3.2370 \(\int \frac{1}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=293 \[ -\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \sqrt{a+b x+c x^2} (2 c d-b e)}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(e*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (5*e*(2*c*d
- b*e)*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(
44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + b*x + c*x^2])/(24*(c*
d^2 - b*d*e + a*e^2)^3*(d + e*x)) + ((2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(2*b*d + 3*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])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 0.847035, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{e \sqrt{a+b x+c x^2} \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{(2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \sqrt{a+b x+c x^2} (2 c d-b e)}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \sqrt{a+b x+c x^2}}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(e*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (5*e*(2*c*d
- b*e)*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(
44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*Sqrt[a + b*x + c*x^2])/(24*(c*
d^2 - b*d*e + a*e^2)^3*(d + e*x)) + ((2*c*d - b*e)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(2*b*d + 3*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])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

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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)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.56009, size = 315, normalized size = 1.08 \[ \frac{-2 e \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left ((d+e x)^2 \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )+10 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )+8 \left (e (a e-b d)+c d^2\right )^2\right )+3 (d+e x)^3 (2 c d-b e) \log (d+e x) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-3 (d+e x)^3 (2 c d-b e) \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{48 (d+e x)^3 \left (e (a e-b d)+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*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)]*(8*(c*d^2 + e*(-(b*d)
+ a*e))^2 + 10*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x) + (44*c^2*d^2
+ 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e))*(d + e*x)^2) + 3*(2*c*d - b*e)*(8*c^2*d^
2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(d + e*x)^3*Log[d + e*x] - 3*(2*c*d - b*e
)*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(d + e*x)^3*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)]])/(4
8*(c*d^2 + e*(-(b*d) + a*e))^(7/2)*(d + e*x)^3)

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Maple [B]  time = 0.025, size = 1665, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(d/e+x)^3*(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)+5/12/(a*e^2-b*d*e+c*d^2)^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)*b-5/6/e/(a*e^2-b*d*e+c*d^2)^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)*c*d-5/
8*e^2/(a*e^2-b*d*e+c*d^2)^3/(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^2+5/2*e/(a*e^2-b*d*e+c*d^2)^3/(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*c*d-5/2/(a*e^2-b*d*e+c*d^2)^3/
(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^2*d^
2+5/16*e^2/(a*e^2-b*d*e+c*d^2)^3/((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^3-15/8*e/(a
*e^2-b*d*e+c*d^2)^3/((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*c*d+15/4/(a*e^2-b*d*e+
c*d^2)^3/((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^2*d^2-5/2/e/(a*e^2-b*d*e+c*d^2)^3
/((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^3*d^3-3/4/(a*e^2-b*d*e+c*d^2)^2*c/((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+3/2/e/(a*e^2-b*d*e+c*d^2)^2*c^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))*d+2/3*c/(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)

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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)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.6252, 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)^4),x, algorithm="fricas")

[Out]

[-1/96*(4*(72*c^2*d^4*e - 90*b*c*d^3*e^2 - 26*a*b*d*e^4 + 8*a^2*e^5 + (33*b^2 +
20*a*c)*d^2*e^3 + (44*c^2*d^2*e^3 - 44*b*c*d*e^4 + (15*b^2 - 16*a*c)*e^5)*x^2 +
2*(54*c^2*d^3*e^2 - 59*b*c*d^2*e^3 - 5*a*b*e^5 + 2*(10*b^2 - 3*a*c)*d*e^4)*x)*sq
rt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) - 3*(16*c^3*d^6 - 24*b*c^2*d^5*e
+ 6*(3*b^2*c - 4*a*c^2)*d^4*e^2 - (5*b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3
- 24*b*c^2*d^2*e^4 + 6*(3*b^2*c - 4*a*c^2)*d*e^5 - (5*b^3 - 12*a*b*c)*e^6)*x^3 +
3*(16*c^3*d^4*e^2 - 24*b*c^2*d^3*e^3 + 6*(3*b^2*c - 4*a*c^2)*d^2*e^4 - (5*b^3 -
12*a*b*c)*d*e^5)*x^2 + 3*(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(3*b^2*c - 4*a*c^
2)*d^3*e^3 - (5*b^3 - 12*a*b*c)*d^2*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)))/((
c^3*d^9 - 3*b*c^2*d^8*e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*
e^2 - (b^3 + 6*a*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c
^2*d^5*e^4 - 3*a^2*b*d*e^8 + a^3*e^9 + 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*
c)*d^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 -
3*a^2*b*d^2*e^7 + a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e
^5 + 3*(a*b^2 + a^2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d
^3*e^6 + a^3*d^2*e^7 + 3*(b^2*c + a*c^2)*d^6*e^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(
a*b^2 + a^2*c)*d^4*e^5)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), -1/48*(2*(72*c^2*d^4*e
- 90*b*c*d^3*e^2 - 26*a*b*d*e^4 + 8*a^2*e^5 + (33*b^2 + 20*a*c)*d^2*e^3 + (44*c^
2*d^2*e^3 - 44*b*c*d*e^4 + (15*b^2 - 16*a*c)*e^5)*x^2 + 2*(54*c^2*d^3*e^2 - 59*b
*c*d^2*e^3 - 5*a*b*e^5 + 2*(10*b^2 - 3*a*c)*d*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^
2)*sqrt(c*x^2 + b*x + a) + 3*(16*c^3*d^6 - 24*b*c^2*d^5*e + 6*(3*b^2*c - 4*a*c^2
)*d^4*e^2 - (5*b^3 - 12*a*b*c)*d^3*e^3 + (16*c^3*d^3*e^3 - 24*b*c^2*d^2*e^4 + 6*
(3*b^2*c - 4*a*c^2)*d*e^5 - (5*b^3 - 12*a*b*c)*e^6)*x^3 + 3*(16*c^3*d^4*e^2 - 24
*b*c^2*d^3*e^3 + 6*(3*b^2*c - 4*a*c^2)*d^2*e^4 - (5*b^3 - 12*a*b*c)*d*e^5)*x^2 +
3*(16*c^3*d^5*e - 24*b*c^2*d^4*e^2 + 6*(3*b^2*c - 4*a*c^2)*d^3*e^3 - (5*b^3 - 1
2*a*b*c)*d^2*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))))/((c^3*d^9 - 3*b*
c^2*d^8*e - 3*a^2*b*d^4*e^5 + a^3*d^3*e^6 + 3*(b^2*c + a*c^2)*d^7*e^2 - (b^3 + 6
*a*b*c)*d^6*e^3 + 3*(a*b^2 + a^2*c)*d^5*e^4 + (c^3*d^6*e^3 - 3*b*c^2*d^5*e^4 - 3
*a^2*b*d*e^8 + a^3*e^9 + 3*(b^2*c + a*c^2)*d^4*e^5 - (b^3 + 6*a*b*c)*d^3*e^6 + 3
*(a*b^2 + a^2*c)*d^2*e^7)*x^3 + 3*(c^3*d^7*e^2 - 3*b*c^2*d^6*e^3 - 3*a^2*b*d^2*e
^7 + a^3*d*e^8 + 3*(b^2*c + a*c^2)*d^5*e^4 - (b^3 + 6*a*b*c)*d^4*e^5 + 3*(a*b^2
+ a^2*c)*d^3*e^6)*x^2 + 3*(c^3*d^8*e - 3*b*c^2*d^7*e^2 - 3*a^2*b*d^3*e^6 + a^3*d
^2*e^7 + 3*(b^2*c + a*c^2)*d^6*e^3 - (b^3 + 6*a*b*c)*d^5*e^4 + 3*(a*b^2 + a^2*c)
*d^4*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. \[ \int \frac{1}{\left (d + e x\right )^{4} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(1/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.636361, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^4),x, algorithm="giac")

[Out]

sage0*x