3.2343 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=510 \[ -\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right )}{280 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2048 \left (a e^2-b d e+c d^2\right )^{11/2}}-\frac{3 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b
*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(1024*(c*d^2 - b*d*e + a*e^
2)^5*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))
*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(128*(c*d^2 - b*d*e +
a*e^2)^4*(d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(7*(c*d^2 - b*d*e + a*e^2)*(
d + e*x)^7) - (3*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(5/2))/(28*(c*d^2 - b*d*e + a
*e^2)^2*(d + e*x)^6) - (e*(68*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(17*b*d + 4*a*e))*(a
+ b*x + c*x^2)^(5/2))/(280*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5) + (3*(b^2 - 4*
a*c)^2*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*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])
])/(2048*(c*d^2 - b*d*e + a*e^2)^(11/2))

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Rubi [A]  time = 2.20024, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{e \left (a+b x+c x^2\right )^{5/2} \left (-4 c e (4 a e+17 b d)+21 b^2 e^2+68 c^2 d^2\right )}{280 (d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{128 (d+e x)^4 \left (a e^2-b d e+c d^2\right )^4}-\frac{3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{1024 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^5}+\frac{3 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2048 \left (a e^2-b d e+c d^2\right )^{11/2}}-\frac{3 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{28 (d+e x)^6 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{5/2}}{7 (d+e x)^7 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

(-3*(b^2 - 4*a*c)*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*(b
*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(1024*(c*d^2 - b*d*e + a*e^
2)^5*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))
*(b*d - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(128*(c*d^2 - b*d*e +
a*e^2)^4*(d + e*x)^4) - (e*(a + b*x + c*x^2)^(5/2))/(7*(c*d^2 - b*d*e + a*e^2)*(
d + e*x)^7) - (3*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(5/2))/(28*(c*d^2 - b*d*e + a
*e^2)^2*(d + e*x)^6) - (e*(68*c^2*d^2 + 21*b^2*e^2 - 4*c*e*(17*b*d + 4*a*e))*(a
+ b*x + c*x^2)^(5/2))/(280*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5) + (3*(b^2 - 4*
a*c)^2*(2*c*d - b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*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])
])/(2048*(c*d^2 - b*d*e + a*e^2)^(11/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**(3/2)/(e*x+d)**8,x)

[Out]

Timed out

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Mathematica [A]  time = 6.54627, size = 808, normalized size = 1.58 \[ -\frac{3 (b e-2 c d) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \log (d+e x) \left (b^2-4 a c\right )^2}{2048 \left (c d^2+e (a e-b d)\right )^{11/2}}+\frac{3 (b e-2 c d) \left (8 c^2 d^2+3 b^2 e^2-4 c e (2 b d+a e)\right ) \log \left (-b d-2 c x d+2 a e+b e x+2 \sqrt{c d^2+e (a e-b d)} \sqrt{a+x (b+c x)}\right ) \left (b^2-4 a c\right )^2}{2048 \left (c d^2+e (a e-b d)\right )^{11/2}}+\frac{\sqrt{a+x (b+c x)} \left (\frac{\left (256 c^6 d^6-256 c^5 e (3 b d-10 a e) d^4+64 c^4 e^2 \left (5 b^2 d^2-80 a b e d-247 a^2 e^2\right ) d^2-315 b^6 e^6+1260 b^4 c e^5 (b d+2 a e)-28 b^2 c^2 e^4 \left (61 b^2 d^2+328 a b e d+196 a^2 e^2\right )+32 c^3 e^3 \left (20 b^3 d^3+367 a b^2 e d^2+494 a^2 b e^2 d+64 a^3 e^3\right )\right ) (d+e x)^6}{\left (c d^2+e (a e-b d)\right )^5}+\frac{2 (2 c d-b e) \left (64 c^4 d^4-64 c^3 e (2 b d-9 a e) d^2-105 b^4 e^4+56 b^2 c e^3 (2 b d+13 a e)-16 c^2 e^2 \left (3 b^2 d^2+36 a b e d+73 a^2 e^2\right )\right ) (d+e x)^5}{\left (c d^2+e (a e-b d)\right )^4}+\frac{8 \left (32 c^4 d^4+16 c^3 e (15 a e-4 b d) d^2-21 b^4 e^4+4 b^2 c e^3 (11 b d+31 a e)-4 c^2 e^2 \left (3 b^2 d^2+60 a b e d+32 a^2 e^2\right )\right ) (d+e x)^4}{\left (c d^2+e (a e-b d)\right )^3}+\frac{16 (b e-2 c d) \left (-8 c^2 d^2+9 b^2 e^2+4 c e (2 b d-11 a e)\right ) (d+e x)^3}{\left (c d^2+e (a e-b d)\right )^2}-\frac{128 \left (68 c^2 d^2-68 b c e d+b^2 e^2+64 a c e^2\right ) (d+e x)^2}{c d^2+e (a e-b d)}+6400 (2 c d-b e) (d+e x)-5120 \left (c d^2+e (a e-b d)\right )\right )}{35840 e^3 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(3/2)/(d + e*x)^8,x]

[Out]

(Sqrt[a + x*(b + c*x)]*(-5120*(c*d^2 + e*(-(b*d) + a*e)) + 6400*(2*c*d - b*e)*(d
+ e*x) - (128*(68*c^2*d^2 - 68*b*c*d*e + b^2*e^2 + 64*a*c*e^2)*(d + e*x)^2)/(c*
d^2 + e*(-(b*d) + a*e)) + (16*(-2*c*d + b*e)*(-8*c^2*d^2 + 9*b^2*e^2 + 4*c*e*(2*
b*d - 11*a*e))*(d + e*x)^3)/(c*d^2 + e*(-(b*d) + a*e))^2 + (8*(32*c^4*d^4 - 21*b
^4*e^4 + 16*c^3*d^2*e*(-4*b*d + 15*a*e) + 4*b^2*c*e^3*(11*b*d + 31*a*e) - 4*c^2*
e^2*(3*b^2*d^2 + 60*a*b*d*e + 32*a^2*e^2))*(d + e*x)^4)/(c*d^2 + e*(-(b*d) + a*e
))^3 + (2*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 9*a*e)
+ 56*b^2*c*e^3*(2*b*d + 13*a*e) - 16*c^2*e^2*(3*b^2*d^2 + 36*a*b*d*e + 73*a^2*e
^2))*(d + e*x)^5)/(c*d^2 + e*(-(b*d) + a*e))^4 + ((256*c^6*d^6 - 315*b^6*e^6 - 2
56*c^5*d^4*e*(3*b*d - 10*a*e) + 1260*b^4*c*e^5*(b*d + 2*a*e) + 64*c^4*d^2*e^2*(5
*b^2*d^2 - 80*a*b*d*e - 247*a^2*e^2) - 28*b^2*c^2*e^4*(61*b^2*d^2 + 328*a*b*d*e
+ 196*a^2*e^2) + 32*c^3*e^3*(20*b^3*d^3 + 367*a*b^2*d^2*e + 494*a^2*b*d*e^2 + 64
*a^3*e^3))*(d + e*x)^6)/(c*d^2 + e*(-(b*d) + a*e))^5))/(35840*e^3*(d + e*x)^7) -
(3*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))
*Log[d + e*x])/(2048*(c*d^2 + e*(-(b*d) + a*e))^(11/2)) + (3*(b^2 - 4*a*c)^2*(-2
*c*d + b*e)*(8*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(2*b*d + a*e))*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)]])/(2048*
(c*d^2 + e*(-(b*d) + a*e))^(11/2))

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Maple [B]  time = 0.102, size = 35234, normalized size = 69.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(3/2)/(e*x+d)^8,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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^8,x, algorithm="fricas")

[Out]

Timed out

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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)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.739352, 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)^8,x, algorithm="giac")

[Out]

sage0*x