3.2339 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=307 \[ -\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]

[Out]

-((8*c^2*d^3 - b*e^2*(b*d - 2*a*e) - 2*c*d*e*(3*b*d - 2*a*e) + e*(12*c^2*d^2 + b
^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x)*Sqrt[a + b*x + c*x^2])/(8*e^3*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)^2) - (a + b*x + c*x^2)^(3/2)/(3*e*(d + e*x)^3) + (c^(3/2)*Arc
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^4 - ((2*c*d - b*e)*(8*c^2
*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*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])])/(16*e^4*(c*d^2 - b*d*e +
a*e^2)^(3/2))

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Rubi [A]  time = 0.919856, antiderivative size = 307, 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 \[ -\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^4 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{8 e^3 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-((8*c^2*d^3 - b*e^2*(b*d - 2*a*e) - 2*c*d*e*(3*b*d - 2*a*e) + e*(12*c^2*d^2 + b
^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*x)*Sqrt[a + b*x + c*x^2])/(8*e^3*(c*d^2 - b*d*e
+ a*e^2)*(d + e*x)^2) - (a + b*x + c*x^2)^(3/2)/(3*e*(d + e*x)^3) + (c^(3/2)*Arc
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e^4 - ((2*c*d - b*e)*(8*c^2
*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*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])])/(16*e^4*(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((c*x**2+b*x+a)**(3/2)/(e*x+d)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 2.37311, size = 334, normalized size = 1.09 \[ \frac{-\frac{2 e \sqrt{a+x (b+c x)} \left (\frac{(d+e x)^2 \left (4 c e (8 a e-11 b d)+3 b^2 e^2+44 c^2 d^2\right )}{e (a e-b d)+c d^2}+8 \left (e (a e-b d)+c d^2\right )-14 (d+e x) (2 c d-b e)\right )}{(d+e x)^3}-\frac{3 (b e-2 c d) \log (d+e x) \left (4 c e (2 b d-3 a e)+b^2 e^2-8 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{3 (b e-2 c d) \left (4 c e (2 b d-3 a e)+b^2 e^2-8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+48 c^{3/2} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{48 e^4} \]

Antiderivative was successfully verified.

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

[Out]

((-2*e*Sqrt[a + x*(b + c*x)]*(8*(c*d^2 + e*(-(b*d) + a*e)) - 14*(2*c*d - b*e)*(d
+ e*x) + ((44*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-11*b*d + 8*a*e))*(d + e*x)^2)/(c*d^
2 + e*(-(b*d) + a*e))))/(d + e*x)^3 - (3*(-2*c*d + b*e)*(-8*c^2*d^2 + b^2*e^2 +
4*c*e*(2*b*d - 3*a*e))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) + 48*c^(3/
2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] + (3*(-2*c*d + b*e)*(-8*c^2*
d^2 + b^2*e^2 + 4*c*e*(2*b*d - 3*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)]])/(c*d^2 + e*(-(b*d) + a*e)
)^(3/2))/(48*e^4)

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Maple [B]  time = 0.026, size = 10401, normalized size = 33.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 17.0627, 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)^4,x, algorithm="giac")

[Out]

sage0*x