^4*c^3*d*e^6 + 231*b^5*c^2*e^7)*g)*x^2 + 16*(23552*c^7*d^6*e
- 78496*b*c^6*d^5*e^2 + 97424*b^2*c^5*d^4*e^3 - 60288*b^3*c^4*d^3*e^4 + 21168*b
^4*c^3*d^2*e^5 - 3990*b^5*c^2*d*e^6 + 315*b^6*c*e^7)*f + (212992*c^7*d^7 - 87340
8*b*c^6*d^6*e + 1519680*b^2*c^5*d^5*e^2 - 1433392*b^3*c^4*d^4*e^3 + 790176*b^4*c
^3*d^3*e^4 - 256788*b^5*c^2*d^2*e^5 + 45780*b^6*c*d*e^6 - 3465*b^7*e^7)*g - 2*(1
6*(7840*c^7*d^5*e^2 + 1392*b*c^6*d^4*e^3 - 13984*b^2*c^5*d^3*e^4 + 5760*b^3*c^4*
d^2*e^5 - 1218*b^4*c^3*d*e^6 + 105*b^5*c^2*e^7)*f - (60480*c^7*d^6*e - 208832*b*
c^6*d^5*e^2 + 278416*b^2*c^5*d^4*e^3 - 188384*b^3*c^4*d^3*e^4 + 70668*b^4*c^3*d^
2*e^5 - 14028*b^5*c^2*d*e^6 + 1155*b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*sqrt(c) - 315*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^
6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14*
b^6*c^2*d*e^7 - b^7*c*e^8)*f + (768*c^8*d^8 - 4096*b*c^7*d^7*e + 8960*b^2*c^6*d^
6*e^2 - 10752*b^3*c^5*d^5*e^3 + 7840*b^4*c^4*d^4*e^4 - 3584*b^5*c^3*d^3*e^5 + 10
08*b^6*c^2*d^2*e^6 - 160*b^7*c*d*e^7 + 11*b^8*e^8)*g)*arctan(1/2*(2*c*e*x + b*e)
/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c))))/(c^(13/2)*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.325317, size = 1, normalized size = 0. \[ \mathit{Done} \]

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

[Out]

Done