3.2369 \(\int \frac{1}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=209 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.607131, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{3 e \sqrt{a+b x+c x^2} (2 c d-b e)}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 113.116, size = 192, normalized size = 0.92 \[ \frac{3 e \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{4 \left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{e \sqrt{a + b x + c x^{2}}}{2 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )} - \frac{\left (- \frac{a c e^{2}}{2} + \frac{3 b^{2} e^{2}}{8} - b c d e + c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*e*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)/(4*(d + e*x)*(a*e**2 - b*d*e + c*d**2)*
*2) - e*sqrt(a + b*x + c*x**2)/(2*(d + e*x)**2*(a*e**2 - b*d*e + c*d**2)) - (-a*
c*e**2/2 + 3*b**2*e**2/8 - b*c*d*e + 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)))/(a*e**2 - b*d*e
+ c*d**2)**(5/2)

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Mathematica [A]  time = 0.590859, size = 232, normalized size = 1.11 \[ \frac{(d+e x)^2 \log (d+e x) \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right )-(d+e x)^2 \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )-2 e \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} (e (2 a e-5 b d-3 b e x)+2 c d (4 d+3 e x))}{8 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 959, normalized size = 4.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/e/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2)+3/4*e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-3/2/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)
*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d-3/8*e/(a*
e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2
+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^2+3/2/(a*e^2-b*d*e+c*d^2)
^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(
d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e
^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b*c*d-3/2/e/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b
*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d
^2)/e^2)^(1/2))/(d/e+x))*c^2*d^2+1/2/e*c/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2))/(d/e+x))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.39513, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(8*c*d^2*e - 5*b*d*e^2 + 2*a*e^3 + 3*(2*c*d*e^2 - b*e^3)*x)*sqrt(c*d^2
- b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) + (8*c^2*d^4 - 8*b*c*d^3*e + (3*b^2 - 4*
a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + (3*b^2 - 4*a*c)*e^4)*x^2 + 2*(8*c^
2*d^3*e - 8*b*c*d^2*e^2 + (3*b^2 - 4*a*c)*d*e^3)*x)*log(((8*a*b*d*e - 8*a^2*e^2
- (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c
*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) + 4*(b*c*
d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e -
a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d
^2)))/((c^2*d^6 - 2*b*c*d^5*e - 2*a*b*d^3*e^3 + a^2*d^2*e^4 + (b^2 + 2*a*c)*d^4*
e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a*c)*d^2*e
^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + (b^2 + 2*a*
c)*d^3*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/8*(2*(8*c*d^2*e - 5*b*d*e^2 + 2*
a*e^3 + 3*(2*c*d*e^2 - b*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x +
a) + (8*c^2*d^4 - 8*b*c*d^3*e + (3*b^2 - 4*a*c)*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*
c*d*e^3 + (3*b^2 - 4*a*c)*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + (3*b^2 - 4
*a*c)*d*e^3)*x)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d -
b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))))/((c^2*d^6 - 2*b*c*d^5
*e - 2*a*b*d^3*e^3 + a^2*d^2*e^4 + (b^2 + 2*a*c)*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*
d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*
b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3)*x)*sqrt(-c*d^2
+ b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.565781, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x