3.2352 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=492 \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 e^6 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{96 e^3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{64 e^5 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{5 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^6}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]

[Out]

(5*(64*c^3*d^3 + b^3*e^3 + 4*b*c*e^2*(4*b*d - 5*a*e) - 16*c^2*d*e*(5*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*e^5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)) - (5*(16*c^2*d^3 - b*e^2*(b*d - 4*
a*e) - 4*c*d*e*(3*b*d - a*e) + 3*e*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x
)*(a + b*x + c*x^2)^(3/2))/(96*e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (a + b
*x + c*x^2)^(5/2)/(4*e*(d + e*x)^4) - (5*c^(3/2)*(2*c*d - b*e)*ArcTanh[(b + 2*c*
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*e^6) + (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*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*e^6*(c*d^2 - b*d*e + a*e^2)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.88318, antiderivative size = 492, 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 \[ \frac{5 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{128 e^6 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (3 e x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 c d e (3 b d-a e)-b e^2 (b d-4 a e)+16 c^2 d^3\right )}{96 e^3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (4 b d-5 a e)+b^3 e^3+64 c^3 d^3\right )}{64 e^5 (d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{5 c^{3/2} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^6}-\frac{\left (a+b x+c x^2\right )^{5/2}}{4 e (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(5*(64*c^3*d^3 + b^3*e^3 + 4*b*c*e^2*(4*b*d - 5*a*e) - 16*c^2*d*e*(5*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*e^5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)) - (5*(16*c^2*d^3 - b*e^2*(b*d - 4*
a*e) - 4*c*d*e*(3*b*d - a*e) + 3*e*(8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x
)*(a + b*x + c*x^2)^(3/2))/(96*e^3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (a + b
*x + c*x^2)^(5/2)/(4*e*(d + e*x)^4) - (5*c^(3/2)*(2*c*d - b*e)*ArcTanh[(b + 2*c*
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*e^6) + (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*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*e^6*(c*d^2 - b*d*e + a*e^2)^(3/2))

_______________________________________________________________________________________

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)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 4.31806, size = 507, normalized size = 1.03 \[ \frac{\frac{15 \log (d+e x) \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{15 \left (48 c^2 e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-8 b^2 c e^3 (2 b d-3 a e)-64 c^3 d^2 e (4 b d-3 a e)-b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+2 e \sqrt{a+x (b+c x)} \left (\frac{(2 c d-b e) \left (556 a c e^2+15 b^2 e^2-616 b c d e+616 c^2 d^2\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )}-\frac{2 \left (4 c e (27 a e-86 b d)+59 b^2 e^2+344 c^2 d^2\right )}{(d+e x)^2}+\frac{136 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^3}-\frac{48 \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^4}+192 c^2\right )+960 c^{3/2} (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{384 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(192*c^2 - (48*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x
)^4 + (136*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x)^3 - (2*(344*c^2*d
^2 + 59*b^2*e^2 + 4*c*e*(-86*b*d + 27*a*e)))/(d + e*x)^2 + ((2*c*d - b*e)*(616*c
^2*d^2 - 616*b*c*d*e + 15*b^2*e^2 + 556*a*c*e^2))/((c*d^2 + e*(-(b*d) + a*e))*(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[d + e*x]
)/(c*d^2 + e*(-(b*d) + a*e))^(3/2) + 960*c^(3/2)*(-2*c*d + b*e)*Log[b + 2*c*x +
2*Sqrt[c]*Sqrt[a + x*(b + c*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*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)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(384*e^6)

_______________________________________________________________________________________

Maple [B]  time = 0.047, size = 28635, normalized size = 58.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

_______________________________________________________________________________________

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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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)^5,x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

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)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError