3.1564 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=462 \[ \frac{\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^5 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{8 e^4 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{4 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^5} \]

[Out]

((64*c^3*d^3 + b^3*e^3 + 4*b*c*e^2*(4*b*d - 5*a*e) - 16*c^2*d*e*(5*b*d - 4*a*e)
+ 2*c*e*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*x^2])
/(8*e^4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)) - ((16*c^2*d^3 - b*e^2*(b*d - 4*a*e)
- 4*c*d*e*(3*b*d - a*e) + 3*e*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*(a
+ b*x + c*x^2)^(3/2))/(12*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (4*c^(3/2)*
(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^5 + ((12
8*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e)
+ 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*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])])/(16*e^5*(c*d^2
- b*d*e + a*e^2)^(3/2))

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Rubi [A]  time = 1.42657, antiderivative size = 462, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^5 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{12 e^2 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{8 e^4 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{4 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^5} \]

Antiderivative was successfully verified.

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

[Out]

((64*c^3*d^3 + b^3*e^3 + 4*b*c*e^2*(4*b*d - 5*a*e) - 16*c^2*d*e*(5*b*d - 4*a*e)
+ 2*c*e*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*x^2])
/(8*e^4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)) - ((16*c^2*d^3 - b*e^2*(b*d - 4*a*e)
- 4*c*d*e*(3*b*d - a*e) + 3*e*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x)*(a
+ b*x + c*x^2)^(3/2))/(12*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (4*c^(3/2)*
(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^5 + ((12
8*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e)
+ 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2))*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])])/(16*e^5*(c*d^2
- b*d*e + a*e^2)^(3/2))

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

[Out]

Timed out

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Mathematica [A]  time = 2.54592, size = 481, normalized size = 1.04 \[ \frac{\frac{3 \log (d+e x) \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{3 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+2 e \sqrt{a+x (b+c x)} \left (\frac{(2 c d-b e) \left (4 c e (23 a e-26 b d)+3 b^2 e^2+104 c^2 d^2\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )}-\frac{2 \left (4 c e (3 a e-10 b d)+7 b^2 e^2+40 c^2 d^2\right )}{(d+e x)^2}+\frac{8 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^3}+48 c^2\right )-192 c^{3/2} (2 c d-b e) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{48 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(48*c^2 + (8*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))
)/(d + e*x)^3 - (2*(40*c^2*d^2 + 7*b^2*e^2 + 4*c*e*(-10*b*d + 3*a*e)))/(d + e*x)
^2 + ((2*c*d - b*e)*(104*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-26*b*d + 23*a*e)))/((c*d^
2 + e*(-(b*d) + a*e))*(d + e*x))) + (3*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b
*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e +
a^2*e^2))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) - 192*c^(3/2)*(2*c*d -
b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] - (3*(128*c^4*d^4 - b^4*e
^4 - 8*b^2*c*e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*
b^2*d^2 - 4*a*b*d*e + a^2*e^2))*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)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)
)/(48*e^5)

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Maple [B]  time = 0.03, size = 15982, normalized size = 34.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError