3.2288 \(\int \frac{1}{\sqrt{d+e x} \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=428 \[ \frac{\sqrt{c} \left (-2 c e \left (-d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(
c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) + (Sqrt[c]*(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]])/(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))
- (Sqrt[c]*(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 + S
qrt[b^2 - 4*a*c])*e]])/(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 = 3.64563, antiderivative size = 428, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{c} \left (-2 c e \left (-d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(
c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) + (Sqrt[c]*(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]])/(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))
- (Sqrt[c]*(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 + S
qrt[b^2 - 4*a*c])*e]])/(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(1/(e*x+d)**(1/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.2921, size = 389, normalized size = 0.91 \[ \frac{\frac{\sqrt{2} \sqrt{c} \left (-2 c e \left (d \sqrt{b^2-4 a c}+6 a e-4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )-8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{\sqrt{2} \sqrt{c} \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}-b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{2 \sqrt{d+e x} \left (2 c (a e+c d x)+b^2 (-e)+b c (d-e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}}{2 \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.092, size = 1114, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2
))*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^
2))^(1/2))+4*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d+(-4*a*c*e^
2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan
h(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b-8*e
*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2)
)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2
)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d-6*e*c^2/(-e^2*(4*a*
c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e
+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e
+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)+2*e*c/(-e^
2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c
*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))
+4*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)
^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^
(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b-8*e*c^3/(-e^2*(4
*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b
*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e
-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d+6*e*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4
*a*c-b^2)/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a
*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c
-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{2} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^2 + b*x + a)^2*sqrt(e*x + d)), 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(1/((c*x^2 + b*x + a)^2*sqrt(e*x + d)),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(1/(e*x+d)**(1/2)/(c*x**2+b*x+a)**2,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(1/((c*x^2 + b*x + a)^2*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out