3.2289 \(\int \frac{1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=604 \[ -\frac{e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 10.9386, antiderivative size = 604, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((e*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*
e + a*e^2)^2*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)*Sqrt[d + e*x]*(a + b*x + c*x^2)) + (Sqrt[c]*
(8*c^3*d^3 + 3*b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4
*a*c]*d - 16*a*e) - 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e + 5*a*Sqrt[
b^2 - 4*a*c]*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)^2) - (Sqrt[c]*(8*c^3*d^3 + 3*b^2*(b - Sqrt[b^2 -
4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 16*a*e) - 2*c*e^2*(b^2*d
- b*Sqrt[b^2 - 4*a*c]*d + 8*a*b*e - 5*a*Sqrt[b^2 - 4*a*c]*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)^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)**(3/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 6.3641, size = 690, normalized size = 1.14 \[ \sqrt{d+e x} \left (\frac{-3 a b c e^2+4 a c^2 d e-2 a c^2 e^2 x+b^3 e^2-2 b^2 c d e+b^2 c e^2 x+b c^2 d^2-2 b c^2 d e x+2 c^3 d^2 x}{\left (4 a c-b^2\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 e^3}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}\right )-\frac{\left (-2 c^3 d^2 e \sqrt{b^2-4 a c}+2 b c^2 d e^2 \sqrt{b^2-4 a c}+10 a c^2 e^3 \sqrt{b^2-4 a c}-3 b^2 c e^3 \sqrt{b^2-4 a c}-16 a b c^2 e^3+32 a c^3 d e^2+3 b^3 c e^3-2 b^2 c^2 d e^2-12 b c^3 d^2 e+8 c^4 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )^2}-\frac{\left (-2 c^3 d^2 e \sqrt{b^2-4 a c}+2 b c^2 d e^2 \sqrt{b^2-4 a c}+10 a c^2 e^3 \sqrt{b^2-4 a c}-3 b^2 c e^3 \sqrt{b^2-4 a c}+16 a b c^2 e^3-32 a c^3 d e^2-3 b^3 c e^3+2 b^2 c^2 d e^2+12 b c^3 d^2 e-8 c^4 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d} \left (-a e^2+b d e-c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*((-2*e^3)/((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)) + (b*c^2*d^2 - 2*b
^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 - 3*a*b*c*e^2 + 2*c^3*d^2*x - 2*b*c^2*d*e*x + b
^2*c*e^2*x - 2*a*c^2*e^2*x)/((-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x +
c*x^2))) - ((8*c^4*d^3 - 12*b*c^3*d^2*e - 2*c^3*Sqrt[b^2 - 4*a*c]*d^2*e - 2*b^2
*c^2*d*e^2 + 32*a*c^3*d*e^2 + 2*b*c^2*Sqrt[b^2 - 4*a*c]*d*e^2 + 3*b^3*c*e^3 - 16
*a*b*c^2*e^3 - 3*b^2*c*Sqrt[b^2 - 4*a*c]*e^3 + 10*a*c^2*Sqrt[b^2 - 4*a*c]*e^3)*A
rcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])
/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*(-
(c*d^2) + b*d*e - a*e^2)^2) - ((-8*c^4*d^3 + 12*b*c^3*d^2*e - 2*c^3*Sqrt[b^2 - 4
*a*c]*d^2*e + 2*b^2*c^2*d*e^2 - 32*a*c^3*d*e^2 + 2*b*c^2*Sqrt[b^2 - 4*a*c]*d*e^2
- 3*b^3*c*e^3 + 16*a*b*c^2*e^3 - 3*b^2*c*Sqrt[b^2 - 4*a*c]*e^3 + 10*a*c^2*Sqrt[
b^2 - 4*a*c]*e^3)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqr
t[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - b*e + Sqrt
[b^2 - 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2)^2)

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Maple [B]  time = 0.107, size = 10471, normalized size = 17.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{2}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^2 + b*x + a)^2*(e*x + d)^(3/2)), 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*(e*x + d)^(3/2)),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)**(3/2)/(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*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out