3.2338 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=236 \[ \frac{3 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^4 \sqrt{a e^2-b d e+c d^2}}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^4}+\frac{3 \sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.665522, antiderivative size = 236, 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 \[ \frac{3 \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^4 \sqrt{a e^2-b d e+c d^2}}-\frac{3 \sqrt{c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^4}+\frac{3 \sqrt{a+b x+c x^2} (-b e+4 c d+2 c e x)}{4 e^3 (d+e x)}-\frac{\left (a+b x+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 139.829, size = 224, normalized size = 0.95 \[ \frac{3 \sqrt{c} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 e^{4}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{2}} - \frac{3 \sqrt{a + b x + c x^{2}} \left (b e - 4 c d - 2 c e x\right )}{4 e^{3} \left (d + e x\right )} - \frac{3 \left (4 a c e^{2} + b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{8 e^{4} \sqrt{a e^{2} - b d e + c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.597366, size = 272, normalized size = 1.15 \[ \frac{\frac{3 \log (d+e x) \left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{3 \left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{2 e \sqrt{a+x (b+c x)} \left (e (2 a e+3 b d+5 b e x)-2 c \left (6 d^2+9 d e x+2 e^2 x^2\right )\right )}{(d+e x)^2}+12 \sqrt{c} (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{8 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 7299, normalized size = 30.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.6981, 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)^3,x, algorithm="fricas")

[Out]

[-1/16*(12*(2*c*d^3 - b*d^2*e + (2*c*d*e^2 - b*e^3)*x^2 + 2*(2*c*d^2*e - b*d*e^2
)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt
(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(4*c*e^3*x^2 + 12*c*d^2*e - 3
*b*d*e^2 - 2*a*e^3 + (18*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(
c*x^2 + b*x + a) - 3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d
^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 +
(b^2 + 4*a*c)*d*e^3)*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)))/((e^6*x^2 + 2*d*e^5*x
+ d^2*e^4)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/16*(24*(2*c*d^3 - b*d^2*e + (2*c*d*
e^2 - b*e^3)*x^2 + 2*(2*c*d^2*e - b*d*e^2)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(-
c)*arctan(1/2*(2*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) - 4*(4*c*e^3*x^2 + 1
2*c*d^2*e - 3*b*d*e^2 - 2*a*e^3 + (18*c*d*e^2 - 5*b*e^3)*x)*sqrt(c*d^2 - b*d*e +
a*e^2)*sqrt(c*x^2 + b*x + a) - 3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e
^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*
b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*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)))/((e^6*x^
2 + 2*d*e^5*x + d^2*e^4)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/8*(6*(2*c*d^3 - b*d^2*
e + (2*c*d*e^2 - b*e^3)*x^2 + 2*(2*c*d^2*e - b*d*e^2)*x)*sqrt(-c*d^2 + b*d*e - a
*e^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x +
b)*sqrt(c) - 4*a*c) - 2*(4*c*e^3*x^2 + 12*c*d^2*e - 3*b*d*e^2 - 2*a*e^3 + (18*c*
d*e^2 - 5*b*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(8*c^
2*d^4 - 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (b^
2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (b^2 + 4*a*c)*d*e^3)*x)*a
rctan(-1/2*sqrt(-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))))/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(
-c*d^2 + b*d*e - a*e^2)), -1/8*(12*(2*c*d^3 - b*d^2*e + (2*c*d*e^2 - b*e^3)*x^2
+ 2*(2*c*d^2*e - b*d*e^2)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(-c)*arctan(1/2*(2
*c*x + b)/(sqrt(c*x^2 + b*x + a)*sqrt(-c))) - 2*(4*c*e^3*x^2 + 12*c*d^2*e - 3*b*
d*e^2 - 2*a*e^3 + (18*c*d*e^2 - 5*b*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*
x^2 + b*x + a) + 3*(8*c^2*d^4 - 8*b*c*d^3*e + (b^2 + 4*a*c)*d^2*e^2 + (8*c^2*d^2
*e^2 - 8*b*c*d*e^3 + (b^2 + 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (
b^2 + 4*a*c)*d*e^3)*x)*arctan(-1/2*sqrt(-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))))/((e^6*x^2 + 2*
d*e^5*x + d^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)/(e*x + d)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError