3.1563 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=309 \[ \frac{3 \sqrt{c} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 e^5}-\frac{3 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^5 \sqrt{a e^2-b d e+c d^2}}-\frac{3 \sqrt{a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{4 e^4 (d+e x)}+\frac{\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.897393, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{3 \sqrt{c} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 e^5}-\frac{3 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^5 \sqrt{a e^2-b d e+c d^2}}-\frac{3 \sqrt{a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{4 e^4 (d+e x)}+\frac{\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 168.77, size = 311, normalized size = 1.01 \[ \frac{3 \sqrt{c} \left (4 a c e^{2} + 3 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{4 e^{5}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (2 b e - 8 c d - 4 c e x\right )}{4 e^{2} \left (d + e x\right )^{2}} - \frac{3 \sqrt{a + b x + c x^{2}} \left (8 a c e^{2} + 2 b^{2} e^{2} - 24 b c d e + 32 c^{2} d^{2} - 8 c e x \left (b e - 2 c d\right )\right )}{8 e^{4} \left (d + e x\right )} - \frac{3 \left (b e - 2 c d\right ) \left (12 a c e^{2} + b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{8 e^{5} \sqrt{a e^{2} - b d e + c d^{2}}} \]

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)**3,x)

[Out]

3*sqrt(c)*(4*a*c*e**2 + 3*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*atanh((b + 2*c*
x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(4*e**5) - (a + b*x + c*x**2)**(3/2)*(2*b
*e - 8*c*d - 4*c*e*x)/(4*e**2*(d + e*x)**2) - 3*sqrt(a + b*x + c*x**2)*(8*a*c*e*
*2 + 2*b**2*e**2 - 24*b*c*d*e + 32*c**2*d**2 - 8*c*e*x*(b*e - 2*c*d))/(8*e**4*(d
+ e*x)) - 3*(b*e - 2*c*d)*(12*a*c*e**2 + b**2*e**2 - 16*b*c*d*e + 16*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**5*sqrt(a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 1.05752, size = 365, normalized size = 1.18 \[ \frac{-\frac{3 (2 c d-b e) \log (d+e x) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right )}{\sqrt{e (a e-b d)+c d^2}}+6 \sqrt{c} \left (4 c e (a e-4 b d)+3 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+\frac{3 (2 c d-b e) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{2 e \sqrt{a+x (b+c x)} \left (-2 c e \left (b \left (18 d^2+28 d e x+7 e^2 x^2\right )-2 a e (d+2 e x)\right )+b e^2 (2 a e+3 b d+5 b e x)+4 c^2 \left (12 d^3+18 d^2 e x+4 d e^2 x^2-e^3 x^3\right )\right )}{(d+e x)^2}}{8 e^5} \]

Antiderivative was successfully verified.

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

[Out]

((-2*e*Sqrt[a + x*(b + c*x)]*(b*e^2*(3*b*d + 2*a*e + 5*b*e*x) + 4*c^2*(12*d^3 +
18*d^2*e*x + 4*d*e^2*x^2 - e^3*x^3) - 2*c*e*(-2*a*e*(d + 2*e*x) + b*(18*d^2 + 28
*d*e*x + 7*e^2*x^2))))/(d + e*x)^2 - (3*(2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 + 4*
c*e*(-4*b*d + 3*a*e))*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] + 6*Sqrt[c]*(
16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*b*d + a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a
+ x*(b + c*x)]] + (3*(2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*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)]])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(8*e^5)

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Maple [B]  time = 0.023, size = 10362, normalized size = 33.5 \[ \text{output too large to display} \]

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)^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)*(2*c*x + b)/(e*x + d)^3,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)^3,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)**3,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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError