3.2350 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=331 \[ -\frac{5 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 \sqrt{c} e^6}+\frac{5 \sqrt{a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^6}+\frac{5 \sqrt{a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{8 e^5}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

[Out]

(5*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a
+ b*x + c*x^2])/(8*e^5) + (5*(8*c*d - 3*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))
/(12*e^3*(d + e*x)) - (a + b*x + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*(2*c*d - b*
e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(16*Sqrt[c]*e^6) + (5*Sqrt[c*d^2 - b*d*e + a*e^2]*(16*
c^2*d^2 + 3*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])])/(8*e^6)

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Rubi [A]  time = 1.29761, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{5 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{16 \sqrt{c} e^6}+\frac{5 \sqrt{a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^6}+\frac{5 \sqrt{a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{8 e^5}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{12 e^3 (d+e x)}-\frac{\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(5*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a
+ b*x + c*x^2])/(8*e^5) + (5*(8*c*d - 3*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))
/(12*e^3*(d + e*x)) - (a + b*x + c*x^2)^(5/2)/(2*e*(d + e*x)^2) - (5*(2*c*d - b*
e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]
*Sqrt[a + b*x + c*x^2])])/(16*Sqrt[c]*e^6) + (5*Sqrt[c*d^2 - b*d*e + a*e^2]*(16*
c^2*d^2 + 3*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])])/(8*e^6)

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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)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.46775, size = 458, normalized size = 1.38 \[ \frac{\frac{30 \log (d+e x) \left (c e^2 \left (4 a^2 e^2-20 a b d e+19 b^2 d^2\right )+3 b^2 e^3 (a e-b d)-4 c^2 d^2 e (8 b d-5 a e)+16 c^3 d^4\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{30 \left (c e^2 \left (4 a^2 e^2-20 a b d e+19 b^2 d^2\right )+3 b^2 e^3 (a e-b d)-4 c^2 d^2 e (8 b d-5 a e)+16 c^3 d^4\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}+2 e \sqrt{a+x (b+c x)} \left (\frac{54 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{d+e x}-\frac{12 \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+56 a c e^2+33 b^2 e^2+2 c e x (13 b e-18 c d)-162 b c d e+144 c^2 d^2+8 c^2 e^2 x^2\right )-\frac{15 (2 c d-b e) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}}{48 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(144*c^2*d^2 - 162*b*c*d*e + 33*b^2*e^2 + 56*a*c*e^2
+ 2*c*e*(-18*c*d + 13*b*e)*x + 8*c^2*e^2*x^2 - (12*(c*d^2 + e*(-(b*d) + a*e))^2)
/(d + e*x)^2 + (54*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x)) + (30*(1
6*c^3*d^4 - 4*c^2*d^2*e*(8*b*d - 5*a*e) + 3*b^2*e^3*(-(b*d) + a*e) + c*e^2*(19*b
^2*d^2 - 20*a*b*d*e + 4*a^2*e^2))*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] -
(15*(2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*Log[b + 2*c*x
+ 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sqrt[c] - (30*(16*c^3*d^4 - 4*c^2*d^2*e*(8*
b*d - 5*a*e) + 3*b^2*e^3*(-(b*d) + a*e) + c*e^2*(19*b^2*d^2 - 20*a*b*d*e + 4*a^2
*e^2))*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*S
qrt[a + x*(b + c*x)]])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(48*e^6)

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Maple [B]  time = 0.026, size = 14002, normalized size = 42.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,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((c*x^2 + b*x + a)^(5/2)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 131.447, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/96*(30*(16*c^2*d^4 - 16*b*c*d^3*e + (3*b^2 + 4*a*c)*d^2*e^2 + (16*c^2*d^2*e^2
- 16*b*c*d*e^3 + (3*b^2 + 4*a*c)*e^4)*x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 +
(3*b^2 + 4*a*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*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 -
4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e
)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d
^2)) + 4*(8*c^2*e^5*x^4 + 240*c^2*d^4*e - 300*b*c*d^3*e^2 - 30*a*b*d*e^4 - 12*a^
2*e^5 + 5*(15*b^2 + 28*a*c)*d^2*e^3 - 2*(10*c^2*d*e^4 - 13*b*c*e^5)*x^3 + (80*c^
2*d^2*e^3 - 110*b*c*d*e^4 + (33*b^2 + 56*a*c)*e^5)*x^2 + 2*(180*c^2*d^3*e^2 - 23
0*b*c*d^2*e^3 - 27*a*b*e^5 + 10*(6*b^2 + 11*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a)
*sqrt(c) - 15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 6*(3*b^2*c + 4*a*c^2)*d^3*e^2 - (b^
3 + 12*a*b*c)*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^
2)*d*e^4 - (b^3 + 12*a*b*c)*e^5)*x^2 + 2*(32*c^3*d^4*e - 48*b*c^2*d^3*e^2 + 6*(3
*b^2*c + 4*a*c^2)*d^2*e^3 - (b^3 + 12*a*b*c)*d*e^4)*x)*log(-4*(2*c^2*x + b*c)*sq
rt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/((e^8*x^2 +
2*d*e^7*x + d^2*e^6)*sqrt(c)), 1/48*(15*(16*c^2*d^4 - 16*b*c*d^3*e + (3*b^2 + 4*
a*c)*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*d*e^3 + (3*b^2 + 4*a*c)*e^4)*x^2 + 2*(16
*c^2*d^3*e - 16*b*c*d^2*e^2 + (3*b^2 + 4*a*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e
^2)*sqrt(-c)*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 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x +
a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)
*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(8*c^2*e^5*x^4 + 240*c^2*d^4*e - 300*b*c
*d^3*e^2 - 30*a*b*d*e^4 - 12*a^2*e^5 + 5*(15*b^2 + 28*a*c)*d^2*e^3 - 2*(10*c^2*d
*e^4 - 13*b*c*e^5)*x^3 + (80*c^2*d^2*e^3 - 110*b*c*d*e^4 + (33*b^2 + 56*a*c)*e^5
)*x^2 + 2*(180*c^2*d^3*e^2 - 230*b*c*d^2*e^3 - 27*a*b*e^5 + 10*(6*b^2 + 11*a*c)*
d*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 6*(
3*b^2*c + 4*a*c^2)*d^3*e^2 - (b^3 + 12*a*b*c)*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c
^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5)*x^2 + 2*(32*c^3
*d^4*e - 48*b*c^2*d^3*e^2 + 6*(3*b^2*c + 4*a*c^2)*d^2*e^3 - (b^3 + 12*a*b*c)*d*e
^4)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((e^8*x^2 + 2
*d*e^7*x + d^2*e^6)*sqrt(-c)), -1/96*(60*(16*c^2*d^4 - 16*b*c*d^3*e + (3*b^2 + 4
*a*c)*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*d*e^3 + (3*b^2 + 4*a*c)*e^4)*x^2 + 2*(1
6*c^2*d^3*e - 16*b*c*d^2*e^2 + (3*b^2 + 4*a*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a
*e^2)*sqrt(c)*arctan(-1/2*(b*d - 2*a*e + (2*c*d - b*e)*x)/(sqrt(-c*d^2 + b*d*e -
a*e^2)*sqrt(c*x^2 + b*x + a))) - 4*(8*c^2*e^5*x^4 + 240*c^2*d^4*e - 300*b*c*d^3
*e^2 - 30*a*b*d*e^4 - 12*a^2*e^5 + 5*(15*b^2 + 28*a*c)*d^2*e^3 - 2*(10*c^2*d*e^4
- 13*b*c*e^5)*x^3 + (80*c^2*d^2*e^3 - 110*b*c*d*e^4 + (33*b^2 + 56*a*c)*e^5)*x^
2 + 2*(180*c^2*d^3*e^2 - 230*b*c*d^2*e^3 - 27*a*b*e^5 + 10*(6*b^2 + 11*a*c)*d*e^
4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 15*(32*c^3*d^5 - 48*b*c^2*d^4*e + 6*(3*b^2
*c + 4*a*c^2)*d^3*e^2 - (b^3 + 12*a*b*c)*d^2*e^3 + (32*c^3*d^3*e^2 - 48*b*c^2*d^
2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5)*x^2 + 2*(32*c^3*d^4*
e - 48*b*c^2*d^3*e^2 + 6*(3*b^2*c + 4*a*c^2)*d^2*e^3 - (b^3 + 12*a*b*c)*d*e^4)*x
)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*
a*c)*sqrt(c)))/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(c)), -1/48*(30*(16*c^2*d^4
- 16*b*c*d^3*e + (3*b^2 + 4*a*c)*d^2*e^2 + (16*c^2*d^2*e^2 - 16*b*c*d*e^3 + (3*b
^2 + 4*a*c)*e^4)*x^2 + 2*(16*c^2*d^3*e - 16*b*c*d^2*e^2 + (3*b^2 + 4*a*c)*d*e^3)
*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(-c)*arctan(-1/2*(b*d - 2*a*e + (2*c*d - b*
e)*x)/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a))) - 2*(8*c^2*e^5*x^4 +
240*c^2*d^4*e - 300*b*c*d^3*e^2 - 30*a*b*d*e^4 - 12*a^2*e^5 + 5*(15*b^2 + 28*a*
c)*d^2*e^3 - 2*(10*c^2*d*e^4 - 13*b*c*e^5)*x^3 + (80*c^2*d^2*e^3 - 110*b*c*d*e^4
+ (33*b^2 + 56*a*c)*e^5)*x^2 + 2*(180*c^2*d^3*e^2 - 230*b*c*d^2*e^3 - 27*a*b*e^
5 + 10*(6*b^2 + 11*a*c)*d*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 15*(32*c^3*d^
5 - 48*b*c^2*d^4*e + 6*(3*b^2*c + 4*a*c^2)*d^3*e^2 - (b^3 + 12*a*b*c)*d^2*e^3 +
(32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b
*c)*e^5)*x^2 + 2*(32*c^3*d^4*e - 48*b*c^2*d^3*e^2 + 6*(3*b^2*c + 4*a*c^2)*d^2*e^
3 - (b^3 + 12*a*b*c)*d*e^4)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x
+ a)*c)))/((e^8*x^2 + 2*d*e^7*x + d^2*e^6)*sqrt(-c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError