∖int(b + 2cx)(d + ex)2∖left(a + bx + cx2∖right)5∕2∖,dx

3.1566 \(\int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=289 \[ -\frac{5 e \left (b^2-4 a c\right )^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8192 c^5}-\frac{5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3072 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{192 c^3}+\frac{\left (a+b x+c x^2\right )^{7/2} \left (-2 c e (16 a e+9 b d)+9 b^2 e^2+14 c e x (2 c d-b e)+32 c^2 d^2\right )}{504 c^2}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2} \]

[Out]

(5*(b^2 - 4*a*c)^3*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^5)

- (5*(b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(3072

*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(192

*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/9 + ((32*c^2*d^2 + 9*b^2*e^2 - 2

*c*e*(9*b*d + 16*a*e) + 14*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(504*c^

2) - (5*(b^2 - 4*a*c)^4*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +

b*x + c*x^2])])/(16384*c^(11/2))

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Rubi [A] time = 1.07872, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{5 e \left (b^2-4 a c\right )^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16384 c^{11/2}}+\frac{5 e \left (b^2-4 a c\right )^3 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{8192 c^5}-\frac{5 e \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{3072 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{192 c^3}+\frac{\left (a+b x+c x^2\right )^{7/2} \left (-2 c e (16 a e+9 b d)+9 b^2 e^2+14 c e x (2 c d-b e)+32 c^2 d^2\right )}{504 c^2}+\frac{2}{9} (d+e x)^2 \left (a+b x+c x^2\right )^{7/2} \]

Antiderivative was successfully veriﬁed.

[In] Int[(b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^3*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(8192*c^5)

- (5*(b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(3072

*c^4) + ((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(192

*c^3) + (2*(d + e*x)^2*(a + b*x + c*x^2)^(7/2))/9 + ((32*c^2*d^2 + 9*b^2*e^2 - 2

*c*e*(9*b*d + 16*a*e) + 14*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/2))/(504*c^

2) - (5*(b^2 - 4*a*c)^4*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a +

b*x + c*x^2])])/(16384*c^(11/2))

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Rubi in Sympy [A] time = 72.6094, size = 287, normalized size = 0.99 \[ \frac{2 \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9} + \frac{\left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (- 32 a c e^{2} + 9 b^{2} e^{2} - 18 b c d e + 32 c^{2} d^{2} - 14 c e x \left (b e - 2 c d\right )\right )}{504 c^{2}} - \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{192 c^{3}} + \frac{5 e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3072 c^{4}} - \frac{5 e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{3} \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{8192 c^{5}} + \frac{5 e \left (- 4 a c + b^{2}\right )^{4} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16384 c^{\frac{11}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((2*c*x+b)*(e*x+d)**2*(c*x**2+b*x+a)**(5/2),x)

[Out]

2*(d + e*x)**2*(a + b*x + c*x**2)**(7/2)/9 + (a + b*x + c*x**2)**(7/2)*(-32*a*c*

e**2 + 9*b**2*e**2 - 18*b*c*d*e + 32*c**2*d**2 - 14*c*e*x*(b*e - 2*c*d))/(504*c*

*2) - e*(b + 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(5/2)/(192

*c**3) + 5*e*(b + 2*c*x)*(-4*a*c + b**2)**2*(b*e - 2*c*d)*(a + b*x + c*x**2)**(3

/2)/(3072*c**4) - 5*e*(b + 2*c*x)*(-4*a*c + b**2)**3*(b*e - 2*c*d)*sqrt(a + b*x

+ c*x**2)/(8192*c**5) + 5*e*(-4*a*c + b**2)**4*(b*e - 2*c*d)*atanh((b + 2*c*x)/(

2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(16384*c**(11/2))

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Mathematica [B] time = 1.78332, size = 593, normalized size = 2.05 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-32768 a^4 c^4 e^2+64 a^3 c^3 \left (837 b^2 e^2-2 b c e (837 d+187 e x)+4 c^2 \left (576 d^2+315 d e x+64 e^2 x^2\right )\right )+48 a^2 c^2 \left (-511 b^4 e^2+2 b^3 c e (511 d+141 e x)-4 b^2 c^2 e x (141 d+46 e x)+16 b c^3 x \left (576 d^2+663 d e x+227 e^2 x^2\right )+32 c^4 x^2 \left (288 d^2+413 d e x+160 e^2 x^2\right )\right )+4 a c \left (1155 b^6 e^2-42 b^5 c e (55 d+17 e x)+12 b^4 c^2 e x (119 d+44 e x)-32 b^3 c^3 e x^2 (33 d+13 e x)+192 b^2 c^4 x^2 \left (576 d^2+815 d e x+314 e^2 x^2\right )+768 b c^5 x^3 \left (288 d^2+451 d e x+185 e^2 x^2\right )+512 c^6 x^4 \left (216 d^2+357 d e x+152 e^2 x^2\right )\right )-315 b^8 e^2+210 b^7 c e (3 d+e x)-84 b^6 c^2 e x (5 d+2 e x)+48 b^5 c^3 e x^2 (7 d+3 e x)-32 b^4 c^4 e x^3 (9 d+4 e x)+256 b^3 c^5 x^3 \left (576 d^2+897 d e x+367 e^2 x^2\right )+1536 b^2 c^6 x^4 \left (288 d^2+475 d e x+202 e^2 x^2\right )+2048 b c^7 x^5 \left (216 d^2+369 d e x+161 e^2 x^2\right )+4096 c^8 x^6 \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+315 e \left (b^2-4 a c\right )^4 (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{1032192 c^{11/2}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(b + 2*c*x)*(d + e*x)^2*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-315*b^8*e^2 - 32768*a^4*c^4*e^2 + 210*b^7*c*e

*(3*d + e*x) - 84*b^6*c^2*e*x*(5*d + 2*e*x) + 48*b^5*c^3*e*x^2*(7*d + 3*e*x) - 3

2*b^4*c^4*e*x^3*(9*d + 4*e*x) + 4096*c^8*x^6*(36*d^2 + 63*d*e*x + 28*e^2*x^2) +

2048*b*c^7*x^5*(216*d^2 + 369*d*e*x + 161*e^2*x^2) + 1536*b^2*c^6*x^4*(288*d^2 +

475*d*e*x + 202*e^2*x^2) + 256*b^3*c^5*x^3*(576*d^2 + 897*d*e*x + 367*e^2*x^2)

+ 64*a^3*c^3*(837*b^2*e^2 - 2*b*c*e*(837*d + 187*e*x) + 4*c^2*(576*d^2 + 315*d*e

*x + 64*e^2*x^2)) + 48*a^2*c^2*(-511*b^4*e^2 - 4*b^2*c^2*e*x*(141*d + 46*e*x) +

2*b^3*c*e*(511*d + 141*e*x) + 32*c^4*x^2*(288*d^2 + 413*d*e*x + 160*e^2*x^2) + 1

6*b*c^3*x*(576*d^2 + 663*d*e*x + 227*e^2*x^2)) + 4*a*c*(1155*b^6*e^2 - 32*b^3*c^

3*e*x^2*(33*d + 13*e*x) - 42*b^5*c*e*(55*d + 17*e*x) + 12*b^4*c^2*e*x*(119*d + 4

4*e*x) + 512*c^6*x^4*(216*d^2 + 357*d*e*x + 152*e^2*x^2) + 768*b*c^5*x^3*(288*d^

2 + 451*d*e*x + 185*e^2*x^2) + 192*b^2*c^4*x^2*(576*d^2 + 815*d*e*x + 314*e^2*x^

2))) + 315*(b^2 - 4*a*c)^4*e*(-2*c*d + b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x

*(b + c*x)]])/(1032192*c^(11/2))

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Maple [B] time = 0.02, size = 1358, normalized size = 4.7 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((2*c*x+b)*(e*x+d)^2*(c*x^2+b*x+a)^(5/2),x)

[Out]

5/64*a^4/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b*e^2+1/56*b^2/c^2*

(c*x^2+b*x+a)^(7/2)*e^2-1/192*b^4/c^3*(c*x^2+b*x+a)^(5/2)*e^2+5/3072*b^6/c^4*(c*

x^2+b*x+a)^(3/2)*e^2-5/8192*b^8/c^5*(c*x^2+b*x+a)^(1/2)*e^2+5/16384*b^9/c^(11/2)

*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^2+1/2*x*(c*x^2+b*x+a)^(7/2)*d*e-4

/63*e^2/c*a*(c*x^2+b*x+a)^(7/2)+15/2048*b^6/c^4*(c*x^2+b*x+a)^(1/2)*a*e^2+2/9*e^

2*x^2*(c*x^2+b*x+a)^(7/2)-5/1024*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a

)^(1/2))*a*e^2+1/48*a/c^2*(c*x^2+b*x+a)^(5/2)*b^2*e^2-1/28*b/c*(c*x^2+b*x+a)^(7/

2)*d*e+1/96*b^3/c^2*(c*x^2+b*x+a)^(5/2)*d*e-5/1536*b^5/c^3*(c*x^2+b*x+a)^(3/2)*d

*e-5/48*a^2*(c*x^2+b*x+a)^(3/2)*x*d*e-1/12*a*(c*x^2+b*x+a)^(5/2)*x*d*e-5/8192*b^

8/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e+5/96*a^2/c*(c*x^2+b*x+

a)^(3/2)*x*b*e^2-15/256*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2*e^2+5/2048*b^6/c^3*(c*

x^2+b*x+a)^(1/2)*x*d*e-5/192*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x*a*e^2-1/24*a/c*(c*x^2

+b*x+a)^(5/2)*b*d*e-5/64*a^3/c*(c*x^2+b*x+a)^(1/2)*b*d*e+15/1024*b^5/c^3*(c*x^2+

b*x+a)^(1/2)*x*a*e^2-5/768*b^4/c^2*(c*x^2+b*x+a)^(3/2)*x*d*e+5/192*b^3/c^2*(c*x^

2+b*x+a)^(3/2)*a*d*e+1/24*a/c*(c*x^2+b*x+a)^(5/2)*x*b*e^2-15/256*b^4/c^(5/2)*ln(

(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*e+5/512*b^6/c^(7/2)*ln((1/2*b+c*x

)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e+5/64*a^3/c*(c*x^2+b*x+a)^(1/2)*x*b*e^2+1/48

*b^2/c*(c*x^2+b*x+a)^(5/2)*x*d*e+5/32*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+

b*x+a)^(1/2))*a^3*d*e+15/256*b^3/c^2*(c*x^2+b*x+a)^(1/2)*a^2*d*e-5/96*a^2/c*(c*x

^2+b*x+a)^(3/2)*b*d*e-15/1024*b^5/c^3*(c*x^2+b*x+a)^(1/2)*a*d*e+2/7*(c*x^2+b*x+a

)^(7/2)*d^2-1/36*x*(c*x^2+b*x+a)^(7/2)/c*b*e^2-5/64*b^3/c^(5/2)*ln((1/2*b+c*x)/c

^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3*e^2+15/512*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c

*x^2+b*x+a)^(1/2))*a^2*e^2-5/4096*b^7/c^4*(c*x^2+b*x+a)^(1/2)*x*e^2-15/512*b^4/c

^3*(c*x^2+b*x+a)^(1/2)*a^2*e^2-5/32*a^4/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*

x+a)^(1/2))*d*e+5/4096*b^7/c^4*(c*x^2+b*x+a)^(1/2)*d*e-1/96*b^3/c^2*(c*x^2+b*x+a

)^(5/2)*x*e^2+5/1536*b^5/c^3*(c*x^2+b*x+a)^(3/2)*x*e^2-5/384*b^4/c^3*(c*x^2+b*x+

a)^(3/2)*a*e^2-5/32*a^3*(c*x^2+b*x+a)^(1/2)*x*d*e+5/192*a^2/c^2*(c*x^2+b*x+a)^(3

/2)*b^2*e^2+5/128*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b^2*e^2-15/512*b^4/c^2*(c*x^2+b*x+

a)^(1/2)*x*a*d*e+5/96*b^2/c*(c*x^2+b*x+a)^(3/2)*x*a*d*e+15/128*b^2/c*(c*x^2+b*x+

a)^(1/2)*x*a^2*d*e

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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)^(5/2)*(2*c*x + b)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 0.612821, 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)^(5/2)*(2*c*x + b)*(e*x + d)^2,x, algorithm="fricas")

[Out]

[1/2064384*(4*(114688*c^8*e^2*x^8 + 147456*a^3*c^5*d^2 + 14336*(18*c^8*d*e + 23*

b*c^7*e^2)*x^7 + 1024*(144*c^8*d^2 + 738*b*c^7*d*e + (303*b^2*c^6 + 304*a*c^7)*e

^2)*x^6 + 256*(1728*b*c^7*d^2 + 6*(475*b^2*c^6 + 476*a*c^7)*d*e + (367*b^3*c^5 +

2220*a*b*c^6)*e^2)*x^5 + 128*(3456*(b^2*c^6 + a*c^7)*d^2 + 6*(299*b^3*c^5 + 180

4*a*b*c^6)*d*e - (b^4*c^4 - 1884*a*b^2*c^5 - 1920*a^2*c^6)*e^2)*x^4 + 16*(9216*(

b^3*c^5 + 6*a*b*c^6)*d^2 - 6*(3*b^4*c^4 - 6520*a*b^2*c^5 - 6608*a^2*c^6)*d*e + (

9*b^5*c^3 - 104*a*b^3*c^4 + 10896*a^2*b*c^5)*e^2)*x^3 + 6*(105*b^7*c - 1540*a*b^

5*c^2 + 8176*a^2*b^3*c^3 - 17856*a^3*b*c^4)*d*e - (315*b^8 - 4620*a*b^6*c + 2452

8*a^2*b^4*c^2 - 53568*a^3*b^2*c^3 + 32768*a^4*c^4)*e^2 + 8*(55296*(a*b^2*c^5 + a

^2*c^6)*d^2 + 6*(7*b^5*c^3 - 88*a*b^3*c^4 + 10608*a^2*b*c^5)*d*e - (21*b^6*c^2 -

264*a*b^4*c^3 + 1104*a^2*b^2*c^4 - 2048*a^3*c^5)*e^2)*x^2 + 2*(221184*a^2*b*c^5

*d^2 - 6*(35*b^6*c^2 - 476*a*b^4*c^3 + 2256*a^2*b^2*c^4 - 6720*a^3*c^5)*d*e + (1

05*b^7*c - 1428*a*b^5*c^2 + 6768*a^2*b^3*c^3 - 11968*a^3*b*c^4)*e^2)*x)*sqrt(c*x

^2 + b*x + a)*sqrt(c) - 315*(2*(b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*

b^2*c^4 + 256*a^4*c^5)*d*e - (b^9 - 16*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^

3 + 256*a^4*b*c^4)*e^2)*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), 1/1032192*(2*(114688*c^8*e^2*x^8

+ 147456*a^3*c^5*d^2 + 14336*(18*c^8*d*e + 23*b*c^7*e^2)*x^7 + 1024*(144*c^8*d^2

+ 738*b*c^7*d*e + (303*b^2*c^6 + 304*a*c^7)*e^2)*x^6 + 256*(1728*b*c^7*d^2 + 6*

(475*b^2*c^6 + 476*a*c^7)*d*e + (367*b^3*c^5 + 2220*a*b*c^6)*e^2)*x^5 + 128*(345

6*(b^2*c^6 + a*c^7)*d^2 + 6*(299*b^3*c^5 + 1804*a*b*c^6)*d*e - (b^4*c^4 - 1884*a

*b^2*c^5 - 1920*a^2*c^6)*e^2)*x^4 + 16*(9216*(b^3*c^5 + 6*a*b*c^6)*d^2 - 6*(3*b^

4*c^4 - 6520*a*b^2*c^5 - 6608*a^2*c^6)*d*e + (9*b^5*c^3 - 104*a*b^3*c^4 + 10896*

a^2*b*c^5)*e^2)*x^3 + 6*(105*b^7*c - 1540*a*b^5*c^2 + 8176*a^2*b^3*c^3 - 17856*a

^3*b*c^4)*d*e - (315*b^8 - 4620*a*b^6*c + 24528*a^2*b^4*c^2 - 53568*a^3*b^2*c^3

+ 32768*a^4*c^4)*e^2 + 8*(55296*(a*b^2*c^5 + a^2*c^6)*d^2 + 6*(7*b^5*c^3 - 88*a*

b^3*c^4 + 10608*a^2*b*c^5)*d*e - (21*b^6*c^2 - 264*a*b^4*c^3 + 1104*a^2*b^2*c^4

- 2048*a^3*c^5)*e^2)*x^2 + 2*(221184*a^2*b*c^5*d^2 - 6*(35*b^6*c^2 - 476*a*b^4*c

^3 + 2256*a^2*b^2*c^4 - 6720*a^3*c^5)*d*e + (105*b^7*c - 1428*a*b^5*c^2 + 6768*a

^2*b^3*c^3 - 11968*a^3*b*c^4)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 315*(2*(b

^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*d*e - (b^9

- 16*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 + 256*a^4*b*c^4)*e^2)*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. \[ \int \left (b + 2 c x\right ) \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((2*c*x+b)*(e*x+d)**2*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((b + 2*c*x)*(d + e*x)**2*(a + b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A] time = 0.29173, size = 1112, normalized size = 3.85 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((c*x^2 + b*x + a)^(5/2)*(2*c*x + b)*(e*x + d)^2,x, algorithm="giac")

[Out]

1/516096*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(8*c^3*x*e^2 + (18*c^11*d*e

+ 23*b*c^10*e^2)/c^8)*x + (144*c^11*d^2 + 738*b*c^10*d*e + 303*b^2*c^9*e^2 + 30

4*a*c^10*e^2)/c^8)*x + (1728*b*c^10*d^2 + 2850*b^2*c^9*d*e + 2856*a*c^10*d*e + 3

67*b^3*c^8*e^2 + 2220*a*b*c^9*e^2)/c^8)*x + (3456*b^2*c^9*d^2 + 3456*a*c^10*d^2

+ 1794*b^3*c^8*d*e + 10824*a*b*c^9*d*e - b^4*c^7*e^2 + 1884*a*b^2*c^8*e^2 + 1920

*a^2*c^9*e^2)/c^8)*x + (9216*b^3*c^8*d^2 + 55296*a*b*c^9*d^2 - 18*b^4*c^7*d*e +

39120*a*b^2*c^8*d*e + 39648*a^2*c^9*d*e + 9*b^5*c^6*e^2 - 104*a*b^3*c^7*e^2 + 10

896*a^2*b*c^8*e^2)/c^8)*x + (55296*a*b^2*c^8*d^2 + 55296*a^2*c^9*d^2 + 42*b^5*c^

6*d*e - 528*a*b^3*c^7*d*e + 63648*a^2*b*c^8*d*e - 21*b^6*c^5*e^2 + 264*a*b^4*c^6

*e^2 - 1104*a^2*b^2*c^7*e^2 + 2048*a^3*c^8*e^2)/c^8)*x + (221184*a^2*b*c^8*d^2 -

210*b^6*c^5*d*e + 2856*a*b^4*c^6*d*e - 13536*a^2*b^2*c^7*d*e + 40320*a^3*c^8*d*

e + 105*b^7*c^4*e^2 - 1428*a*b^5*c^5*e^2 + 6768*a^2*b^3*c^6*e^2 - 11968*a^3*b*c^

7*e^2)/c^8)*x + (147456*a^3*c^8*d^2 + 630*b^7*c^4*d*e - 9240*a*b^5*c^5*d*e + 490

56*a^2*b^3*c^6*d*e - 107136*a^3*b*c^7*d*e - 315*b^8*c^3*e^2 + 4620*a*b^6*c^4*e^2

- 24528*a^2*b^4*c^5*e^2 + 53568*a^3*b^2*c^6*e^2 - 32768*a^4*c^7*e^2)/c^8) + 5/1

6384*(2*b^8*c*d*e - 32*a*b^6*c^2*d*e + 192*a^2*b^4*c^3*d*e - 512*a^3*b^2*c^4*d*e

+ 512*a^4*c^5*d*e - b^9*e^2 + 16*a*b^7*c*e^2 - 96*a^2*b^5*c^2*e^2 + 256*a^3*b^3

*c^3*e^2 - 256*a^4*b*c^4*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt

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