3.1547 \(\int (b+2 c x) (d+e x)^3 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=312 \[ -\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{256 c^4}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (6 c e x \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )-24 c^2 d e (16 a e+3 b d)+12 b c e^2 (11 a e+10 b d)-35 b^3 e^3+64 c^3 d^3\right )}{480 c^3}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{10 c} \]

[Out]

((b^2 - 4*a*c)*e*(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])/(256*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2
))/(10*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(3/2))/3 + ((64*c^3*d^3 - 35*b^3*e^3
+ 12*b*c*e^2*(10*b*d + 11*a*e) - 24*c^2*d*e*(3*b*d + 16*a*e) + 6*c*e*(8*c^2*d^2
+ 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(480*c^3) - ((b
^2 - 4*a*c)^2*e*(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])])/(512*c^(9/2))

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Rubi [A]  time = 1.00018, antiderivative size = 312, 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 \[ -\frac{e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{512 c^{9/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{256 c^4}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (6 c e x \left (-4 c e (5 a e+2 b d)+7 b^2 e^2+8 c^2 d^2\right )-24 c^2 d e (16 a e+3 b d)+12 b c e^2 (11 a e+10 b d)-35 b^3 e^3+64 c^3 d^3\right )}{480 c^3}+\frac{1}{3} (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}+\frac{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{10 c} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)*e*(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])/(256*c^4) + ((2*c*d - b*e)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2
))/(10*c) + ((d + e*x)^3*(a + b*x + c*x^2)^(3/2))/3 + ((64*c^3*d^3 - 35*b^3*e^3
+ 12*b*c*e^2*(10*b*d + 11*a*e) - 24*c^2*d*e*(3*b*d + 16*a*e) + 6*c*e*(8*c^2*d^2
+ 7*b^2*e^2 - 4*c*e*(2*b*d + 5*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(480*c^3) - ((b
^2 - 4*a*c)^2*e*(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])])/(512*c^(9/2))

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Rubi in Sympy [A]  time = 132.342, size = 330, normalized size = 1.06 \[ \frac{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3} - \frac{\left (d + e x\right )^{2} \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{10 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 99 a b c e^{3} + 288 a c^{2} d e^{2} + \frac{105 b^{3} e^{3}}{4} - 90 b^{2} c d e^{2} + 54 b c^{2} d^{2} e - 48 c^{3} d^{3} - \frac{9 c e x \left (- 20 a c e^{2} + 7 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{2}\right )}{360 c^{3}} + \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + 7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{256 c^{4}} - \frac{e \left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c e^{2} + 7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(1/2),x)

[Out]

(d + e*x)**3*(a + b*x + c*x**2)**(3/2)/3 - (d + e*x)**2*(b*e - 2*c*d)*(a + b*x +
c*x**2)**(3/2)/(10*c) - (a + b*x + c*x**2)**(3/2)*(-99*a*b*c*e**3 + 288*a*c**2*
d*e**2 + 105*b**3*e**3/4 - 90*b**2*c*d*e**2 + 54*b*c**2*d**2*e - 48*c**3*d**3 -
9*c*e*x*(-20*a*c*e**2 + 7*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/2)/(360*c**3) + e
*(b + 2*c*x)*(-4*a*c + b**2)*sqrt(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)/(256*c**4) - e*(-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)))/(512*c**(9/2))

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Mathematica [A]  time = 0.585615, size = 372, normalized size = 1.19 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-48 a^2 c^2 e^2 (-27 b e+64 c d+10 c e x)+8 a c \left (-95 b^3 e^3+6 b^2 c e^2 (50 d+9 e x)-12 b c^2 e \left (25 d^2+14 d e x+3 e^2 x^2\right )+8 c^3 \left (40 d^3+45 d^2 e x+24 d e^2 x^2+5 e^3 x^3\right )\right )+105 b^5 e^3-10 b^4 c e^2 (36 d+7 e x)+8 b^3 c^2 e \left (45 d^2+30 d e x+7 e^2 x^2\right )-48 b^2 c^3 e x \left (5 d^2+4 d e x+e^2 x^2\right )+64 b c^4 x \left (40 d^3+75 d^2 e x+54 d e^2 x^2+14 e^3 x^3\right )+128 c^5 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )-15 e \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{7680 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(105*b^5*e^3 - 10*b^4*c*e^2*(36*d + 7*e*x) - 48
*a^2*c^2*e^2*(64*c*d - 27*b*e + 10*c*e*x) - 48*b^2*c^3*e*x*(5*d^2 + 4*d*e*x + e^
2*x^2) + 8*b^3*c^2*e*(45*d^2 + 30*d*e*x + 7*e^2*x^2) + 128*c^5*x^2*(20*d^3 + 45*
d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + 64*b*c^4*x*(40*d^3 + 75*d^2*e*x + 54*d*e^
2*x^2 + 14*e^3*x^3) + 8*a*c*(-95*b^3*e^3 + 6*b^2*c*e^2*(50*d + 9*e*x) - 12*b*c^2
*e*(25*d^2 + 14*d*e*x + 3*e^2*x^2) + 8*c^3*(40*d^3 + 45*d^2*e*x + 24*d*e^2*x^2 +
5*e^3*x^3))) - 15*(b^2 - 4*a*c)^2*e*(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)]])/(7680*c^(9/2))

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Maple [B]  time = 0.02, size = 992, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^3*(c*x^2+b*x+a)^(1/2),x)

[Out]

7/80*b^2/c^2*x*(c*x^2+b*x+a)^(3/2)*e^3-3/4*a^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*d^2*e-1/4*b/c*(c*x^2+b*x+a)^(3/2)*d^2*e+1/3*e^3*x^3*(c*x^2+b*
x+a)^(3/2)-1/4*e^3/c*a*x*(c*x^2+b*x+a)^(3/2)+3/4*b/c^(3/2)*a^2*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*d*e^2+3/8*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*a*d^2*e-3/8*a/c*(c*x^2+b*x+a)^(1/2)*b*d^2*e+3/8*b^2/c^2*a*(c*x^2+b*
x+a)^(1/2)*d*e^2+1/8*e^3/c*a^2*(c*x^2+b*x+a)^(1/2)*x+1/16*e^3/c^2*a^2*(c*x^2+b*x
+a)^(1/2)*b+3/16*b^2/c*(c*x^2+b*x+a)^(1/2)*x*d^2*e-3/8*b^3/c^(5/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d*e^2+3/64*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))*d*e^2+3/32*b^3/c^2*(c*x^2+b*x+a)^(1/2)*d^2*e-3/4*a*(c*x^2+b*
x+a)^(1/2)*x*d^2*e-1/8*b^3/c^3*a*(c*x^2+b*x+a)^(1/2)*e^3-9/32*b^2/c^(5/2)*a^2*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*e^3+11/40*a/c^2*(c*x^2+b*x+a)^(3/2)*b*
e^3-3/64*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2*e+1/4*b^2/c
^2*(c*x^2+b*x+a)^(3/2)*d*e^2+7/128*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*e^3-3/32*b^4/c^
3*(c*x^2+b*x+a)^(1/2)*d*e^2+15/128*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))*a*e^3-1/10*x^2*(c*x^2+b*x+a)^(3/2)/c*b*e^3-4/5*a/c*(c*x^2+b*x+a)^(3/2
)*d*e^2+3/4*b/c*a*(c*x^2+b*x+a)^(1/2)*x*d*e^2-7/96*b^3/c^3*(c*x^2+b*x+a)^(3/2)*e
^3+7/256*b^5/c^4*(c*x^2+b*x+a)^(1/2)*e^3-7/512*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*e^3+1/8*e^3/c^(3/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x
+a)^(1/2))+6/5*d*e^2*x^2*(c*x^2+b*x+a)^(3/2)+3/2*d^2*e*x*(c*x^2+b*x+a)^(3/2)-3/1
0*b/c*x*(c*x^2+b*x+a)^(3/2)*d*e^2-3/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*d*e^2-1/4*b
^2/c^2*a*(c*x^2+b*x+a)^(1/2)*x*e^3+2/3*d^3*(c*x^2+b*x+a)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.390917, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/15360*(4*(1280*c^5*e^3*x^5 + 2560*a*c^4*d^3 + 128*(36*c^5*d*e^2 + 7*b*c^4*e^3
)*x^4 + 120*(3*b^3*c^2 - 20*a*b*c^3)*d^2*e - 24*(15*b^4*c - 100*a*b^2*c^2 + 128*
a^2*c^3)*d*e^2 + (105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^2)*e^3 + 16*(360*c^5*d^2*
e + 216*b*c^4*d*e^2 - (3*b^2*c^3 - 20*a*c^4)*e^3)*x^3 + 8*(320*c^5*d^3 + 600*b*c
^4*d^2*e - 24*(b^2*c^3 - 8*a*c^4)*d*e^2 + (7*b^3*c^2 - 36*a*b*c^3)*e^3)*x^2 + 2*
(1280*b*c^4*d^3 - 120*(b^2*c^3 - 12*a*c^4)*d^2*e + 24*(5*b^3*c^2 - 28*a*b*c^3)*d
*e^2 - (35*b^4*c - 216*a*b^2*c^2 + 240*a^2*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sq
rt(c) - 15*(24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^2*e - 24*(b^5*c - 8*a*b^3*
c^2 + 16*a^2*b*c^3)*d*e^2 + (7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*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^(9/2), 1/7680*(2*(1280*c^5*e^3*x^5 + 2560*a*c^4*d^3 + 128*(3
6*c^5*d*e^2 + 7*b*c^4*e^3)*x^4 + 120*(3*b^3*c^2 - 20*a*b*c^3)*d^2*e - 24*(15*b^4
*c - 100*a*b^2*c^2 + 128*a^2*c^3)*d*e^2 + (105*b^5 - 760*a*b^3*c + 1296*a^2*b*c^
2)*e^3 + 16*(360*c^5*d^2*e + 216*b*c^4*d*e^2 - (3*b^2*c^3 - 20*a*c^4)*e^3)*x^3 +
8*(320*c^5*d^3 + 600*b*c^4*d^2*e - 24*(b^2*c^3 - 8*a*c^4)*d*e^2 + (7*b^3*c^2 -
36*a*b*c^3)*e^3)*x^2 + 2*(1280*b*c^4*d^3 - 120*(b^2*c^3 - 12*a*c^4)*d^2*e + 24*(
5*b^3*c^2 - 28*a*b*c^3)*d*e^2 - (35*b^4*c - 216*a*b^2*c^2 + 240*a^2*c^3)*e^3)*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*e - 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*e^2 + (7*b^6 - 60*a*b^4*c + 144*
a^2*b^2*c^2 - 64*a^3*c^3)*e^3)*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. \[ \int \left (b + 2 c x\right ) \left (d + e x\right )^{3} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(e*x+d)**3*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.287426, size = 683, normalized size = 2.19 \[ \frac{1}{3840} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{3} + \frac{36 \, c^{6} d e^{2} + 7 \, b c^{5} e^{3}}{c^{5}}\right )} x + \frac{360 \, c^{6} d^{2} e + 216 \, b c^{5} d e^{2} - 3 \, b^{2} c^{4} e^{3} + 20 \, a c^{5} e^{3}}{c^{5}}\right )} x + \frac{320 \, c^{6} d^{3} + 600 \, b c^{5} d^{2} e - 24 \, b^{2} c^{4} d e^{2} + 192 \, a c^{5} d e^{2} + 7 \, b^{3} c^{3} e^{3} - 36 \, a b c^{4} e^{3}}{c^{5}}\right )} x + \frac{1280 \, b c^{5} d^{3} - 120 \, b^{2} c^{4} d^{2} e + 1440 \, a c^{5} d^{2} e + 120 \, b^{3} c^{3} d e^{2} - 672 \, a b c^{4} d e^{2} - 35 \, b^{4} c^{2} e^{3} + 216 \, a b^{2} c^{3} e^{3} - 240 \, a^{2} c^{4} e^{3}}{c^{5}}\right )} x + \frac{2560 \, a c^{5} d^{3} + 360 \, b^{3} c^{3} d^{2} e - 2400 \, a b c^{4} d^{2} e - 360 \, b^{4} c^{2} d e^{2} + 2400 \, a b^{2} c^{3} d e^{2} - 3072 \, a^{2} c^{4} d e^{2} + 105 \, b^{5} c e^{3} - 760 \, a b^{3} c^{2} e^{3} + 1296 \, a^{2} b c^{3} e^{3}}{c^{5}}\right )} + \frac{{\left (24 \, b^{4} c^{2} d^{2} e - 192 \, a b^{2} c^{3} d^{2} e + 384 \, a^{2} c^{4} d^{2} e - 24 \, b^{5} c d e^{2} + 192 \, a b^{3} c^{2} d e^{2} - 384 \, a^{2} b c^{3} d e^{2} + 7 \, b^{6} e^{3} - 60 \, a b^{4} c e^{3} + 144 \, a^{2} b^{2} c^{2} e^{3} - 64 \, a^{3} c^{3} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{512 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*e^3 + (36*c^6*d*e^2 + 7*b*c^5*e
^3)/c^5)*x + (360*c^6*d^2*e + 216*b*c^5*d*e^2 - 3*b^2*c^4*e^3 + 20*a*c^5*e^3)/c^
5)*x + (320*c^6*d^3 + 600*b*c^5*d^2*e - 24*b^2*c^4*d*e^2 + 192*a*c^5*d*e^2 + 7*b
^3*c^3*e^3 - 36*a*b*c^4*e^3)/c^5)*x + (1280*b*c^5*d^3 - 120*b^2*c^4*d^2*e + 1440
*a*c^5*d^2*e + 120*b^3*c^3*d*e^2 - 672*a*b*c^4*d*e^2 - 35*b^4*c^2*e^3 + 216*a*b^
2*c^3*e^3 - 240*a^2*c^4*e^3)/c^5)*x + (2560*a*c^5*d^3 + 360*b^3*c^3*d^2*e - 2400
*a*b*c^4*d^2*e - 360*b^4*c^2*d*e^2 + 2400*a*b^2*c^3*d*e^2 - 3072*a^2*c^4*d*e^2 +
105*b^5*c*e^3 - 760*a*b^3*c^2*e^3 + 1296*a^2*b*c^3*e^3)/c^5) + 1/512*(24*b^4*c^
2*d^2*e - 192*a*b^2*c^3*d^2*e + 384*a^2*c^4*d^2*e - 24*b^5*c*d*e^2 + 192*a*b^3*c
^2*d*e^2 - 384*a^2*b*c^3*d*e^2 + 7*b^6*e^3 - 60*a*b^4*c*e^3 + 144*a^2*b^2*c^2*e^
3 - 64*a^3*c^3*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/
c^(9/2)