3.1553 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=280 \[ -\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^3 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^3} \]

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

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Rubi [A]  time = 0.759696, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^3 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{\sqrt{a+b x+c x^2} \left (e x \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )-2 c d e (3 b d-2 a e)-b e^2 (b d-2 a e)+8 c^2 d^3\right )}{4 e^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 142.613, size = 272, normalized size = 0.97 \[ \frac{2 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{e^{3}} - \frac{\sqrt{a + b x + c x^{2}} \left (\frac{d \left (b e - 2 c d\right )^{2}}{2} + \frac{e x \left (8 a c e^{2} + b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{2} + \left (b e + 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )\right )}{2 e^{2} \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (b e - 2 c d\right ) \left (- 12 a c e^{2} + b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{8 e^{3} \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

2*c**(3/2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/e**3 - sqrt(a +
b*x + c*x**2)*(d*(b*e - 2*c*d)**2/2 + e*x*(8*a*c*e**2 + b**2*e**2 - 12*b*c*d*e
+ 12*c**2*d**2)/2 + (b*e + 2*c*d)*(a*e**2 - b*d*e + c*d**2))/(2*e**2*(d + e*x)**
2*(a*e**2 - b*d*e + c*d**2)) + (b*e - 2*c*d)*(-12*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**3*(a*e**2 - b*d*e + c*d**2)**(3/2))

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Mathematica [A]  time = 1.17289, size = 318, normalized size = 1.14 \[ \frac{-\frac{(2 c d-b e) \log (d+e x) \left (4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{(2 c d-b e) \left (4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+16 c^{3/2} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-\frac{2 e \sqrt{a+x (b+c x)} \left (2 c e (d+2 e x) (2 a e-3 b d)+b e^2 (2 a e-b d+b e x)+4 c^2 d^2 (2 d+3 e x)\right )}{(d+e x)^2 \left (e (a e-b d)+c d^2\right )}}{8 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((-2*e*Sqrt[a + x*(b + c*x)]*(2*c*e*(-3*b*d + 2*a*e)*(d + 2*e*x) + 4*c^2*d^2*(2*
d + 3*e*x) + b*e^2*(-(b*d) + 2*a*e + b*e*x)))/((c*d^2 + e*(-(b*d) + a*e))*(d + e
*x)^2) - ((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) + 16*c^(3/2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqr
t[a + x*(b + c*x)]] + ((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))/(8*e^3)

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Maple [B]  time = 0.017, size = 5046, normalized size = 18. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(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(sqrt(c*x^2 + b*x + a)*(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(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d)^3,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)

[Out]

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

[Out]

Exception raised: TypeError