3.2286 \(\int \frac{(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=363 \[ \frac{\left (-2 c e \left (-d \sqrt{b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (-2 c e \left (d \sqrt{b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 3.14828, antiderivative size = 363, 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 \[ \frac{\left (-2 c e \left (-d \sqrt{b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (-2 c e \left (d \sqrt{b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [A]  time = 1.11383, size = 357, normalized size = 0.98 \[ \frac{\left (2 c e \left (d \sqrt{b^2-4 a c}+2 a e-4 b d\right )+b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}-\frac{\left (-2 c e \left (d \sqrt{b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x} (2 a e-b d+b e x-2 c d x)}{\left (b^2-4 a c\right ) (a+x (b+c x))} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/((b^2 - 4*a*c)*(a + x*(b + c*
x))) + ((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 + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d + (-b + Sqr
t[b^2 - 4*a*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 - 2*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]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c
*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Maple [B]  time = 0.079, size = 6226, normalized size = 17.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.281324, size = 3235, normalized size = 8.91 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqr
t((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*
e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 1
2*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*
c^3 - 64*a^3*c^4))*log(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 + 2*sqrt(e^
6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4
*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 -
64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e
^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)
*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a
*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b
^2 + 4*a*c)*e^5)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)
*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c
^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a
^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c
- 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt(1/2)*((b^4 - 8*a*b^2*c
+ 16*a^2*c^2)*e^4 + 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^
3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 1
2*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2
*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 + (b^6*c - 12*a*b^4*c^2
+ 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2
*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2
- 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)*sqrt((32*c^3*d^3 - 48*b*
c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a*b
^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*
b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*lo
g(sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 - 2*sqrt(e^6/(b^6*c^2 - 12*a*b^4
*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sq
rt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12*a*b*c)
*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2
*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e^4 + (3*b^2 + 4*a*c)*e^5)*sqrt
(e*x + d)) - sqrt(1/2)*(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c
)*x)*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a*c^2)*d*e^2 - (b^3 + 12
*a*b*c)*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*sqrt(e^6/(b^6
*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6*c - 12*a*b^4*c^2 + 48*
a^2*b^2*c^3 - 64*a^3*c^4))*log(-sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^4 -
2*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*
b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3 - 48*b*c^2*d^2*e + 6*(3*b^2*c + 4*a
*c^2)*d*e^2 - (b^3 + 12*a*b*c)*e^3 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64
*a^3*c^4)*sqrt(e^6/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)))/(b^6
*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + (16*c^2*d^2*e^3 - 16*b*c*d*e
^4 + (3*b^2 + 4*a*c)*e^5)*sqrt(e*x + d)) - 2*(b*d - 2*a*e + (2*c*d - b*e)*x)*sqr
t(e*x + d))/(a*b^2 - 4*a^2*c + (b^2*c - 4*a*c^2)*x^2 + (b^3 - 4*a*b*c)*x)

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

[Out]

Timed out