3.2330 \(\int \frac{\sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx\)
Optimal. Leaf size=308 \[ \frac{\sqrt{a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]
[Out]
((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*
Sqrt[a + b*x + c*x^2])/(64*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) - (e*(a + b*x
+ c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (5*e*(2*c*d - b*e)*(a
+ b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) - ((b^2 - 4*a*c
)*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*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])])/(128*(c*d^2 - b
*d*e + a*e^2)^(7/2))
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Rubi [A] time = 1.04284, antiderivative size = 308, 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
\[ \frac{\sqrt{a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac{\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}-\frac{5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (a+b x+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x + c*x^2]/(d + e*x)^5,x]
[Out]
((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b*d - 2*a*e + (2*c*d - b*e)*x)*
Sqrt[a + b*x + c*x^2])/(64*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^2) - (e*(a + b*x
+ c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) - (5*e*(2*c*d - b*e)*(a
+ b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3) - ((b^2 - 4*a*c
)*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*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])])/(128*(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((c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)
[Out]
Timed out
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Mathematica [A] time = 1.30008, size = 382, normalized size = 1.24 \[ \frac{-2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (-2 (d+e x)^2 \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )-(d+e x)^3 (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )-8 (d+e x) (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2+48 \left (e (a e-b d)+c d^2\right )^3\right )-3 e \left (b^2-4 a c\right ) (d+e x)^4 \log (d+e x) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )+3 e \left (b^2-4 a c\right ) (d+e x)^4 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{384 e (d+e x)^4 \left (e (a e-b d)+c d^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^5,x]
[Out]
(-2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]*(48*(c*d^2 + e*(-(b*d)
+ a*e))^3 - 8*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x) - 2*(c*d^2 +
e*(-(b*d) + a*e))*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*(d + e*x)^2 -
(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*(d + e*x)^3) - 3
*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(d + e*x)^4*Log[
d + e*x] + 3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(d +
e*x)^4*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)]])/(384*e*(c*d^2 + e*(-(b*d) + a*e))^(7/2)*(d + e*x)^4)
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Maple [B] time = 0.03, size = 7991, normalized size = 25.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x)
[Out]
result too large to display
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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(sqrt(c*x^2 + b*x + a)/(e*x + d)^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 7.9395, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^5,x, algorithm="fricas")
[Out]
[1/768*(4*(48*b*c^2*d^5 + 136*a^2*b*d*e^4 - 48*a^3*e^5 - 16*(3*b^2*c + 14*a*c^2)
*d^4*e + 15*(b^3 + 20*a*b*c)*d^3*e^2 - 2*(59*a*b^2 + 76*a^2*c)*d^2*e^3 + (16*c^3
*d^3*e^2 - 24*b*c^2*d^2*e^3 + 2*(19*b^2*c - 52*a*c^2)*d*e^4 - (15*b^3 - 52*a*b*c
)*e^5)*x^3 + (64*c^3*d^4*e - 104*b*c^2*d^3*e^2 + 20*(7*b^2*c - 16*a*c^2)*d^2*e^3
- (55*b^3 - 164*a*b*c)*d*e^4 + 2*(5*a*b^2 - 12*a^2*c)*e^5)*x^2 + (96*c^3*d^5 -
176*b*c^2*d^4*e - 8*a^2*b*e^5 + 2*(99*b^2*c - 148*a*c^2)*d^3*e^2 - (73*b^3 - 124
*a*b*c)*d^2*e^3 + 4*(9*a*b^2 - 8*a^2*c)*d*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sq
rt(c*x^2 + b*x + a) + 3*(16*(b^2*c^2 - 4*a*c^3)*d^6 - 16*(b^3*c - 4*a*b*c^2)*d^5
*e + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^4*e^2 + (16*(b^2*c^2 - 4*a*c^3)*d^2*e^4
- 16*(b^3*c - 4*a*b*c^2)*d*e^5 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*e^6)*x^4 + 4
*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c - 4*a*b*c^2)*d^2*e^4 + (5*b^4 - 24*
a*b^2*c + 16*a^2*c^2)*d*e^5)*x^3 + 6*(16*(b^2*c^2 - 4*a*c^3)*d^4*e^2 - 16*(b^3*c
- 4*a*b*c^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^2*e^4)*x^2 + 4*(16*(
b^2*c^2 - 4*a*c^3)*d^5*e - 16*(b^3*c - 4*a*b*c^2)*d^4*e^2 + (5*b^4 - 24*a*b^2*c
+ 16*a^2*c^2)*d^3*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^3*d^10 - 3*b*c^2
*d^9*e - 3*a^2*b*d^5*e^5 + a^3*d^4*e^6 + 3*(b^2*c + a*c^2)*d^8*e^2 - (b^3 + 6*a*
b*c)*d^7*e^3 + 3*(a*b^2 + a^2*c)*d^6*e^4 + (c^3*d^6*e^4 - 3*b*c^2*d^5*e^5 - 3*a^
2*b*d*e^9 + a^3*e^10 + 3*(b^2*c + a*c^2)*d^4*e^6 - (b^3 + 6*a*b*c)*d^3*e^7 + 3*(
a*b^2 + a^2*c)*d^2*e^8)*x^4 + 4*(c^3*d^7*e^3 - 3*b*c^2*d^6*e^4 - 3*a^2*b*d^2*e^8
+ a^3*d*e^9 + 3*(b^2*c + a*c^2)*d^5*e^5 - (b^3 + 6*a*b*c)*d^4*e^6 + 3*(a*b^2 +
a^2*c)*d^3*e^7)*x^3 + 6*(c^3*d^8*e^2 - 3*b*c^2*d^7*e^3 - 3*a^2*b*d^3*e^7 + a^3*d
^2*e^8 + 3*(b^2*c + a*c^2)*d^6*e^4 - (b^3 + 6*a*b*c)*d^5*e^5 + 3*(a*b^2 + a^2*c)
*d^4*e^6)*x^2 + 4*(c^3*d^9*e - 3*b*c^2*d^8*e^2 - 3*a^2*b*d^4*e^6 + a^3*d^3*e^7 +
3*(b^2*c + a*c^2)*d^7*e^3 - (b^3 + 6*a*b*c)*d^6*e^4 + 3*(a*b^2 + a^2*c)*d^5*e^5
)*x)*sqrt(c*d^2 - b*d*e + a*e^2)), 1/384*(2*(48*b*c^2*d^5 + 136*a^2*b*d*e^4 - 48
*a^3*e^5 - 16*(3*b^2*c + 14*a*c^2)*d^4*e + 15*(b^3 + 20*a*b*c)*d^3*e^2 - 2*(59*a
*b^2 + 76*a^2*c)*d^2*e^3 + (16*c^3*d^3*e^2 - 24*b*c^2*d^2*e^3 + 2*(19*b^2*c - 52
*a*c^2)*d*e^4 - (15*b^3 - 52*a*b*c)*e^5)*x^3 + (64*c^3*d^4*e - 104*b*c^2*d^3*e^2
+ 20*(7*b^2*c - 16*a*c^2)*d^2*e^3 - (55*b^3 - 164*a*b*c)*d*e^4 + 2*(5*a*b^2 - 1
2*a^2*c)*e^5)*x^2 + (96*c^3*d^5 - 176*b*c^2*d^4*e - 8*a^2*b*e^5 + 2*(99*b^2*c -
148*a*c^2)*d^3*e^2 - (73*b^3 - 124*a*b*c)*d^2*e^3 + 4*(9*a*b^2 - 8*a^2*c)*d*e^4)
*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(16*(b^2*c^2 - 4*a*c^
3)*d^6 - 16*(b^3*c - 4*a*b*c^2)*d^5*e + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^4*e^
2 + (16*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 16*(b^3*c - 4*a*b*c^2)*d*e^5 + (5*b^4 - 24
*a*b^2*c + 16*a^2*c^2)*e^6)*x^4 + 4*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c
- 4*a*b*c^2)*d^2*e^4 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*e^5)*x^3 + 6*(16*(b^2
*c^2 - 4*a*c^3)*d^4*e^2 - 16*(b^3*c - 4*a*b*c^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c +
16*a^2*c^2)*d^2*e^4)*x^2 + 4*(16*(b^2*c^2 - 4*a*c^3)*d^5*e - 16*(b^3*c - 4*a*b*
c^2)*d^4*e^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*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)*s
qrt(c*x^2 + b*x + a))))/((c^3*d^10 - 3*b*c^2*d^9*e - 3*a^2*b*d^5*e^5 + a^3*d^4*e
^6 + 3*(b^2*c + a*c^2)*d^8*e^2 - (b^3 + 6*a*b*c)*d^7*e^3 + 3*(a*b^2 + a^2*c)*d^6
*e^4 + (c^3*d^6*e^4 - 3*b*c^2*d^5*e^5 - 3*a^2*b*d*e^9 + a^3*e^10 + 3*(b^2*c + a*
c^2)*d^4*e^6 - (b^3 + 6*a*b*c)*d^3*e^7 + 3*(a*b^2 + a^2*c)*d^2*e^8)*x^4 + 4*(c^3
*d^7*e^3 - 3*b*c^2*d^6*e^4 - 3*a^2*b*d^2*e^8 + a^3*d*e^9 + 3*(b^2*c + a*c^2)*d^5
*e^5 - (b^3 + 6*a*b*c)*d^4*e^6 + 3*(a*b^2 + a^2*c)*d^3*e^7)*x^3 + 6*(c^3*d^8*e^2
- 3*b*c^2*d^7*e^3 - 3*a^2*b*d^3*e^7 + a^3*d^2*e^8 + 3*(b^2*c + a*c^2)*d^6*e^4 -
(b^3 + 6*a*b*c)*d^5*e^5 + 3*(a*b^2 + a^2*c)*d^4*e^6)*x^2 + 4*(c^3*d^9*e - 3*b*c
^2*d^8*e^2 - 3*a^2*b*d^4*e^6 + a^3*d^3*e^7 + 3*(b^2*c + a*c^2)*d^7*e^3 - (b^3 +
6*a*b*c)*d^6*e^4 + 3*(a*b^2 + a^2*c)*d^5*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. \[ \int \frac{\sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)
[Out]
Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**5, x)
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GIAC/XCAS [A] time = 11.4029, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)/(e*x + d)^5,x, algorithm="giac")
[Out]
Done