3.2377 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.681525, antiderivative size = 255, 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 \[ -\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 120.427, size = 243, normalized size = 0.95 \[ \frac{3 e^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} - \frac{e \sqrt{a + b x + c x^{2}} \left (- 8 a c e^{2} + 3 b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}\right )}{\left (d + e x\right ) \left (- 4 a c + b^{2}\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{2 \left (- 2 a c e + b^{2} e - b c d + c x \left (b e - 2 c d\right )\right )}{\left (d + e x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 2.12749, size = 263, normalized size = 1.03 \[ -\frac{\sqrt{a+x (b+c x)} \left (\frac{2 \left (b c \left (c d (d-2 e x)-3 a e^2\right )+2 c^2 \left (a e (2 d-e x)+c d^2 x\right )+b^3 e^2+b^2 c e (e x-2 d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{e^3}{d+e x}\right )}{\left (e (a e-b d)+c d^2\right )^2}+\frac{3 e^2 (2 c d-b e) \log (d+e x)}{2 \left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{3 e^2 (b e-2 c d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{2 \left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 1313, normalized size = 5.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*e^2-b*d*e+c*d^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)-3/2*e^2/(a*e^2-b*d*e+c*d^2)^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)*b+3*e/(a*e^2-b*d*e+c*d^2)^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)*c*d+3*e^2/(a*e^2-b*d*e+c*d^2)^2/(
4*a*c-b^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)*x*b
^2*c-12*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*x*b*c^2*d+12/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*x*c^3*d^2+3/2*e^2
/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*b^3-6*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*b^2*c*d+6/(a*e^2-b*d*e+c*d^2)^2
/(4*a*c-b^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)*b
*c^2*d^2+3/2*e^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-3*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*d-8*c^2/(a*e^2-b*d*e+c*
d^2)/(4*a*c-b^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)*x-4*c/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.898539, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(4*(2*b*c^2*d^3 - 4*(b^2*c - 2*a*c^2)*d^2*e + 2*(b^3 - 3*a*b*c)*d*e^2 + (a
*b^2 - 4*a^2*c)*e^3 + (4*c^3*d^2*e - 4*b*c^2*d*e^2 + (3*b^2*c - 8*a*c^2)*e^3)*x^
2 + (4*c^3*d^3 - 2*b*c^2*d^2*e - 2*(b^2*c - 2*a*c^2)*d*e^2 + (3*b^3 - 10*a*b*c)*
e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(2*(a*b^2*c - 4*a^
2*c^2)*d^2*e^2 - (a*b^3 - 4*a^2*b*c)*d*e^3 + (2*(b^2*c^2 - 4*a*c^3)*d*e^3 - (b^3
*c - 4*a*b*c^2)*e^4)*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e^2 + (b^3*c - 4*a*b*c^2)*
d*e^3 - (b^4 - 4*a*b^2*c)*e^4)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^2 - (b^4 - 6*a
*b^2*c + 8*a^2*c^2)*d*e^3 - (a*b^3 - 4*a^2*b*c)*e^4)*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)))/(((a*b^2*c^2 - 4*a^2*c^3)*d^5 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^4*e + (a*b
^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^2*e^3 + (a^3*b
^2 - 4*a^4*c)*d*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4*e - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e
^2 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^3 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^4
+ (a^2*b^2*c - 4*a^3*c^2)*e^5)*x^3 + ((b^2*c^3 - 4*a*c^4)*d^5 - (b^3*c^2 - 4*a*
b*c^3)*d^4*e - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^2 + (b^5 - 4*a*b^3*c)*d^2
*e^3 - (2*a*b^4 - 9*a^2*b^2*c + 4*a^3*c^2)*d*e^4 + (a^2*b^3 - 4*a^3*b*c)*e^5)*x^
2 + ((b^3*c^2 - 4*a*b*c^3)*d^5 - (2*b^4*c - 9*a*b^2*c^2 + 4*a^2*c^3)*d^4*e + (b^
5 - 4*a*b^3*c)*d^3*e^2 - (a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d^2*e^3 - (a^2*b^3 -
4*a^3*b*c)*d*e^4 + (a^3*b^2 - 4*a^4*c)*e^5)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/
2*(2*(2*b*c^2*d^3 - 4*(b^2*c - 2*a*c^2)*d^2*e + 2*(b^3 - 3*a*b*c)*d*e^2 + (a*b^2
- 4*a^2*c)*e^3 + (4*c^3*d^2*e - 4*b*c^2*d*e^2 + (3*b^2*c - 8*a*c^2)*e^3)*x^2 +
(4*c^3*d^3 - 2*b*c^2*d^2*e - 2*(b^2*c - 2*a*c^2)*d*e^2 + (3*b^3 - 10*a*b*c)*e^3)
*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(2*(a*b^2*c - 4*a^2*c
^2)*d^2*e^2 - (a*b^3 - 4*a^2*b*c)*d*e^3 + (2*(b^2*c^2 - 4*a*c^3)*d*e^3 - (b^3*c
- 4*a*b*c^2)*e^4)*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e^2 + (b^3*c - 4*a*b*c^2)*d*e
^3 - (b^4 - 4*a*b^2*c)*e^4)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^2 - (b^4 - 6*a*b^
2*c + 8*a^2*c^2)*d*e^3 - (a*b^3 - 4*a^2*b*c)*e^4)*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))))/(((a*b^2*c^2 - 4*a^2*c^3)*d^5 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^4*e
+ (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^2*e^3 +
(a^3*b^2 - 4*a^4*c)*d*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4*e - 2*(b^3*c^2 - 4*a*b*c^3)
*d^3*e^2 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^3 - 2*(a*b^3*c - 4*a^2*b*c^2)
*d*e^4 + (a^2*b^2*c - 4*a^3*c^2)*e^5)*x^3 + ((b^2*c^3 - 4*a*c^4)*d^5 - (b^3*c^2
- 4*a*b*c^3)*d^4*e - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*e^2 + (b^5 - 4*a*b^3*
c)*d^2*e^3 - (2*a*b^4 - 9*a^2*b^2*c + 4*a^3*c^2)*d*e^4 + (a^2*b^3 - 4*a^3*b*c)*e
^5)*x^2 + ((b^3*c^2 - 4*a*b*c^3)*d^5 - (2*b^4*c - 9*a*b^2*c^2 + 4*a^2*c^3)*d^4*e
+ (b^5 - 4*a*b^3*c)*d^3*e^2 - (a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*d^2*e^3 - (a^2*
b^3 - 4*a^3*b*c)*d*e^4 + (a^3*b^2 - 4*a^4*c)*e^5)*x)*sqrt(-c*d^2 + b*d*e - a*e^2
))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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