3.2371 \(\int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=286 \[ \frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e (d+e x)^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}+\frac{e \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{4 c^3 \left (b^2-4 a c\right )} \]

[Out]

(-2*(d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c
*x^2]) + (2*e*(2*c*d - b*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(c*(b^2 - 4*a*c))
+ (e*(32*c^3*d^3 - 15*b^3*e^3 + 4*b*c*e^2*(12*b*d + 13*a*e) - 8*c^2*d*e*(5*b*d
+ 16*a*e) + 2*c*e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*x)*Sqrt[a + b*
x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*
d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(7/2))

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Rubi [A]  time = 0.882532, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{2 e (d+e x)^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}+\frac{e \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{4 c^3 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a + b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^3*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c
*x^2]) + (2*e*(2*c*d - b*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(c*(b^2 - 4*a*c))
+ (e*(32*c^3*d^3 - 15*b^3*e^3 + 4*b*c*e^2*(12*b*d + 13*a*e) - 8*c^2*d*e*(5*b*d
+ 16*a*e) + 2*c*e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*x)*Sqrt[a + b*
x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*
d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(7/2))

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Rubi in Sympy [A]  time = 107.244, size = 301, normalized size = 1.05 \[ \frac{2 \left (d + e x\right )^{3} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{2 e \left (d + e x\right )^{2} \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{c \left (- 4 a c + b^{2}\right )} - \frac{e \sqrt{a + b x + c x^{2}} \left (- 39 a b c e^{3} + 96 a c^{2} d e^{2} + \frac{45 b^{3} e^{3}}{4} - 36 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 24 c^{3} d^{3} - \frac{3 c e x \left (- 12 a c e^{2} + 5 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{2}\right )}{3 c^{3} \left (- 4 a c + b^{2}\right )} + \frac{3 e^{2} \left (- 8 a c e^{2} + 10 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

2*(d + e*x)**3*(2*a*e - b*d + x*(b*e - 2*c*d))/((-4*a*c + b**2)*sqrt(a + b*x + c
*x**2)) - 2*e*(d + e*x)**2*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)/(c*(-4*a*c + b**
2)) - e*sqrt(a + b*x + c*x**2)*(-39*a*b*c*e**3 + 96*a*c**2*d*e**2 + 45*b**3*e**3
/4 - 36*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 24*c**3*d**3 - 3*c*e*x*(-12*a*c*e**2
+ 5*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/2)/(3*c**3*(-4*a*c + b**2)) + 3*e**2*(-
8*a*c*e**2 + 10*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)*atanh((b + 2*c*x)/(2*sqrt
(c)*sqrt(a + b*x + c*x**2)))/(16*c**(7/2))

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Mathematica [A]  time = 0.597771, size = 291, normalized size = 1.02 \[ \frac{4 b c \left (-13 a^2 e^4+a c e^2 \left (12 d^2+40 d e x-5 e^2 x^2\right )+2 c^2 d^3 (d-4 e x)\right )+8 c^2 \left (a^2 e^3 (16 d+3 e x)+a c e \left (-8 d^3-12 d^2 e x+8 d e^2 x^2+e^3 x^3\right )+2 c^2 d^4 x\right )+b^3 e^3 (15 a e+c x (5 e x-48 d))-2 b^2 c e^2 \left (a e (24 d+31 e x)+c x \left (-24 d^2+8 d e x+e^2 x^2\right )\right )+15 b^4 e^4 x}{4 c^3 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{8 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a + b*x + c*x^2)^(3/2),x]

[Out]

(15*b^4*e^4*x + b^3*e^3*(15*a*e + c*x*(-48*d + 5*e*x)) + 4*b*c*(-13*a^2*e^4 + 2*
c^2*d^3*(d - 4*e*x) + a*c*e^2*(12*d^2 + 40*d*e*x - 5*e^2*x^2)) - 2*b^2*c*e^2*(a*
e*(24*d + 31*e*x) + c*x*(-24*d^2 + 8*d*e*x + e^2*x^2)) + 8*c^2*(2*c^2*d^4*x + a^
2*e^3*(16*d + 3*e*x) + a*c*e*(-8*d^3 - 12*d^2*e*x + 8*d*e^2*x^2 + e^3*x^3)))/(4*
c^3*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*c
*e*(4*b*d + a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(7/2))

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Maple [B]  time = 0.016, size = 913, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+b*x+a)^(3/2),x)

[Out]

-13/4*e^4*b/c^3*a/(c*x^2+b*x+a)^(1/2)+3/2*e^4*a/c^2*x/(c*x^2+b*x+a)^(1/2)+3*d^2*
e^2*b/c^2/(c*x^2+b*x+a)^(1/2)-6*d^2*e^2*x/c/(c*x^2+b*x+a)^(1/2)-15/8*e^4*b^2/c^3
*x/(c*x^2+b*x+a)^(1/2)+15/16*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-4*d^3*e
/c/(c*x^2+b*x+a)^(1/2)+6*d^2*e^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))+1/2*e^4*x^3/c/(c*x^2+b*x+a)^(1/2)+15/16*e^4*b^3/c^4/(c*x^2+b*x+a)^(1/2)+15/
8*e^4*b^2/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2*d^4*(2*c*x+b)/(4
*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+6*d*e^3*b/c^2*x/(c*x^2+b*x+a)^(1/2)-3*d*e^3*b^4/c^
3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+3*d^2*e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1
/2)-13/4*e^4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-8*d^3*e*b/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)*x-4*d^3*e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+15/8*e^4*b^4/c^
3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+8*d*e^3*a/c^2/(c*x^2+b*x+a)^(1/2)+4*d*e^3*x^
2/c/(c*x^2+b*x+a)^(1/2)-3*d*e^3*b^2/c^3/(c*x^2+b*x+a)^(1/2)-6*d*e^3*b/c^(5/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/4*e^4*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)+
16*d*e^3*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3/2*e^4*a/c^(5/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-6*d*e^3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x
+8*d*e^3*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+6*d^2*e^2*b^2/c/(4*a*c-b^2)/(
c*x^2+b*x+a)^(1/2)*x-13/2*e^4*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x

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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((e*x + d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.462106, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(4*(8*b*c^3*d^4 - 64*a*c^3*d^3*e + 48*a*b*c^2*d^2*e^2 - 2*(b^2*c^2 - 4*a*
c^3)*e^4*x^3 - 16*(3*a*b^2*c - 8*a^2*c^2)*d*e^3 + (15*a*b^3 - 52*a^2*b*c)*e^4 -
(16*(b^2*c^2 - 4*a*c^3)*d*e^3 - 5*(b^3*c - 4*a*b*c^2)*e^4)*x^2 + (16*c^4*d^4 - 3
2*b*c^3*d^3*e + 48*(b^2*c^2 - 2*a*c^3)*d^2*e^2 - 16*(3*b^3*c - 10*a*b*c^2)*d*e^3
+ (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*
(16*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^2 - 16*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (5*a*b^
4 - 24*a^2*b^2*c + 16*a^3*c^2)*e^4 + (16*(b^2*c^3 - 4*a*c^4)*d^2*e^2 - 16*(b^3*c
^2 - 4*a*b*c^3)*d*e^3 + (5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*e^4)*x^2 + (16*(b^
3*c^2 - 4*a*b*c^3)*d^2*e^2 - 16*(b^4*c - 4*a*b^2*c^2)*d*e^3 + (5*b^5 - 24*a*b^3*
c + 16*a^2*b*c^2)*e^4)*x)*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)))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c
^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(c)), -1/8*(2*(8*b*c^3*d^4 - 64*a*c^3*d^3
*e + 48*a*b*c^2*d^2*e^2 - 2*(b^2*c^2 - 4*a*c^3)*e^4*x^3 - 16*(3*a*b^2*c - 8*a^2*
c^2)*d*e^3 + (15*a*b^3 - 52*a^2*b*c)*e^4 - (16*(b^2*c^2 - 4*a*c^3)*d*e^3 - 5*(b^
3*c - 4*a*b*c^2)*e^4)*x^2 + (16*c^4*d^4 - 32*b*c^3*d^3*e + 48*(b^2*c^2 - 2*a*c^3
)*d^2*e^2 - 16*(3*b^3*c - 10*a*b*c^2)*d*e^3 + (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)
*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(16*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e^2
- 16*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*e^4 +
(16*(b^2*c^3 - 4*a*c^4)*d^2*e^2 - 16*(b^3*c^2 - 4*a*b*c^3)*d*e^3 + (5*b^4*c - 2
4*a*b^2*c^2 + 16*a^2*c^3)*e^4)*x^2 + (16*(b^3*c^2 - 4*a*b*c^3)*d^2*e^2 - 16*(b^4
*c - 4*a*b^2*c^2)*d*e^3 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*e^4)*x)*arctan(1/2
*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2
*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((d + e*x)**4/(a + b*x + c*x**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.228261, size = 509, normalized size = 1.78 \[ \frac{{\left ({\left (\frac{2 \,{\left (b^{2} c^{2} e^{4} - 4 \, a c^{3} e^{4}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac{16 \, b^{2} c^{2} d e^{3} - 64 \, a c^{3} d e^{3} - 5 \, b^{3} c e^{4} + 20 \, a b c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{16 \, c^{4} d^{4} - 32 \, b c^{3} d^{3} e + 48 \, b^{2} c^{2} d^{2} e^{2} - 96 \, a c^{3} d^{2} e^{2} - 48 \, b^{3} c d e^{3} + 160 \, a b c^{2} d e^{3} + 15 \, b^{4} e^{4} - 62 \, a b^{2} c e^{4} + 24 \, a^{2} c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{8 \, b c^{3} d^{4} - 64 \, a c^{3} d^{3} e + 48 \, a b c^{2} d^{2} e^{2} - 48 \, a b^{2} c d e^{3} + 128 \, a^{2} c^{2} d e^{3} + 15 \, a b^{3} e^{4} - 52 \, a^{2} b c e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt{c x^{2} + b x + a}} - \frac{3 \,{\left (16 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 5 \, b^{2} e^{4} - 4 \, a c e^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + b*x + a)^(3/2),x, algorithm="giac")

[Out]

1/4*(((2*(b^2*c^2*e^4 - 4*a*c^3*e^4)*x/(b^2*c^3 - 4*a*c^4) + (16*b^2*c^2*d*e^3 -
64*a*c^3*d*e^3 - 5*b^3*c*e^4 + 20*a*b*c^2*e^4)/(b^2*c^3 - 4*a*c^4))*x - (16*c^4
*d^4 - 32*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 - 96*a*c^3*d^2*e^2 - 48*b^3*c*d*e^3 +
160*a*b*c^2*d*e^3 + 15*b^4*e^4 - 62*a*b^2*c*e^4 + 24*a^2*c^2*e^4)/(b^2*c^3 - 4*
a*c^4))*x - (8*b*c^3*d^4 - 64*a*c^3*d^3*e + 48*a*b*c^2*d^2*e^2 - 48*a*b^2*c*d*e^
3 + 128*a^2*c^2*d*e^3 + 15*a*b^3*e^4 - 52*a^2*b*c*e^4)/(b^2*c^3 - 4*a*c^4))/sqrt
(c*x^2 + b*x + a) - 3/8*(16*c^2*d^2*e^2 - 16*b*c*d*e^3 + 5*b^2*e^4 - 4*a*c*e^4)*
ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)