3.1626 \(\int \frac{(b+2 c x) \sqrt{d+e x}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=463 \[ \frac{\sqrt{c} e \left (-2 c e \left (-d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{e \sqrt{d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-Sqrt[d + e*x]/(2*(a + b*x + c*x^2)^2) - (e*Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c
*e + c*(2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x
^2)) + (Sqrt[c]*e*(8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sq
rt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - S
qrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*e*(8*c^2*d^2 - b*(b - S
qrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sq
rt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2
]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a
*e^2))

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Rubi [A]  time = 4.04005, antiderivative size = 463, 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 \[ \frac{\sqrt{c} e \left (-2 c e \left (-d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{e \sqrt{d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[d + e*x]/(2*(a + b*x + c*x^2)^2) - (e*Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c
*e + c*(2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x
^2)) + (Sqrt[c]*e*(8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sq
rt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d -
(b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - S
qrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*e*(8*c^2*d^2 - b*(b - S
qrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sq
rt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2
]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a
*e^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((2*c*x+b)*(e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 3.76995, size = 442, normalized size = 0.95 \[ \frac{1}{8} \left (\frac{\sqrt{2} \sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}+6 a e-4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )-8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d} \left (e (b d-a e)-c d^2\right )}+\frac{\sqrt{2} \sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}-b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (e (b d-a e)-c d^2\right )}-\frac{2 \sqrt{d+e x} \left (\frac{e (a+x (b+c x)) \left (-2 c (a e+c d x)+b^2 e+b c (e x-d)\right )}{\left (b^2-4 a c\right ) \left (e (b d-a e)-c d^2\right )}+2\right )}{(a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[d + e*x]*(2 + (e*(a + x*(b + c*x))*(b^2*e - 2*c*(a*e + c*d*x) + b*c*(-
d + e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e)))))/(a + x*(b + c*x))^2 + (S
qrt[2]*Sqrt[c]*e*(-8*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(-4*b*d + S
qrt[b^2 - 4*a*c]*d + 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d
- b*e + Sqrt[b^2 - 4*a*c]*e]])/((b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d + (-b + Sqrt[b^2
- 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e))) + (Sqrt[2]*Sqrt[c]*e*(8*c^2*d^2 + b*(-b
+ Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/((b^2
- 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e
))))/8

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Maple [B]  time = 0.362, size = 42148, normalized size = 91. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*c*x + b)*sqrt(e*x + d)/(c*x^2 + b*x + a)^3, x)

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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((2*c*x + b)*sqrt(e*x + d)/(c*x^2 + b*x + a)^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)*(e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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

[Out]

Timed out