3.1558 \(\int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=242 \[ \frac{e \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{256 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{96 c^3}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (12 a e+7 b d)+7 b^2 e^2+10 c e x (2 c d-b e)+24 c^2 d^2\right )}{210 c^2}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \]

[Out]

-((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) +
((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(96*c^3) +
(2*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/7 + ((24*c^2*d^2 + 7*b^2*e^2 - 2*c*e*(7*
b*d + 12*a*e) + 10*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(210*c^2) + ((b
^2 - 4*a*c)^3*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])])/(512*c^(9/2))

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Rubi [A]  time = 0.87853, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{e \left (b^2-4 a c\right )^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}-\frac{e \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{256 c^4}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{96 c^3}+\frac{\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (12 a e+7 b d)+7 b^2 e^2+10 c e x (2 c d-b e)+24 c^2 d^2\right )}{210 c^2}+\frac{2}{7} (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)^2*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(256*c^4) +
((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(96*c^3) +
(2*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/7 + ((24*c^2*d^2 + 7*b^2*e^2 - 2*c*e*(7*
b*d + 12*a*e) + 10*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(5/2))/(210*c^2) + ((b
^2 - 4*a*c)^3*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^
2])])/(512*c^(9/2))

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Rubi in Sympy [A]  time = 66.1147, size = 236, normalized size = 0.98 \[ \frac{2 \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7} + \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 24 a c e^{2} + 7 b^{2} e^{2} - 14 b c d e + 24 c^{2} d^{2} - 10 c e x \left (b e - 2 c d\right )\right )}{210 c^{2}} - \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{96 c^{3}} + \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{256 c^{4}} - \frac{e \left (- 4 a c + b^{2}\right )^{3} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} \]

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)**(3/2),x)

[Out]

2*(d + e*x)**2*(a + b*x + c*x**2)**(5/2)/7 + (a + b*x + c*x**2)**(5/2)*(-24*a*c*
e**2 + 7*b**2*e**2 - 14*b*c*d*e + 24*c**2*d**2 - 10*c*e*x*(b*e - 2*c*d))/(210*c*
*2) - e*(b + 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*(a + b*x + c*x**2)**(3/2)/(96*
c**3) + e*(b + 2*c*x)*(-4*a*c + b**2)**2*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)/(2
56*c**4) - e*(-4*a*c + b**2)**3*(b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(
a + b*x + c*x**2)))/(512*c**(9/2))

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Mathematica [A]  time = 0.580407, size = 386, normalized size = 1.6 \[ \frac{\sqrt{a+x (b+c x)} \left (-3072 a^3 c^3 e^2+48 a^2 c^2 \left (77 b^2 e^2-2 b c e (77 d+19 e x)+4 c^2 \left (56 d^2+35 d e x+8 e^2 x^2\right )\right )+32 a c \left (-35 b^4 e^2+7 b^3 c e (10 d+3 e x)-3 b^2 c^2 e x (14 d+5 e x)+2 b c^3 x \left (336 d^2+399 d e x+139 e^2 x^2\right )+4 c^4 x^2 \left (168 d^2+245 d e x+96 e^2 x^2\right )\right )+105 b^6 e^2-70 b^5 c e (3 d+e x)+28 b^4 c^2 e x (5 d+2 e x)-16 b^3 c^3 e x^2 (7 d+3 e x)+32 b^2 c^4 x^2 \left (336 d^2+483 d e x+188 e^2 x^2\right )+256 b c^5 x^3 \left (84 d^2+133 d e x+55 e^2 x^2\right )+512 c^6 x^4 \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )}{26880 c^4}-\frac{e \left (b^2-4 a c\right )^3 (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{512 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(105*b^6*e^2 - 3072*a^3*c^3*e^2 - 70*b^5*c*e*(3*d + e*x)
+ 28*b^4*c^2*e*x*(5*d + 2*e*x) - 16*b^3*c^3*e*x^2*(7*d + 3*e*x) + 512*c^6*x^4*(2
1*d^2 + 35*d*e*x + 15*e^2*x^2) + 256*b*c^5*x^3*(84*d^2 + 133*d*e*x + 55*e^2*x^2)
+ 32*b^2*c^4*x^2*(336*d^2 + 483*d*e*x + 188*e^2*x^2) + 48*a^2*c^2*(77*b^2*e^2 -
2*b*c*e*(77*d + 19*e*x) + 4*c^2*(56*d^2 + 35*d*e*x + 8*e^2*x^2)) + 32*a*c*(-35*
b^4*e^2 + 7*b^3*c*e*(10*d + 3*e*x) - 3*b^2*c^2*e*x*(14*d + 5*e*x) + 4*c^4*x^2*(1
68*d^2 + 245*d*e*x + 96*e^2*x^2) + 2*b*c^3*x*(336*d^2 + 399*d*e*x + 139*e^2*x^2)
)))/(26880*c^4) - ((b^2 - 4*a*c)^3*e*(-2*c*d + b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sq
rt[a + x*(b + c*x)]])/(512*c^(9/2))

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Maple [B]  time = 0.018, size = 895, normalized size = 3.7 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

1/8*b^2/c*(c*x^2+b*x+a)^(1/2)*x*a*d*e-1/96*b^4/c^3*(c*x^2+b*x+a)^(3/2)*e^2+1/256
*b^6/c^4*(c*x^2+b*x+a)^(1/2)*e^2-1/512*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))*e^2+1/30*b^2/c^2*(c*x^2+b*x+a)^(5/2)*e^2+2/3*d*e*x*(c*x^2+b*x+a)^
(5/2)-4/35*e^2/c*a*(c*x^2+b*x+a)^(5/2)-1/21*x*(c*x^2+b*x+a)^(5/2)/c*b*e^2-1/15*b
/c*(c*x^2+b*x+a)^(5/2)*d*e+2/7*x^2*e^2*(c*x^2+b*x+a)^(5/2)-1/8*a^2/c*(c*x^2+b*x+
a)^(1/2)*b*d*e-1/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a*e^2+1/12*a/c*(c*x^2+b*x+a)^(
3/2)*x*b*e^2+3/16*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*d*
e-3/64*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e+1/8*a^2/c*(
c*x^2+b*x+a)^(1/2)*x*b*e^2+1/24*b^2/c*(c*x^2+b*x+a)^(3/2)*x*d*e+1/16*b^3/c^2*(c*
x^2+b*x+a)^(1/2)*a*d*e-1/48*b^3/c^2*(c*x^2+b*x+a)^(3/2)*x*e^2+1/48*b^3/c^2*(c*x^
2+b*x+a)^(3/2)*d*e+1/128*b^5/c^3*(c*x^2+b*x+a)^(1/2)*x*e^2-1/32*b^4/c^3*(c*x^2+b
*x+a)^(1/2)*a*e^2+1/24*a/c^2*(c*x^2+b*x+a)^(3/2)*b^2*e^2-1/12*a/c*(c*x^2+b*x+a)^
(3/2)*b*d*e-1/64*b^4/c^2*(c*x^2+b*x+a)^(1/2)*x*d*e-1/128*b^5/c^3*(c*x^2+b*x+a)^(
1/2)*d*e-3/32*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2*e^2-1/
6*a*(c*x^2+b*x+a)^(3/2)*x*d*e+3/128*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*a*e^2-1/4*a^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*
e+1/256*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e+1/16*a^2/c^2
*(c*x^2+b*x+a)^(1/2)*b^2*e^2+1/8*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a
)^(1/2))*b*e^2+2/5*d^2*(c*x^2+b*x+a)^(5/2)-1/4*a^2*(c*x^2+b*x+a)^(1/2)*x*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)^(3/2)*(2*c*x + b)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.41626, 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)*(2*c*x + b)*(e*x + d)^2,x, algorithm="fricas")

[Out]

[1/107520*(4*(7680*c^6*e^2*x^6 + 10752*a^2*c^4*d^2 + 1280*(14*c^6*d*e + 11*b*c^5
*e^2)*x^5 + 128*(84*c^6*d^2 + 266*b*c^5*d*e + (47*b^2*c^4 + 96*a*c^5)*e^2)*x^4 +
16*(1344*b*c^5*d^2 + 14*(69*b^2*c^4 + 140*a*c^5)*d*e - (3*b^3*c^3 - 556*a*b*c^4
)*e^2)*x^3 - 14*(15*b^5*c - 160*a*b^3*c^2 + 528*a^2*b*c^3)*d*e + (105*b^6 - 1120
*a*b^4*c + 3696*a^2*b^2*c^2 - 3072*a^3*c^3)*e^2 + 8*(1344*(b^2*c^4 + 2*a*c^5)*d^
2 - 14*(b^3*c^3 - 228*a*b*c^4)*d*e + (7*b^4*c^2 - 60*a*b^2*c^3 + 192*a^2*c^4)*e^
2)*x^2 + 2*(10752*a*b*c^4*d^2 + 14*(5*b^4*c^2 - 48*a*b^2*c^3 + 240*a^2*c^4)*d*e
- (35*b^5*c - 336*a*b^3*c^2 + 912*a^2*b*c^3)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(
c) + 105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12
*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*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^(9/2), 1/53760*(2*
(7680*c^6*e^2*x^6 + 10752*a^2*c^4*d^2 + 1280*(14*c^6*d*e + 11*b*c^5*e^2)*x^5 + 1
28*(84*c^6*d^2 + 266*b*c^5*d*e + (47*b^2*c^4 + 96*a*c^5)*e^2)*x^4 + 16*(1344*b*c
^5*d^2 + 14*(69*b^2*c^4 + 140*a*c^5)*d*e - (3*b^3*c^3 - 556*a*b*c^4)*e^2)*x^3 -
14*(15*b^5*c - 160*a*b^3*c^2 + 528*a^2*b*c^3)*d*e + (105*b^6 - 1120*a*b^4*c + 36
96*a^2*b^2*c^2 - 3072*a^3*c^3)*e^2 + 8*(1344*(b^2*c^4 + 2*a*c^5)*d^2 - 14*(b^3*c
^3 - 228*a*b*c^4)*d*e + (7*b^4*c^2 - 60*a*b^2*c^3 + 192*a^2*c^4)*e^2)*x^2 + 2*(1
0752*a*b*c^4*d^2 + 14*(5*b^4*c^2 - 48*a*b^2*c^3 + 240*a^2*c^4)*d*e - (35*b^5*c -
336*a*b^3*c^2 + 912*a^2*b*c^3)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 105*(2*
(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e - (b^7 - 12*a*b^5*c + 4
8*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^2)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 +
b*x + a)*c)))/(sqrt(-c)*c^4)]

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

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)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.286244, size = 721, normalized size = 2.98 \[ \frac{1}{26880} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (6 \, c^{2} x e^{2} + \frac{14 \, c^{8} d e + 11 \, b c^{7} e^{2}}{c^{6}}\right )} x + \frac{84 \, c^{8} d^{2} + 266 \, b c^{7} d e + 47 \, b^{2} c^{6} e^{2} + 96 \, a c^{7} e^{2}}{c^{6}}\right )} x + \frac{1344 \, b c^{7} d^{2} + 966 \, b^{2} c^{6} d e + 1960 \, a c^{7} d e - 3 \, b^{3} c^{5} e^{2} + 556 \, a b c^{6} e^{2}}{c^{6}}\right )} x + \frac{1344 \, b^{2} c^{6} d^{2} + 2688 \, a c^{7} d^{2} - 14 \, b^{3} c^{5} d e + 3192 \, a b c^{6} d e + 7 \, b^{4} c^{4} e^{2} - 60 \, a b^{2} c^{5} e^{2} + 192 \, a^{2} c^{6} e^{2}}{c^{6}}\right )} x + \frac{10752 \, a b c^{6} d^{2} + 70 \, b^{4} c^{4} d e - 672 \, a b^{2} c^{5} d e + 3360 \, a^{2} c^{6} d e - 35 \, b^{5} c^{3} e^{2} + 336 \, a b^{3} c^{4} e^{2} - 912 \, a^{2} b c^{5} e^{2}}{c^{6}}\right )} x + \frac{10752 \, a^{2} c^{6} d^{2} - 210 \, b^{5} c^{3} d e + 2240 \, a b^{3} c^{4} d e - 7392 \, a^{2} b c^{5} d e + 105 \, b^{6} c^{2} e^{2} - 1120 \, a b^{4} c^{3} e^{2} + 3696 \, a^{2} b^{2} c^{4} e^{2} - 3072 \, a^{3} c^{5} e^{2}}{c^{6}}\right )} - \frac{{\left (2 \, b^{6} c d e - 24 \, a b^{4} c^{2} d e + 96 \, a^{2} b^{2} c^{3} d e - 128 \, a^{3} c^{4} d e - b^{7} e^{2} + 12 \, a b^{5} c e^{2} - 48 \, a^{2} b^{3} c^{2} e^{2} + 64 \, a^{3} b c^{3} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{512 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/26880*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(6*c^2*x*e^2 + (14*c^8*d*e + 11*b*
c^7*e^2)/c^6)*x + (84*c^8*d^2 + 266*b*c^7*d*e + 47*b^2*c^6*e^2 + 96*a*c^7*e^2)/c
^6)*x + (1344*b*c^7*d^2 + 966*b^2*c^6*d*e + 1960*a*c^7*d*e - 3*b^3*c^5*e^2 + 556
*a*b*c^6*e^2)/c^6)*x + (1344*b^2*c^6*d^2 + 2688*a*c^7*d^2 - 14*b^3*c^5*d*e + 319
2*a*b*c^6*d*e + 7*b^4*c^4*e^2 - 60*a*b^2*c^5*e^2 + 192*a^2*c^6*e^2)/c^6)*x + (10
752*a*b*c^6*d^2 + 70*b^4*c^4*d*e - 672*a*b^2*c^5*d*e + 3360*a^2*c^6*d*e - 35*b^5
*c^3*e^2 + 336*a*b^3*c^4*e^2 - 912*a^2*b*c^5*e^2)/c^6)*x + (10752*a^2*c^6*d^2 -
210*b^5*c^3*d*e + 2240*a*b^3*c^4*d*e - 7392*a^2*b*c^5*d*e + 105*b^6*c^2*e^2 - 11
20*a*b^4*c^3*e^2 + 3696*a^2*b^2*c^4*e^2 - 3072*a^3*c^5*e^2)/c^6) - 1/512*(2*b^6*
c*d*e - 24*a*b^4*c^2*d*e + 96*a^2*b^2*c^3*d*e - 128*a^3*c^4*d*e - b^7*e^2 + 12*a
*b^5*c*e^2 - 48*a^2*b^3*c^2*e^2 + 64*a^3*b*c^3*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)