3.1555 \(\int \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=307 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (4 a e+b d)+5 b^2 e^2+4 c^2 d^2\right )}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{5 e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{5 e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-5*(b^2 - 4*a*c)*e*(2*c*d - b*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) + ((2*c*d - b*e)*(a + b*x +
c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + ((4*c^2*d^2 + 5*b^2*e^2
- 4*c*e*(b*d + 4*a*e))*(a + b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d
+ e*x)^3) + (5*(b^2 - 4*a*c)^2*e*(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])])/(128*(c*d^2 - b*
d*e + a*e^2)^(7/2))

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Rubi [A]  time = 1.05661, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-4 c e (4 a e+b d)+5 b^2 e^2+4 c^2 d^2\right )}{24 (d+e x)^3 \left (a e^2-b d e+c d^2\right )^2}-\frac{5 e \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{64 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}+\frac{5 e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{4 (d+e x)^4 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^5,x]

[Out]

(-5*(b^2 - 4*a*c)*e*(2*c*d - b*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) + ((2*c*d - b*e)*(a + b*x +
c*x^2)^(3/2))/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + ((4*c^2*d^2 + 5*b^2*e^2
- 4*c*e*(b*d + 4*a*e))*(a + b*x + c*x^2)^(3/2))/(24*(c*d^2 - b*d*e + a*e^2)^2*(d
+ e*x)^3) + (5*(b^2 - 4*a*c)^2*e*(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])])/(128*(c*d^2 - b*
d*e + a*e^2)^(7/2))

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Rubi in Sympy [A]  time = 154.373, size = 291, normalized size = 0.95 \[ \frac{5 e \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{128 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{7}{2}}} - \frac{5 e \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{64 \left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )^{3}} + \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 4 a c e^{2} + \frac{5 b^{2} e^{2}}{4} - b c d e + c^{2} d^{2}\right )}{6 \left (d + e x\right )^{3} \left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{\left (\frac{b e}{4} - \frac{c d}{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{4} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)

[Out]

5*e*(-4*a*c + b**2)**2*(b*e - 2*c*d)*atanh((2*a*e - b*d + x*(b*e - 2*c*d))/(2*sq
rt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2)))/(128*(a*e**2 - b*d*e + c*d*
*2)**(7/2)) - 5*e*(-4*a*c + b**2)*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)*(2*a*e -
b*d + x*(b*e - 2*c*d))/(64*(d + e*x)**2*(a*e**2 - b*d*e + c*d**2)**3) + (a + b*x
+ c*x**2)**(3/2)*(-4*a*c*e**2 + 5*b**2*e**2/4 - b*c*d*e + c**2*d**2)/(6*(d + e*
x)**3*(a*e**2 - b*d*e + c*d**2)**2) - (b*e/4 - c*d/2)*(a + b*x + c*x**2)**(3/2)/
((d + e*x)**4*(a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 1.39269, size = 429, normalized size = 1.4 \[ \frac{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left ((d+e x)^3 \left (-4 c^2 e^2 \left (32 a^2 e^2+36 a b d e-3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )-8 (d+e x) \left (4 c e (4 a e-5 b d)+b^2 e^2+20 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2+2 (d+e x)^2 (2 c d-b e) \left (4 c e (7 a e-2 b d)-5 b^2 e^2+8 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )+48 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3\right )+15 e^3 \left (b^2-4 a c\right )^2 (d+e x)^4 (b e-2 c d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )-15 e^3 \left (b^2-4 a c\right )^2 (d+e x)^4 (b e-2 c d) \log (d+e x)}{384 e^2 (d+e x)^4 \left (e (a e-b d)+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*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*(2*c*d - b*e)*(c*d^2
+ e*(-(b*d) + a*e))^3 - 8*(c*d^2 + e*(-(b*d) + a*e))^2*(20*c^2*d^2 + b^2*e^2 +
4*c*e*(-5*b*d + 4*a*e))*(d + e*x) + 2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(
8*c^2*d^2 - 5*b^2*e^2 + 4*c*e*(-2*b*d + 7*a*e))*(d + e*x)^2 + (32*c^4*d^4 - 15*b
^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) - 4*c^2*e^2*(
-3*b^2*d^2 + 36*a*b*d*e + 32*a^2*e^2))*(d + e*x)^3) - 15*(b^2 - 4*a*c)^2*e^3*(-2
*c*d + b*e)*(d + e*x)^4*Log[d + e*x] + 15*(b^2 - 4*a*c)^2*e^3*(-2*c*d + b*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^2*(c*d^2 + e*(-(b*d) + a*e))^(7/2)*(d + e*x)^4)

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Maple [B]  time = 0.029, size = 10723, normalized size = 34.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(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)*(2*c*x + b)/(e*x + d)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.58194, 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)*(2*c*x + b)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[1/768*(4*(128*a*c^3*d^5 - 48*a^3*b*e^5 - 2*(15*b^3*c + 92*a*b*c^2)*d^4*e + 3*(5
*b^4 + 88*a*b^2*c - 48*a^2*c^2)*d^3*e^2 - 2*(59*a*b^3 + 36*a^2*b*c)*d^2*e^3 + 8*
(17*a^2*b^2 - 4*a^3*c)*d*e^4 + (32*c^4*d^4*e - 64*b*c^3*d^3*e^2 + 12*(b^2*c^2 +
12*a*c^3)*d^2*e^3 + 4*(5*b^3*c - 36*a*b*c^2)*d*e^4 - (15*b^4 - 100*a*b^2*c + 128
*a^2*c^2)*e^5)*x^3 + (128*c^4*d^5 - 272*b*c^3*d^4*e + 16*(5*b^2*c^2 + 36*a*c^3)*
d^3*e^2 + 2*(37*b^3*c - 324*a*b*c^2)*d^2*e^3 - (55*b^4 - 352*a*b^2*c + 272*a^2*c
^2)*d*e^4 + 2*(5*a*b^3 - 28*a^2*b*c)*e^5)*x^2 + (128*b*c^3*d^5 - 4*(91*b^2*c^2 -
68*a*c^3)*d^4*e + 8*(33*b^3*c + 4*a*b*c^2)*d^3*e^2 - (73*b^4 + 60*a*b^2*c + 576
*a^2*c^2)*d^2*e^3 + 4*(9*a*b^3 + 76*a^2*b*c)*d*e^4 - 8*(a^2*b^2 + 16*a^3*c)*e^5)
*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) - 15*(2*(b^4*c - 8*a*b^2*c
^2 + 16*a^2*c^3)*d^5*e - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2 + (2*(b^4*c -
8*a*b^2*c^2 + 16*a^2*c^3)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^4 + 4*
(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)
*d*e^5)*x^3 + 6*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c
+ 16*a^2*b*c^2)*d^2*e^4)*x^2 + 4*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2
- (b^5 - 8*a*b^3*c + 16*a^2*b*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*(128*a*c^3*d^5
- 48*a^3*b*e^5 - 2*(15*b^3*c + 92*a*b*c^2)*d^4*e + 3*(5*b^4 + 88*a*b^2*c - 48*a^
2*c^2)*d^3*e^2 - 2*(59*a*b^3 + 36*a^2*b*c)*d^2*e^3 + 8*(17*a^2*b^2 - 4*a^3*c)*d*
e^4 + (32*c^4*d^4*e - 64*b*c^3*d^3*e^2 + 12*(b^2*c^2 + 12*a*c^3)*d^2*e^3 + 4*(5*
b^3*c - 36*a*b*c^2)*d*e^4 - (15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*e^5)*x^3 + (128
*c^4*d^5 - 272*b*c^3*d^4*e + 16*(5*b^2*c^2 + 36*a*c^3)*d^3*e^2 + 2*(37*b^3*c - 3
24*a*b*c^2)*d^2*e^3 - (55*b^4 - 352*a*b^2*c + 272*a^2*c^2)*d*e^4 + 2*(5*a*b^3 -
28*a^2*b*c)*e^5)*x^2 + (128*b*c^3*d^5 - 4*(91*b^2*c^2 - 68*a*c^3)*d^4*e + 8*(33*
b^3*c + 4*a*b*c^2)*d^3*e^2 - (73*b^4 + 60*a*b^2*c + 576*a^2*c^2)*d^2*e^3 + 4*(9*
a*b^3 + 76*a^2*b*c)*d*e^4 - 8*(a^2*b^2 + 16*a^3*c)*e^5)*x)*sqrt(-c*d^2 + b*d*e -
a*e^2)*sqrt(c*x^2 + b*x + a) - 15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^5*e -
(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^4*e^2 + (2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3
)*d*e^5 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^6)*x^4 + 4*(2*(b^4*c - 8*a*b^2*c^2
+ 16*a^2*c^3)*d^2*e^4 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d*e^5)*x^3 + 6*(2*(b^4*
c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^3*e^3 - (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^2*e^4
)*x^2 + 4*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d^4*e^2 - (b^5 - 8*a*b^3*c + 16*
a^2*b*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)*sqrt(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. \[ \int \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)

[Out]

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**5, x)

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GIAC/XCAS [A]  time = 11.711, 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)*(2*c*x + b)/(e*x + d)^5,x, algorithm="giac")

[Out]

Done