3.2349 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=388 \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac{5 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-4 a e)+4 b c e^2 (12 b d-11 a e)-b^3 e^3+64 c^3 d^3\right )}{64 c e^5}-\frac{5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^6}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)} \]

[Out]

(-5*(64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(12*b*d - 11*a*e) - 16*c^2*d*e*(7*b*d - 4*
a*e) - 2*c*e*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*
x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e - 6*c*e*x)*(a + b*x + c*x^2)^(3/2))/(24*e^3
) - (a + b*x + c*x^2)^(5/2)/(e*(d + e*x)) + (5*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*
e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a
*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*
c^(3/2)*e^6) - (5*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(3/2)*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
*e^6)

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Rubi [A]  time = 1.60457, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{3/2} e^6}-\frac{5 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-4 a e)+4 b c e^2 (12 b d-11 a e)-b^3 e^3+64 c^3 d^3\right )}{64 c e^5}-\frac{5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^6}-\frac{5 \left (a+b x+c x^2\right )^{3/2} (-7 b e+8 c d-6 c e x)}{24 e^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(64*c^3*d^3 - b^3*e^3 + 4*b*c*e^2*(12*b*d - 11*a*e) - 16*c^2*d*e*(7*b*d - 4*
a*e) - 2*c*e*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*
x^2])/(64*c*e^5) - (5*(8*c*d - 7*b*e - 6*c*e*x)*(a + b*x + c*x^2)^(3/2))/(24*e^3
) - (a + b*x + c*x^2)^(5/2)/(e*(d + e*x)) + (5*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*
e^3*(2*b*d - 3*a*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a
*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*
c^(3/2)*e^6) - (5*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^(3/2)*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
*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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.51185, size = 401, normalized size = 1.03 \[ \frac{\frac{15 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2}}+2 e \sqrt{a+x (b+c x)} \left (2 e x \left (4 c e (27 a e-52 b d)+59 b^2 e^2+144 c^2 d^2\right )-\frac{192 \left (e (a e-b d)+c d^2\right )^2}{d+e x}+16 c d e (81 b d-56 a e)+4 b e^2 (139 a e-132 b d)+\frac{15 b^3 e^3}{c}+8 c e^2 x^2 (17 b e-16 c d)-768 c^2 d^3+48 c^2 e^3 x^3\right )+960 (b e-2 c d) \log (d+e x) \left (e (a e-b d)+c d^2\right )^{3/2}+960 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{384 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(-768*c^2*d^3 + (15*b^3*e^3)/c + 16*c*d*e*(81*b*d - 5
6*a*e) + 4*b*e^2*(-132*b*d + 139*a*e) + 2*e*(144*c^2*d^2 + 59*b^2*e^2 + 4*c*e*(-
52*b*d + 27*a*e))*x + 8*c*e^2*(-16*c*d + 17*b*e)*x^2 + 48*c^2*e^3*x^3 - (192*(c*
d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)) + 960*(-2*c*d + b*e)*(c*d^2 + e*(-(b*d) +
a*e))^(3/2)*Log[d + e*x] + (15*(128*c^4*d^4 - b^4*e^4 - 8*b^2*c*e^3*(2*b*d - 3*a
*e) - 64*c^3*d^2*e*(4*b*d - 3*a*e) + 48*c^2*e^2*(3*b^2*d^2 - 4*a*b*d*e + a^2*e^2
))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(3/2) + 960*(2*c*d - b*e)
*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*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^6)

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Maple [B]  time = 0.02, size = 6711, normalized size = 17.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [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)^2,x, algorithm="fricas")

[Out]

Timed out

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [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)^2,x, algorithm="giac")

[Out]

Timed out