3.2332 \(\int (d+e x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=321 \[ \frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}+\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b
+ 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) + ((2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*
e^2 - 4*c*e*(2*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(128*c^4) + (e*(
d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 + 21*b^2*e^2 - 2*c*e
*(49*b*d + 8*a*e) + 30*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(280*c^3) +
(3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))

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Rubi [A]  time = 0.814443, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{1024 c^5}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )}{128 c^4}+\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-2 c e (8 a e+49 b d)+21 b^2 e^2+30 c e x (2 c d-b e)+128 c^2 d^2\right )}{280 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b
+ 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) + ((2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*
e^2 - 4*c*e*(2*b*d + a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(128*c^4) + (e*(
d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) + (e*(128*c^2*d^2 + 21*b^2*e^2 - 2*c*e
*(49*b*d + 8*a*e) + 30*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(280*c^3) +
(3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))

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Rubi in Sympy [A]  time = 80.2735, size = 335, normalized size = 1.04 \[ \frac{e \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7 c} + \frac{e \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 12 a c e^{2} + \frac{63 b^{2} e^{2}}{4} - \frac{147 b c d e}{2} + 96 c^{2} d^{2} - \frac{45 c e x \left (b e - 2 c d\right )}{2}\right )}{210 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 4 a c e^{2} + 3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{128 c^{4}} + \frac{3 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + 3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{1024 c^{5}} - \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \left (- 4 a c e^{2} + 3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)**2*(a + b*x + c*x**2)**(5/2)/(7*c) + e*(a + b*x + c*x**2)**(5/2)*(-1
2*a*c*e**2 + 63*b**2*e**2/4 - 147*b*c*d*e/2 + 96*c**2*d**2 - 45*c*e*x*(b*e - 2*c
*d)/2)/(210*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(3/2)*(-4*a*c*
e**2 + 3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/(128*c**4) + 3*(b + 2*c*x)*(-4*a*c
+ b**2)*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)*(-4*a*c*e**2 + 3*b**2*e**2 - 8*b*c
*d*e + 8*c**2*d**2)/(1024*c**5) - 3*(-4*a*c + b**2)**2*(b*e - 2*c*d)*(-4*a*c*e**
2 + 3*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a +
b*x + c*x**2)))/(2048*c**(11/2))

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Mathematica [A]  time = 0.993328, size = 506, normalized size = 1.58 \[ \frac{-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 b^2 c^2 \left (343 a^2 e^3-2 a c e \left (525 d^2+189 d e x+31 e^2 x^2\right )+2 c^2 x \left (35 d^3+42 d^2 e x+21 d e^2 x^2+4 e^3 x^3\right )\right )+32 b c^3 \left (a^2 e^2 (567 d+73 e x)-2 a c \left (175 d^3+147 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )-4 c^2 x^2 \left (105 d^3+231 d^2 e x+182 d e^2 x^2+50 e^3 x^3\right )\right )-64 c^3 \left (-32 a^3 e^3+a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )+2 a c^2 x \left (175 d^3+336 d^2 e x+245 d e^2 x^2+64 e^3 x^3\right )+4 c^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )-28 b^4 c e \left (c \left (90 d^2+35 d e x+6 e^2 x^2\right )-90 a e^2\right )+16 b^3 c^2 \left (c \left (105 d^3+105 d^2 e x+49 d e^2 x^2+9 e^3 x^3\right )-7 a e^2 (95 d+13 e x)\right )-315 b^6 e^3+210 b^5 c e^2 (7 d+e x)\right )-105 \left (b^2-4 a c\right )^2 (b e-2 c d) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{71680 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-315*b^6*e^3 + 210*b^5*c*e^2*(7*d + e*x) - 28
*b^4*c*e*(-90*a*e^2 + c*(90*d^2 + 35*d*e*x + 6*e^2*x^2)) - 16*b^2*c^2*(343*a^2*e
^3 - 2*a*c*e*(525*d^2 + 189*d*e*x + 31*e^2*x^2) + 2*c^2*x*(35*d^3 + 42*d^2*e*x +
21*d*e^2*x^2 + 4*e^3*x^3)) + 16*b^3*c^2*(-7*a*e^2*(95*d + 13*e*x) + c*(105*d^3
+ 105*d^2*e*x + 49*d*e^2*x^2 + 9*e^3*x^3)) + 32*b*c^3*(a^2*e^2*(567*d + 73*e*x)
- 2*a*c*(175*d^3 + 147*d^2*e*x + 63*d*e^2*x^2 + 11*e^3*x^3) - 4*c^2*x^2*(105*d^3
+ 231*d^2*e*x + 182*d*e^2*x^2 + 50*e^3*x^3)) - 64*c^3*(-32*a^3*e^3 + a^2*c*e*(3
36*d^2 + 105*d*e*x + 16*e^2*x^2) + 4*c^3*x^3*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2
+ 20*e^3*x^3) + 2*a*c^2*x*(175*d^3 + 336*d^2*e*x + 245*d*e^2*x^2 + 64*e^3*x^3))
) - 105*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a
*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(71680*c^(11/2))

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Maple [B]  time = 0.018, size = 1437, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*d*e^2*a/c^2*(c*x^2+b*x+a)^(3/2)*b-3/16*d*e^2*a^2/c*(c*x^2+b*x+a)^(1/2)*x-3
/32*d*e^2*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b+7/32*d*e^2*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x
-9/256*d^2*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/128*e^3*
b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-21/256*d*e^2*b^4/c^3
*(c*x^2+b*x+a)^(1/2)*x+3/16*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a-3/8*d^2*e*b/c*(c
*x^2+b*x+a)^(3/2)*x+1/16*e^3*b/c^2*a*(c*x^2+b*x+a)^(3/2)*x+3/32*e^3*b/c^2*a^2*(c
*x^2+b*x+a)^(1/2)*x-3/32*e^3*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a-1/8*d*e^2*a/c*(c*x^
2+b*x+a)^(3/2)*x+9/64*d^2*e*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x+27/64*d*e^2*b^2/c^(5/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+9/32*d^2*e*b^3/c^(5/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-3/28*e^3*b/c^2*x*(c*x^2+b*x+a)^(5/2)-3/16
*d^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*a+1/8*d^3/c*(c*x^2+
b*x+a)^(3/2)*b+3/8*d^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3
/5*d^2*e*(c*x^2+b*x+a)^(5/2)/c+3/128*d^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*b^4+1/7*e^3*x^2*(c*x^2+b*x+a)^(5/2)/c-9/2048*e^3*b^7/c^(11/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-3/64*e^3*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x-9/
32*d^2*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a-9/16*d^2*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))*a^2-45/256*d*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a-3/32*d^3/c*(c*x^2+b*x+a)^(1/2)*x*b^2+1/2*d*e^2*x*(c*x^2+b*x+a
)^(5/2)/c-7/20*d*e^2*b/c^2*(c*x^2+b*x+a)^(5/2)+3/40*e^3*b^2/c^3*(c*x^2+b*x+a)^(5
/2)-3/128*e^3*b^4/c^4*(c*x^2+b*x+a)^(3/2)+9/1024*e^3*b^6/c^5*(c*x^2+b*x+a)^(1/2)
-2/35*e^3*a/c^2*(c*x^2+b*x+a)^(5/2)+3/8*d^3*(c*x^2+b*x+a)^(1/2)*x*a-3/64*d^3/c^2
*(c*x^2+b*x+a)^(1/2)*b^3+1/4*d^3*(c*x^2+b*x+a)^(3/2)*x-3/16*d^2*e*b^2/c^2*(c*x^2
+b*x+a)^(3/2)+9/128*d^2*e*b^4/c^3*(c*x^2+b*x+a)^(1/2)+3/8*d*e^2*b^2/c^2*(c*x^2+b
*x+a)^(1/2)*x*a-9/16*d^2*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a+9/512*e^3*b^5/c^4*(c*x^2+
b*x+a)^(1/2)*x-3/64*e^3*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a+1/32*e^3*b^2/c^3*a*(c*x^2+
b*x+a)^(3/2)+3/64*e^3*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)+3/32*e^3*b/c^(5/2)*a^3*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+21/512*e^3*b^5/c^(9/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*a+7/64*d*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)-21/512*d*e^
2*b^5/c^4*(c*x^2+b*x+a)^(1/2)+21/1024*d*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))-3/16*d*e^2*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))+3/16*d^3/c*(c*x^2+b*x+a)^(1/2)*b*a

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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*x^2 + b*x + a)^(3/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.376228, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/143360*(4*(5120*c^6*e^3*x^6 + 1280*(14*c^6*d*e^2 + 5*b*c^5*e^3)*x^5 + 128*(16
8*c^6*d^2*e + 182*b*c^5*d*e^2 + (b^2*c^4 + 64*a*c^5)*e^3)*x^4 - 560*(3*b^3*c^3 -
20*a*b*c^4)*d^3 + 168*(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*d^2*e - 14*(10
5*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*d*e^2 + (315*b^6 - 2520*a*b^4*c + 5488
*a^2*b^2*c^2 - 2048*a^3*c^3)*e^3 + 16*(560*c^6*d^3 + 1848*b*c^5*d^2*e + 14*(3*b^
2*c^4 + 140*a*c^5)*d*e^2 - (9*b^3*c^3 - 44*a*b*c^4)*e^3)*x^3 + 8*(1680*b*c^5*d^3
+ 168*(b^2*c^4 + 32*a*c^5)*d^2*e - 14*(7*b^3*c^3 - 36*a*b*c^4)*d*e^2 + (21*b^4*
c^2 - 124*a*b^2*c^3 + 128*a^2*c^4)*e^3)*x^2 + 2*(560*(b^2*c^4 + 20*a*c^5)*d^3 -
168*(5*b^3*c^3 - 28*a*b*c^4)*d^2*e + 14*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^
4)*d*e^2 - (105*b^5*c - 728*a*b^3*c^2 + 1168*a^2*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(c) + 105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3 - 24*(b^5*c^2 -
8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e + 2*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3
- 64*a^3*c^4)*d*e^2 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^3)
*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a
*c)*sqrt(c)))/c^(11/2), 1/71680*(2*(5120*c^6*e^3*x^6 + 1280*(14*c^6*d*e^2 + 5*b*
c^5*e^3)*x^5 + 128*(168*c^6*d^2*e + 182*b*c^5*d*e^2 + (b^2*c^4 + 64*a*c^5)*e^3)*
x^4 - 560*(3*b^3*c^3 - 20*a*b*c^4)*d^3 + 168*(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a
^2*c^4)*d^2*e - 14*(105*b^5*c - 760*a*b^3*c^2 + 1296*a^2*b*c^3)*d*e^2 + (315*b^6
- 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^3)*e^3 + 16*(560*c^6*d^3 + 1848*
b*c^5*d^2*e + 14*(3*b^2*c^4 + 140*a*c^5)*d*e^2 - (9*b^3*c^3 - 44*a*b*c^4)*e^3)*x
^3 + 8*(1680*b*c^5*d^3 + 168*(b^2*c^4 + 32*a*c^5)*d^2*e - 14*(7*b^3*c^3 - 36*a*b
*c^4)*d*e^2 + (21*b^4*c^2 - 124*a*b^2*c^3 + 128*a^2*c^4)*e^3)*x^2 + 2*(560*(b^2*
c^4 + 20*a*c^5)*d^3 - 168*(5*b^3*c^3 - 28*a*b*c^4)*d^2*e + 14*(35*b^4*c^2 - 216*
a*b^2*c^3 + 240*a^2*c^4)*d*e^2 - (105*b^5*c - 728*a*b^3*c^2 + 1168*a^2*b*c^3)*e^
3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 105*(16*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c
^5)*d^3 - 24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e + 2*(7*b^6*c - 60*a*b^
4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^2 - (3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c
^2 - 64*a^3*b*c^3)*e^3)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c
)))/(sqrt(-c)*c^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3*(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.223187, size = 965, normalized size = 3.01 \[ \frac{1}{35840} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \, c x e^{3} + \frac{14 \, c^{7} d e^{2} + 5 \, b c^{6} e^{3}}{c^{6}}\right )} x + \frac{168 \, c^{7} d^{2} e + 182 \, b c^{6} d e^{2} + b^{2} c^{5} e^{3} + 64 \, a c^{6} e^{3}}{c^{6}}\right )} x + \frac{560 \, c^{7} d^{3} + 1848 \, b c^{6} d^{2} e + 42 \, b^{2} c^{5} d e^{2} + 1960 \, a c^{6} d e^{2} - 9 \, b^{3} c^{4} e^{3} + 44 \, a b c^{5} e^{3}}{c^{6}}\right )} x + \frac{1680 \, b c^{6} d^{3} + 168 \, b^{2} c^{5} d^{2} e + 5376 \, a c^{6} d^{2} e - 98 \, b^{3} c^{4} d e^{2} + 504 \, a b c^{5} d e^{2} + 21 \, b^{4} c^{3} e^{3} - 124 \, a b^{2} c^{4} e^{3} + 128 \, a^{2} c^{5} e^{3}}{c^{6}}\right )} x + \frac{560 \, b^{2} c^{5} d^{3} + 11200 \, a c^{6} d^{3} - 840 \, b^{3} c^{4} d^{2} e + 4704 \, a b c^{5} d^{2} e + 490 \, b^{4} c^{3} d e^{2} - 3024 \, a b^{2} c^{4} d e^{2} + 3360 \, a^{2} c^{5} d e^{2} - 105 \, b^{5} c^{2} e^{3} + 728 \, a b^{3} c^{3} e^{3} - 1168 \, a^{2} b c^{4} e^{3}}{c^{6}}\right )} x - \frac{1680 \, b^{3} c^{4} d^{3} - 11200 \, a b c^{5} d^{3} - 2520 \, b^{4} c^{3} d^{2} e + 16800 \, a b^{2} c^{4} d^{2} e - 21504 \, a^{2} c^{5} d^{2} e + 1470 \, b^{5} c^{2} d e^{2} - 10640 \, a b^{3} c^{3} d e^{2} + 18144 \, a^{2} b c^{4} d e^{2} - 315 \, b^{6} c e^{3} + 2520 \, a b^{4} c^{2} e^{3} - 5488 \, a^{2} b^{2} c^{3} e^{3} + 2048 \, a^{3} c^{4} e^{3}}{c^{6}}\right )} - \frac{3 \,{\left (16 \, b^{4} c^{3} d^{3} - 128 \, a b^{2} c^{4} d^{3} + 256 \, a^{2} c^{5} d^{3} - 24 \, b^{5} c^{2} d^{2} e + 192 \, a b^{3} c^{3} d^{2} e - 384 \, a^{2} b c^{4} d^{2} e + 14 \, b^{6} c d e^{2} - 120 \, a b^{4} c^{2} d e^{2} + 288 \, a^{2} b^{2} c^{3} d e^{2} - 128 \, a^{3} c^{4} d e^{2} - 3 \, b^{7} e^{3} + 28 \, a b^{5} c e^{3} - 80 \, a^{2} b^{3} c^{2} e^{3} + 64 \, a^{3} b c^{3} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*c*x*e^3 + (14*c^7*d*e^2 + 5*b*c
^6*e^3)/c^6)*x + (168*c^7*d^2*e + 182*b*c^6*d*e^2 + b^2*c^5*e^3 + 64*a*c^6*e^3)/
c^6)*x + (560*c^7*d^3 + 1848*b*c^6*d^2*e + 42*b^2*c^5*d*e^2 + 1960*a*c^6*d*e^2 -
9*b^3*c^4*e^3 + 44*a*b*c^5*e^3)/c^6)*x + (1680*b*c^6*d^3 + 168*b^2*c^5*d^2*e +
5376*a*c^6*d^2*e - 98*b^3*c^4*d*e^2 + 504*a*b*c^5*d*e^2 + 21*b^4*c^3*e^3 - 124*a
*b^2*c^4*e^3 + 128*a^2*c^5*e^3)/c^6)*x + (560*b^2*c^5*d^3 + 11200*a*c^6*d^3 - 84
0*b^3*c^4*d^2*e + 4704*a*b*c^5*d^2*e + 490*b^4*c^3*d*e^2 - 3024*a*b^2*c^4*d*e^2
+ 3360*a^2*c^5*d*e^2 - 105*b^5*c^2*e^3 + 728*a*b^3*c^3*e^3 - 1168*a^2*b*c^4*e^3)
/c^6)*x - (1680*b^3*c^4*d^3 - 11200*a*b*c^5*d^3 - 2520*b^4*c^3*d^2*e + 16800*a*b
^2*c^4*d^2*e - 21504*a^2*c^5*d^2*e + 1470*b^5*c^2*d*e^2 - 10640*a*b^3*c^3*d*e^2
+ 18144*a^2*b*c^4*d*e^2 - 315*b^6*c*e^3 + 2520*a*b^4*c^2*e^3 - 5488*a^2*b^2*c^3*
e^3 + 2048*a^3*c^4*e^3)/c^6) - 3/2048*(16*b^4*c^3*d^3 - 128*a*b^2*c^4*d^3 + 256*
a^2*c^5*d^3 - 24*b^5*c^2*d^2*e + 192*a*b^3*c^3*d^2*e - 384*a^2*b*c^4*d^2*e + 14*
b^6*c*d*e^2 - 120*a*b^4*c^2*d*e^2 + 288*a^2*b^2*c^3*d*e^2 - 128*a^3*c^4*d*e^2 -
3*b^7*e^3 + 28*a*b^5*c*e^3 - 80*a^2*b^3*c^2*e^3 + 64*a^3*b*c^3*e^3)*ln(abs(-2*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)