3.2337 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=227 \[ \frac{3 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c} e^4}-\frac{3 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^4}-\frac{3 \sqrt{a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.706138, antiderivative size = 227, 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 \[ \frac{3 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c} e^4}-\frac{3 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^4}-\frac{3 \sqrt{a+b x+c x^2} (-3 b e+4 c d-2 c e x)}{4 e^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 101.117, size = 218, normalized size = 0.96 \[ - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )} + \frac{3 \sqrt{a + b x + c x^{2}} \left (3 b e - 4 c d + 2 c e x\right )}{4 e^{3}} - \frac{3 \left (b e - 2 c d\right ) \sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{4}} + \frac{3 \left (4 a c e^{2} + b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{c} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.50723, size = 253, normalized size = 1.11 \[ \frac{\frac{3 \left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}-\frac{2 e \sqrt{a+x (b+c x)} \left (e (4 a e-9 b d-5 b e x)+2 c \left (6 d^2+3 d e x-e^2 x^2\right )\right )}{d+e x}+12 (b e-2 c d) \log (d+e x) \sqrt{e (a e-b d)+c d^2}+12 (2 c d-b e) \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{8 e^4} \]

Antiderivative was successfully verified.

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

[Out]

((-2*e*Sqrt[a + x*(b + c*x)]*(e*(-9*b*d + 4*a*e - 5*b*e*x) + 2*c*(6*d^2 + 3*d*e*
x - e^2*x^2)))/(d + e*x) + 12*(-2*c*d + b*e)*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Log[
d + e*x] + (3*(8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))*Log[b + 2*c*x + 2*Sqr
t[c]*Sqrt[a + x*(b + c*x)]])/Sqrt[c] + 12*(2*c*d - b*e)*Sqrt[c*d^2 + e*(-(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)]*S
qrt[a + x*(b + c*x)]])/(8*e^4)

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Maple [B]  time = 0.018, size = 3450, normalized size = 15.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8/e/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2
+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)*b^3*d+27/8/e^2/(a
*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d^2*b^2-6/e^3/(a*e^2-b*d*e
+c*d^2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(3/2)*d^3*b-9/2/e/(a*e^2-b*d*e+c*d^2)*ln((
1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2))*c^(1/2)*d*a*b+9/2/e^2/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2
*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2))*c^(3/2)*d^2*a+3/2*c^(1/2)/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*c*d)/
e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2))*a^2-1/(a*e^2-b*d*e+c*d^2)/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*
e^2-b*d*e+c*d^2)/e^2)^(5/2)+1/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b+c/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*
c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x+9/4/(a*e^2-b*d*e+c*d^2)*(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*a*b-3/2/e^2/(a*e^2-b*d
*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c
*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+
x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^3*d^2-9/e^2/(a*e^2-b*d*e+c*d^2)/((
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x
)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*b*d^2*c+3/8/(a*e^2-b*d*e+c*d^2)/c^(1/2)*ln((1/
2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2))*a*b^2-3/2/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/
e+x))*a^2*b-1/e/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*
d*e+c*d^2)/e^2)^(3/2)*c*d+3/2*c/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*a+3/e^5/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*
e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)
/e^2)^(1/2))/(d/e+x))*c^3*d^5-3/e/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e
*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*a*c*d+3/2/e^2/(a*e^2-b*d*e+c*d^2)*(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^2*d^2+21/4/e^2/(
a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)*b*d^2*c-3/2/e/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2
-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c*d+3/e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d
^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
)/(d/e+x))*a^2*c*d+3/e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2
*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*b
^2*d+6/e^3/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*
e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^
2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2*d^3*c-9/4/e
/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*b^2*d+6/e^3/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*
e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*
(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*a*c^2*d
^3-15/2/e^4/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d
*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b*d^4*c^2-3/e^
3/(a*e^2-b*d*e+c*d^2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*c^2*d^3+3/e^4/(a*e^2-b*d*e+c*d^2)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/
2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))*c^(5/2)*d^
4

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.869, 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)^2,x, algorithm="fricas")

[Out]

[-1/16*(12*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*s
qrt(c)*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 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b
*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*
x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4*(2*c*e^3*x^2 - 12*c*d^2*e + 9*b*d*e^2 - 4*a*e^
3 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*(8*c^2*d^3 - 8*b*
c*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*e^3)*
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)))/((e^5*x + d*e^4)*sqrt(c)), 1/16*(24*(2*c*d^2 - b*d*e + (2*c*d*e
- b*e^2)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c)*arctan(-1/2*(b*d - 2*a*e + (2*c
*d - b*e)*x)/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a))) + 4*(2*c*e^3*
x^2 - 12*c*d^2*e + 9*b*d*e^2 - 4*a*e^3 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(c) + 3*(8*c^2*d^3 - 8*b*c*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*
e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*e^3)*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)))/((e^5*x + d*e^4)*sqrt(c)),
-1/8*(6*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt
(-c)*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 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d
- 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)
/(e^2*x^2 + 2*d*e*x + d^2)) - 2*(2*c*e^3*x^2 - 12*c*d^2*e + 9*b*d*e^2 - 4*a*e^3
- (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(8*c^2*d^3 - 8*b*c
*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*e^3)*x
)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((e^5*x + d*e^4)*s
qrt(-c)), 1/8*(12*(2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(-c*d^2 + b*d*e -
a*e^2)*sqrt(-c)*arctan(-1/2*(b*d - 2*a*e + (2*c*d - b*e)*x)/(sqrt(-c*d^2 + b*d*e
- a*e^2)*sqrt(c*x^2 + b*x + a))) + 2*(2*c*e^3*x^2 - 12*c*d^2*e + 9*b*d*e^2 - 4*
a*e^3 - (6*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(8*c^2*d^3 -
8*b*c*d^2*e + (b^2 + 4*a*c)*d*e^2 + (8*c^2*d^2*e - 8*b*c*d*e^2 + (b^2 + 4*a*c)*
e^3)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((e^5*x + d*
e^4)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [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)^2,x, algorithm="giac")

[Out]

Timed out