3.2341 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=296 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c
*x^2])/(128*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(b*d - 2*a*e
+ (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(16*(c*d^2 - b*d*e + a*e^2)^2*(d +
e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) +
(3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt
[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(256*(c*d^2 - b*d*e + a*e^2)^(7
/2))

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Rubi [A]  time = 0.624991, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{256 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{5 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{16 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c
*x^2])/(128*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) + ((2*c*d - b*e)*(b*d - 2*a*e
+ (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(16*(c*d^2 - b*d*e + a*e^2)^2*(d +
e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) +
(3*(b^2 - 4*a*c)^2*(2*c*d - b*e)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt
[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(256*(c*d^2 - b*d*e + a*e^2)^(7
/2))

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Rubi in Sympy [A]  time = 100.715, size = 275, normalized size = 0.93 \[ - \frac{e \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{5 \left (d + e x\right )^{5} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (\frac{b e}{2} - c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{128 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{7}{2}}} - \frac{3 \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{128 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )^{3}} + \frac{\left (\frac{b e}{2} - c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{8 \left (d + e x\right )^{4} \left (a e^{2} - b d e + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**6,x)

[Out]

-e*(a + b*x + c*x**2)**(5/2)/(5*(d + e*x)**5*(a*e**2 - b*d*e + c*d**2)) + 3*(-4*
a*c + b**2)**2*(b*e/2 - c*d)*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sqrt(a + b
*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2)))/(128*(a*e**2 - b*d*e + c*d**2)**(7/
2)) - 3*(-4*a*c + b**2)*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)*(2*a*e - b*d + x*(b
*e - 2*c*d))/(128*(d + e*x)**2*(a*e**2 - b*d*e + c*d**2)**3) + (b*e/2 - c*d)*(a
+ b*x + c*x**2)**(3/2)*(2*a*e - b*d + x*(b*e - 2*c*d))/(8*(d + e*x)**4*(a*e**2 -
b*d*e + c*d**2)**2)

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Mathematica [A]  time = 2.43581, size = 457, normalized size = 1.54 \[ \frac{-2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (-(d+e x)^4 \left (-4 c^2 e^2 \left (32 a^2 e^2+36 a b d e-3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )+8 (d+e x)^2 \left (4 c e (8 a e-9 b d)+b^2 e^2+36 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2-2 (d+e x)^3 (2 c d-b e) \left (4 c e (7 a e-2 b d)-5 b^2 e^2+8 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )-176 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3+128 \left (e (a e-b d)+c d^2\right )^4\right )+15 e^3 \left (b^2-4 a c\right )^2 (d+e x)^5 (b e-2 c d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )-15 e^3 \left (b^2-4 a c\right )^2 (d+e x)^5 (b e-2 c d) \log (d+e x)}{1280 e^3 (d+e x)^5 \left (e (a e-b d)+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

(-2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(128*(c*d^2 + e*(-(b*d)
+ a*e))^4 - 176*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e*x) + 8*(c*d^2
+ e*(-(b*d) + a*e))^2*(36*c^2*d^2 + b^2*e^2 + 4*c*e*(-9*b*d + 8*a*e))*(d + e*x)
^2 - 2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(8*c^2*d^2 - 5*b^2*e^2 + 4*c*e*(
-2*b*d + 7*a*e))*(d + e*x)^3 - (32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d -
9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) - 4*c^2*e^2*(-3*b^2*d^2 + 36*a*b*d*e + 32*a^
2*e^2))*(d + e*x)^4) - 15*(b^2 - 4*a*c)^2*e^3*(-2*c*d + b*e)*(d + e*x)^5*Log[d +
e*x] + 15*(b^2 - 4*a*c)^2*e^3*(-2*c*d + b*e)*(d + e*x)^5*Log[-(b*d) + 2*a*e - 2
*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(1280*
e^3*(c*d^2 + e*(-(b*d) + a*e))^(7/2)*(d + e*x)^5)

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Maple [B]  time = 0.041, size = 20477, normalized size = 69.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(e*x+d)^6,x)

[Out]

result too large to display

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 30.0402, 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)^6,x, algorithm="fricas")

[Out]

[1/2560*(4*(336*a^3*b*d*e^4 - 128*a^4*e^5 - 10*(3*b^3*c - 20*a*b*c^2)*d^5 + 3*(5
*b^4 - 40*a*b^2*c - 176*a^2*c^2)*d^4*e + 2*(5*a*b^3 + 348*a^2*b*c)*d^3*e^2 - 8*(
31*a^2*b^2 + 52*a^3*c)*d^2*e^3 + (32*c^4*d^4*e - 64*b*c^3*d^3*e^2 + 12*(b^2*c^2
+ 12*a*c^3)*d^2*e^3 + 4*(5*b^3*c - 36*a*b*c^2)*d*e^4 - (15*b^4 - 100*a*b^2*c + 1
28*a^2*c^2)*e^5)*x^4 + 2*(80*c^4*d^5 - 168*b*c^3*d^4*e + 2*(23*b^2*c^2 + 180*a*c
^3)*d^3*e^2 + (47*b^3*c - 396*a*b*c^2)*d^2*e^3 - (35*b^4 - 226*a*b^2*c + 200*a^2
*c^2)*d*e^4 + (5*a*b^3 - 28*a^2*b*c)*e^5)*x^3 + 2*(120*b*c^3*d^5 - 2*(167*b^2*c^
2 - 116*a*c^3)*d^4*e + (233*b^3*c + 76*a*b*c^2)*d^3*e^2 - 2*(32*b^4 + 23*a*b^2*c
+ 308*a^2*c^2)*d^2*e^3 + (23*a*b^3 + 316*a^2*b*c)*d*e^4 - 4*(a^2*b^2 + 32*a^3*c
)*e^5)*x^2 - 2*(88*a^3*b*e^5 - 10*(b^2*c^2 + 20*a*c^3)*d^5 + (75*b^3*c + 268*a*b
*c^2)*d^4*e - (35*b^4 + 486*a*b^2*c - 360*a^2*c^2)*d^3*e^2 + (233*a*b^3 + 76*a^2
*b*c)*d^2*e^3 - 16*(16*a^2*b^2 - 5*a^3*c)*d*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*
sqrt(c*x^2 + b*x + a) - 15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*
a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b
^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3
)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2
*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2*e^4)*x^3 + 10*
(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)
*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c
+ 16*a^2*b*c^2)*d^4*e^2)*x)*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (
8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b
^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) + 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a
^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c
)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^3*d^11 - 3*b*
c^2*d^10*e - 3*a^2*b*d^6*e^5 + a^3*d^5*e^6 + 3*(b^2*c + a*c^2)*d^9*e^2 - (b^3 +
6*a*b*c)*d^8*e^3 + 3*(a*b^2 + a^2*c)*d^7*e^4 + (c^3*d^6*e^5 - 3*b*c^2*d^5*e^6 -
3*a^2*b*d*e^10 + a^3*e^11 + 3*(b^2*c + a*c^2)*d^4*e^7 - (b^3 + 6*a*b*c)*d^3*e^8
+ 3*(a*b^2 + a^2*c)*d^2*e^9)*x^5 + 5*(c^3*d^7*e^4 - 3*b*c^2*d^6*e^5 - 3*a^2*b*d^
2*e^9 + a^3*d*e^10 + 3*(b^2*c + a*c^2)*d^5*e^6 - (b^3 + 6*a*b*c)*d^4*e^7 + 3*(a*
b^2 + a^2*c)*d^3*e^8)*x^4 + 10*(c^3*d^8*e^3 - 3*b*c^2*d^7*e^4 - 3*a^2*b*d^3*e^8
+ a^3*d^2*e^9 + 3*(b^2*c + a*c^2)*d^6*e^5 - (b^3 + 6*a*b*c)*d^5*e^6 + 3*(a*b^2 +
a^2*c)*d^4*e^7)*x^3 + 10*(c^3*d^9*e^2 - 3*b*c^2*d^8*e^3 - 3*a^2*b*d^4*e^7 + a^3
*d^3*e^8 + 3*(b^2*c + a*c^2)*d^7*e^4 - (b^3 + 6*a*b*c)*d^6*e^5 + 3*(a*b^2 + a^2*
c)*d^5*e^6)*x^2 + 5*(c^3*d^10*e - 3*b*c^2*d^9*e^2 - 3*a^2*b*d^5*e^6 + a^3*d^4*e^
7 + 3*(b^2*c + a*c^2)*d^8*e^3 - (b^3 + 6*a*b*c)*d^7*e^4 + 3*(a*b^2 + a^2*c)*d^6*
e^5)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/1280*(2*(336*a^3*b*d*e^4 - 128*a^4*e^5 -
10*(3*b^3*c - 20*a*b*c^2)*d^5 + 3*(5*b^4 - 40*a*b^2*c - 176*a^2*c^2)*d^4*e + 2*
(5*a*b^3 + 348*a^2*b*c)*d^3*e^2 - 8*(31*a^2*b^2 + 52*a^3*c)*d^2*e^3 + (32*c^4*d^
4*e - 64*b*c^3*d^3*e^2 + 12*(b^2*c^2 + 12*a*c^3)*d^2*e^3 + 4*(5*b^3*c - 36*a*b*c
^2)*d*e^4 - (15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*e^5)*x^4 + 2*(80*c^4*d^5 - 168*
b*c^3*d^4*e + 2*(23*b^2*c^2 + 180*a*c^3)*d^3*e^2 + (47*b^3*c - 396*a*b*c^2)*d^2*
e^3 - (35*b^4 - 226*a*b^2*c + 200*a^2*c^2)*d*e^4 + (5*a*b^3 - 28*a^2*b*c)*e^5)*x
^3 + 2*(120*b*c^3*d^5 - 2*(167*b^2*c^2 - 116*a*c^3)*d^4*e + (233*b^3*c + 76*a*b*
c^2)*d^3*e^2 - 2*(32*b^4 + 23*a*b^2*c + 308*a^2*c^2)*d^2*e^3 + (23*a*b^3 + 316*a
^2*b*c)*d*e^4 - 4*(a^2*b^2 + 32*a^3*c)*e^5)*x^2 - 2*(88*a^3*b*e^5 - 10*(b^2*c^2
+ 20*a*c^3)*d^5 + (75*b^3*c + 268*a*b*c^2)*d^4*e - (35*b^4 + 486*a*b^2*c - 360*a
^2*c^2)*d^3*e^2 + (233*a*b^3 + 76*a^2*b*c)*d^2*e^3 - 16*(16*a^2*b^2 - 5*a^3*c)*d
*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) - 15*(2*(b^4*c - 8*a
*b^2*c^2 + 16*a^2*c^3)*d^6 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5*e + (2*(b^4*c
- 8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^5 +
5*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^
2)*d*e^5)*x^4 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^
3*c + 16*a^2*b*c^2)*d^2*e^4)*x^3 + 10*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*
e^2 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^3*e^3)*x^2 + 5*(2*(b^4*c - 8*a*b^2*c^2
+ 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2)*x)*arctan(-1/2*s
qrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*
e^2)*sqrt(c*x^2 + b*x + a))))/((c^3*d^11 - 3*b*c^2*d^10*e - 3*a^2*b*d^6*e^5 + a^
3*d^5*e^6 + 3*(b^2*c + a*c^2)*d^9*e^2 - (b^3 + 6*a*b*c)*d^8*e^3 + 3*(a*b^2 + a^2
*c)*d^7*e^4 + (c^3*d^6*e^5 - 3*b*c^2*d^5*e^6 - 3*a^2*b*d*e^10 + a^3*e^11 + 3*(b^
2*c + a*c^2)*d^4*e^7 - (b^3 + 6*a*b*c)*d^3*e^8 + 3*(a*b^2 + a^2*c)*d^2*e^9)*x^5
+ 5*(c^3*d^7*e^4 - 3*b*c^2*d^6*e^5 - 3*a^2*b*d^2*e^9 + a^3*d*e^10 + 3*(b^2*c + a
*c^2)*d^5*e^6 - (b^3 + 6*a*b*c)*d^4*e^7 + 3*(a*b^2 + a^2*c)*d^3*e^8)*x^4 + 10*(c
^3*d^8*e^3 - 3*b*c^2*d^7*e^4 - 3*a^2*b*d^3*e^8 + a^3*d^2*e^9 + 3*(b^2*c + a*c^2)
*d^6*e^5 - (b^3 + 6*a*b*c)*d^5*e^6 + 3*(a*b^2 + a^2*c)*d^4*e^7)*x^3 + 10*(c^3*d^
9*e^2 - 3*b*c^2*d^8*e^3 - 3*a^2*b*d^4*e^7 + a^3*d^3*e^8 + 3*(b^2*c + a*c^2)*d^7*
e^4 - (b^3 + 6*a*b*c)*d^6*e^5 + 3*(a*b^2 + a^2*c)*d^5*e^6)*x^2 + 5*(c^3*d^10*e -
3*b*c^2*d^9*e^2 - 3*a^2*b*d^5*e^6 + a^3*d^4*e^7 + 3*(b^2*c + a*c^2)*d^8*e^3 - (
b^3 + 6*a*b*c)*d^7*e^4 + 3*(a*b^2 + a^2*c)*d^6*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e
^2))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.63792, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x