3.2333 \(\int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=257 \[ \frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}+\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]

[Out]

-((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*Sqrt[
a + b*x + c*x^2])/(512*c^4) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b
+ 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d - b*e)*(a + b*x + c*x
^2)^(5/2))/(60*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*
c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[
c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2))

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Rubi [A]  time = 0.730328, antiderivative size = 257, 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 \[ \frac{\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}+\frac{7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b + 2*c*x)*Sqrt[
a + b*x + c*x^2])/(512*c^4) + ((24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*(b
+ 2*c*x)*(a + b*x + c*x^2)^(3/2))/(192*c^3) + (7*e*(2*c*d - b*e)*(a + b*x + c*x
^2)^(5/2))/(60*c^2) + (e*(d + e*x)*(a + b*x + c*x^2)^(5/2))/(6*c) + ((b^2 - 4*a*
c)^2*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[
c]*Sqrt[a + b*x + c*x^2])])/(1024*c^(9/2))

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Rubi in Sympy [A]  time = 59.1985, size = 260, normalized size = 1.01 \[ \frac{e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{6 c} - \frac{7 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{60 c^{2}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 4 a c e^{2} + 7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{192 c^{3}} - \frac{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + 7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{512 c^{4}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c e^{2} + 7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)*(a + b*x + c*x**2)**(5/2)/(6*c) - 7*e*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(5/2)/(60*c**2) + (b + 2*c*x)*(a + b*x + c*x**2)**(3/2)*(-4*a*c*e**2 + 7*b**
2*e**2 - 24*b*c*d*e + 24*c**2*d**2)/(192*c**3) - (b + 2*c*x)*(-4*a*c + b**2)*sqr
t(a + b*x + c*x**2)*(-4*a*c*e**2 + 7*b**2*e**2 - 24*b*c*d*e + 24*c**2*d**2)/(512
*c**4) + (-4*a*c + b**2)**2*(-4*a*c*e**2 + 7*b**2*e**2 - 24*b*c*d*e + 24*c**2*d*
*2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(1024*c**(9/2))

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Mathematica [A]  time = 0.586702, size = 334, normalized size = 1.3 \[ \frac{15 \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 b c^2 \left (-81 a^2 e^2+6 a c \left (25 d^2+14 d e x+3 e^2 x^2\right )+4 c^2 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )\right )-32 c^3 \left (3 a^2 e (32 d+5 e x)+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+8 b^3 c \left (c \left (45 d^2+30 d e x+7 e^2 x^2\right )-95 a e^2\right )+48 b^2 c^2 \left (a e (50 d+9 e x)-c x \left (5 d^2+4 d e x+e^2 x^2\right )\right )+105 b^5 e^2-10 b^4 c e (36 d+7 e x)\right )}{15360 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(105*b^5*e^2 - 10*b^4*c*e*(36*d + 7*e*x) + 48*
b^2*c^2*(a*e*(50*d + 9*e*x) - c*x*(5*d^2 + 4*d*e*x + e^2*x^2)) + 8*b^3*c*(-95*a*
e^2 + c*(45*d^2 + 30*d*e*x + 7*e^2*x^2)) - 16*b*c^2*(-81*a^2*e^2 + 6*a*c*(25*d^2
+ 14*d*e*x + 3*e^2*x^2) + 4*c^2*x^2*(45*d^2 + 66*d*e*x + 26*e^2*x^2)) - 32*c^3*
(3*a^2*e*(32*d + 5*e*x) + 4*c^2*x^3*(15*d^2 + 24*d*e*x + 10*e^2*x^2) + 2*a*c*x*(
75*d^2 + 96*d*e*x + 35*e^2*x^2))) + 15*(b^2 - 4*a*c)^2*(24*c^2*d^2 + 7*b^2*e^2 -
4*c*e*(6*b*d + a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(15360*c
^(9/2))

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Maple [B]  time = 0.015, size = 922, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*d*e*b/c*(c*x^2+b*x+a)^(1/2)*x*a+7/1024*e^2*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))+1/8*d^2/c*(c*x^2+b*x+a)^(3/2)*b+3/8*d^2*(c*x^2+b*x+a)^(1
/2)*x*a-3/64*d^2/c^2*(c*x^2+b*x+a)^(1/2)*b^3+3/8*d^2/c^(1/2)*ln((1/2*b+c*x)/c^(1
/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/128*d^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))*b^4+2/5*d*e*(c*x^2+b*x+a)^(5/2)/c-1/16*e^2*a^3/c^(3/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/6*e^2*x*(c*x^2+b*x+a)^(5/2)/c-7/60*e^2*b/c^2*(c*
x^2+b*x+a)^(5/2)+7/192*e^2*b^3/c^3*(c*x^2+b*x+a)^(3/2)-7/512*e^2*b^5/c^4*(c*x^2+
b*x+a)^(1/2)-1/4*d*e*b/c*(c*x^2+b*x+a)^(3/2)*x+3/32*d*e*b^3/c^2*(c*x^2+b*x+a)^(1
/2)*x-3/16*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a-3/8*d*e*b/c^(3/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/16*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))*a+1/8*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a-1/24*e^2*a/c*(c*x^2+b*
x+a)^(3/2)*x-1/8*d*e*b^2/c^2*(c*x^2+b*x+a)^(3/2)+3/64*d*e*b^4/c^3*(c*x^2+b*x+a)^
(1/2)-3/128*d*e*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+7/96*e^2
*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x-7/256*e^2*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x+1/16*e^2*
b^3/c^3*(c*x^2+b*x+a)^(1/2)*a+9/64*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^2-15/256*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^2/c*(c*x^2+b*x+a)^(1/2)*x*b^2-1/16*e^2*a^2/c*(c*x^2+b*x+a)^(1/2)*
x-1/32*e^2*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b+3/16*d^2/c*(c*x^2+b*x+a)^(1/2)*b*a+1/4*
d^2*(c*x^2+b*x+a)^(3/2)*x-1/48*e^2*a/c^2*(c*x^2+b*x+a)^(3/2)*b-3/16*d^2/c^(3/2)*
ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312853, 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)^2,x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*c^5*e^2*x^5 + 128*(24*c^5*d*e + 13*b*c^4*e^2)*x^4 + 16*(120*c^
5*d^2 + 264*b*c^4*d*e + (3*b^2*c^3 + 140*a*c^4)*e^2)*x^3 - 120*(3*b^3*c^2 - 20*a
*b*c^3)*d^2 + 24*(15*b^4*c - 100*a*b^2*c^2 + 128*a^2*c^3)*d*e - (105*b^5 - 760*a
*b^3*c + 1296*a^2*b*c^2)*e^2 + 8*(360*b*c^4*d^2 + 24*(b^2*c^3 + 32*a*c^4)*d*e -
(7*b^3*c^2 - 36*a*b*c^3)*e^2)*x^2 + 2*(120*(b^2*c^3 + 20*a*c^4)*d^2 - 24*(5*b^3*
c^2 - 28*a*b*c^3)*d*e + (35*b^4*c - 216*a*b^2*c^2 + 240*a^2*c^3)*e^2)*x)*sqrt(c*
x^2 + b*x + a)*sqrt(c) - 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 -
64*a^3*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), 1/15360*(2*(1280*c^5*e^2*x^5 + 128*(24*c^
5*d*e + 13*b*c^4*e^2)*x^4 + 16*(120*c^5*d^2 + 264*b*c^4*d*e + (3*b^2*c^3 + 140*a
*c^4)*e^2)*x^3 - 120*(3*b^3*c^2 - 20*a*b*c^3)*d^2 + 24*(15*b^4*c - 100*a*b^2*c^2
+ 128*a^2*c^3)*d*e - (105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*e^2 + 8*(360*b*c^
4*d^2 + 24*(b^2*c^3 + 32*a*c^4)*d*e - (7*b^3*c^2 - 36*a*b*c^3)*e^2)*x^2 + 2*(120
*(b^2*c^3 + 20*a*c^4)*d^2 - 24*(5*b^3*c^2 - 28*a*b*c^3)*d*e + (35*b^4*c - 216*a*
b^2*c^2 + 240*a^2*c^3)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 15*(24*(b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*d^2 - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e +
(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*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. \[ \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220995, size = 625, normalized size = 2.43 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{2} + \frac{24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac{120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac{360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac{120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac{360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} c^{3} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*e^2 + (24*c^6*d*e + 13*b*c^5*e^
2)/c^5)*x + (120*c^6*d^2 + 264*b*c^5*d*e + 3*b^2*c^4*e^2 + 140*a*c^5*e^2)/c^5)*x
+ (360*b*c^5*d^2 + 24*b^2*c^4*d*e + 768*a*c^5*d*e - 7*b^3*c^3*e^2 + 36*a*b*c^4*
e^2)/c^5)*x + (120*b^2*c^4*d^2 + 2400*a*c^5*d^2 - 120*b^3*c^3*d*e + 672*a*b*c^4*
d*e + 35*b^4*c^2*e^2 - 216*a*b^2*c^3*e^2 + 240*a^2*c^4*e^2)/c^5)*x - (360*b^3*c^
3*d^2 - 2400*a*b*c^4*d^2 - 360*b^4*c^2*d*e + 2400*a*b^2*c^3*d*e - 3072*a^2*c^4*d
*e + 105*b^5*c*e^2 - 760*a*b^3*c^2*e^2 + 1296*a^2*b*c^3*e^2)/c^5) - 1/1024*(24*b
^4*c^2*d^2 - 192*a*b^2*c^3*d^2 + 384*a^2*c^4*d^2 - 24*b^5*c*d*e + 192*a*b^3*c^2*
d*e - 384*a^2*b*c^3*d*e + 7*b^6*e^2 - 60*a*b^4*c*e^2 + 144*a^2*b^2*c^2*e^2 - 64*
a^3*c^3*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)