3.2380 \(\int \frac{(d+e x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=389 \[ -\frac{8 (d+e x)^2 \left (8 a^2 c e^3-x (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-2 b c d \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{e \sqrt{a+b x+c x^2} \left (8 c^2 e^2 \left (-16 a^2 e^2-25 a b d e+b^2 d^2\right )+2 c e x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac{2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{7/2}} \]
[Out]
(-2*(d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(3/2)) - (8*(d + e*x)^2*(8*a^2*c*e^3 - 2*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(3*c*
d^2*e - a*e^3) - (2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*x))/(
3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (e*(64*c^4*d^4 - 15*b^4*e^4 - 16*c^
3*d^2*e*(7*b*d - 16*a*e) + 10*b^2*c*e^3*(3*b*d + 10*a*e) + 8*c^2*e^2*(b^2*d^2 -
25*a*b*d*e - 16*a^2*e^2) + 2*c*e*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2
*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*c^3*(b^2 - 4*a*c)^2) + (5*e^4*(2*c*d
- b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(7/2))
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Rubi [A] time = 1.14154, antiderivative size = 389, 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
\[ -\frac{8 (d+e x)^2 \left (8 a^2 c e^3-x (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )+b^2 \left (3 c d^2 e-a e^3\right )-2 b c d \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{e \sqrt{a+b x+c x^2} \left (8 c^2 e^2 \left (-16 a^2 e^2-25 a b d e+b^2 d^2\right )+2 c e x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )+10 b^2 c e^3 (10 a e+3 b d)-16 c^3 d^2 e (7 b d-16 a e)-15 b^4 e^4+64 c^4 d^4\right )}{3 c^3 \left (b^2-4 a c\right )^2}-\frac{2 (d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^5/(a + b*x + c*x^2)^(5/2),x]
[Out]
(-2*(d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^(3/2)) - (8*(d + e*x)^2*(8*a^2*c*e^3 - 2*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(3*c*
d^2*e - a*e^3) - (2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*x))/(
3*c*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) - (e*(64*c^4*d^4 - 15*b^4*e^4 - 16*c^
3*d^2*e*(7*b*d - 16*a*e) + 10*b^2*c*e^3*(3*b*d + 10*a*e) + 8*c^2*e^2*(b^2*d^2 -
25*a*b*d*e - 16*a^2*e^2) + 2*c*e*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2
*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*c^3*(b^2 - 4*a*c)^2) + (5*e^4*(2*c*d
- b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(7/2))
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Rubi in Sympy [A] time = 158.093, size = 422, normalized size = 1.08 \[ \frac{2 \left (d + e x\right )^{4} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (d + e x\right )^{2} \left (- 16 a^{2} c e^{3} + 2 a b^{2} e^{3} + 12 a b c d e^{2} - 6 b^{2} c d^{2} e + 4 b c^{2} d^{3} + 2 x \left (b e - 2 c d\right ) \left (- 6 a c e^{2} + b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )\right )}{3 c \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} + \frac{2 e \sqrt{a + b x + c x^{2}} \left (64 a^{2} c^{2} e^{4} - 50 a b^{2} c e^{4} + 100 a b c^{2} d e^{3} - 128 a c^{3} d^{2} e^{2} + \frac{15 b^{4} e^{4}}{2} - 15 b^{3} c d e^{3} - 4 b^{2} c^{2} d^{2} e^{2} + 56 b c^{3} d^{3} e - 32 c^{4} d^{4} - c e x \left (b e - 2 c d\right ) \left (- 28 a c e^{2} + 5 b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right )\right )}{3 c^{3} \left (- 4 a c + b^{2}\right )^{2}} - \frac{5 e^{4} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**5/(c*x**2+b*x+a)**(5/2),x)
[Out]
2*(d + e*x)**4*(2*a*e - b*d + x*(b*e - 2*c*d))/(3*(-4*a*c + b**2)*(a + b*x + c*x
**2)**(3/2)) + 4*(d + e*x)**2*(-16*a**2*c*e**3 + 2*a*b**2*e**3 + 12*a*b*c*d*e**2
- 6*b**2*c*d**2*e + 4*b*c**2*d**3 + 2*x*(b*e - 2*c*d)*(-6*a*c*e**2 + b**2*e**2
+ 2*b*c*d*e - 2*c**2*d**2))/(3*c*(-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2)) + 2*
e*sqrt(a + b*x + c*x**2)*(64*a**2*c**2*e**4 - 50*a*b**2*c*e**4 + 100*a*b*c**2*d*
e**3 - 128*a*c**3*d**2*e**2 + 15*b**4*e**4/2 - 15*b**3*c*d*e**3 - 4*b**2*c**2*d*
*2*e**2 + 56*b*c**3*d**3*e - 32*c**4*d**4 - c*e*x*(b*e - 2*c*d)*(-28*a*c*e**2 +
5*b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))/(3*c**3*(-4*a*c + b**2)**2) - 5*e**4*(b*
e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2*c**(7/2))
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Mathematica [A] time = 2.27718, size = 564, normalized size = 1.45 \[ \frac{b^4 e^4 \left (15 a^2 e-30 a c x (2 d+3 e x)+c^2 x^3 (3 e x-40 d)\right )-2 b^3 c \left (15 a^2 e^4 (d+7 e x)+2 a c e^4 x^2 (37 e x-45 d)+c^2 d^2 \left (d^3+15 d^2 e x-30 d e^2 x^2-10 e^3 x^3\right )\right )-4 b^2 c \left (25 a^3 e^5-3 a^2 c e^4 x (35 d+4 e x)+a c^2 e \left (5 d^4-60 d^3 e x+30 d^2 e^2 x^2-70 d e^3 x^3+6 e^4 x^4\right )+c^3 d^3 x \left (-3 d^2+30 d e x-10 e^2 x^2\right )\right )+8 b c^2 \left (a^3 e^4 (25 d+39 e x)+4 a^2 c e^2 \left (5 d^3-15 d^2 e x+8 e^3 x^3\right )+3 a c^2 d^2 \left (d^3-5 d^2 e x+10 d e^2 x^2-10 e^3 x^3\right )+2 c^3 d^4 x^2 (3 d-5 e x)\right )+16 c^2 \left (8 a^4 e^5+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+a^2 c^2 e \left (-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right )+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+2 c^4 d^5 x^3\right )+10 b^5 e^4 x (3 a e+c x (2 e x-3 d))+15 b^6 e^5 x^2}{3 c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac{5 e^4 (2 c d-b e) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{2 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^5/(a + b*x + c*x^2)^(5/2),x]
[Out]
(15*b^6*e^5*x^2 + 10*b^5*e^4*x*(3*a*e + c*x*(-3*d + 2*e*x)) + b^4*e^4*(15*a^2*e
+ c^2*x^3*(-40*d + 3*e*x) - 30*a*c*x*(2*d + 3*e*x)) - 2*b^3*c*(15*a^2*e^4*(d + 7
*e*x) + 2*a*c*e^4*x^2*(-45*d + 37*e*x) + c^2*d^2*(d^3 + 15*d^2*e*x - 30*d*e^2*x^
2 - 10*e^3*x^3)) + 8*b*c^2*(2*c^3*d^4*x^2*(3*d - 5*e*x) + a^3*e^4*(25*d + 39*e*x
) + 3*a*c^2*d^2*(d^3 - 5*d^2*e*x + 10*d*e^2*x^2 - 10*e^3*x^3) + 4*a^2*c*e^2*(5*d
^3 - 15*d^2*e*x + 8*e^3*x^3)) + 16*c^2*(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*
(3*d^2 + 10*e^2*x^2) + a^3*c*e^3*(-20*d^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(
-5*d^4 - 30*d^2*e^2*x^2 - 20*d*e^3*x^3 + 3*e^4*x^4)) - 4*b^2*c*(25*a^3*e^5 - 3*a
^2*c*e^4*x*(35*d + 4*e*x) + c^3*d^3*x*(-3*d^2 + 30*d*e*x - 10*e^2*x^2) + a*c^2*e
*(5*d^4 - 60*d^3*e*x + 30*d^2*e^2*x^2 - 70*d*e^3*x^3 + 6*e^4*x^4)))/(3*c^3*(b^2
- 4*a*c)^2*(a + x*(b + c*x))^(3/2)) + (5*e^4*(2*c*d - b*e)*Log[b + 2*c*x + 2*Sqr
t[c]*Sqrt[a + x*(b + c*x)]])/(2*c^(7/2))
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Maple [B] time = 0.022, size = 2395, normalized size = 6.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^5/(c*x^2+b*x+a)^(5/2),x)
[Out]
5/6*e^5*b/c^2*x^3/(c*x^2+b*x+a)^(3/2)+5/2*e^5*b/c^3*x/(c*x^2+b*x+a)^(1/2)+5/2*d*
e^4/c^3*b/(c*x^2+b*x+a)^(1/2)+16/3*d^5*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b+4/3
*d^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+32/3*d^5*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)*x-5/3*d*e^4*x^3/c/(c*x^2+b*x+a)^(3/2)-5/48*d*e^4*b^3/c^4/(c*x^2+b*x+a)^(
3/2)-5*d*e^4/c^2*x/(c*x^2+b*x+a)^(1/2)-20/3*d^2*e^3*a/c^2/(c*x^2+b*x+a)^(3/2)-5/
16*e^5*b^3/c^4*x/(c*x^2+b*x+a)^(3/2)+20*d*e^4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+a
)^(1/2)*x-40/3*d^4*e*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-10*d^2*e^3*x^2/c/(c*x
^2+b*x+a)^(3/2)+5/12*d^2*e^3*b^2/c^3/(c*x^2+b*x+a)^(3/2)+5/96*e^5*b^6/c^5/(4*a*c
-b^2)/(c*x^2+b*x+a)^(3/2)-5/48*d*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-5/2
*d^2*e^3*b/c^2*x/(c*x^2+b*x+a)^(3/2)+5/12*d^2*e^3*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x
+a)^(3/2)+10/3*d^2*e^3*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/6*d*e^4*b^5/c
^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/3*d*e^4*b/c^3*a/(c*x^2+b*x+a)^(3/2)+5/2*d
*e^4/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+5/6*d^3*e^2*b^3/c^2/(4*a*c-b^2)/(c*
x^2+b*x+a)^(3/2)+40/3*d^3*e^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+20/3*d^3*e
^2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+10/3*d^3*e^2*a/c/(4*a*c-b^2)/(c*x^2+b
*x+a)^(3/2)*b+20/3*d^2*e^3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-5*d^2*e^3*b
^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+5/2*d*e^4*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)
+5/8*d*e^4*b^2/c^3*x/(c*x^2+b*x+a)^(3/2)-19/12*e^5*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+
b*x+a)^(3/2)*x+160/3*d^3*e^2*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-80/3*d^4*e*
b*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-40*d^2*e^3*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+
b*x+a)^(1/2)+5/3*d^3*e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+20/3*d^3*e^2*a/
(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+80/3*d^3*e^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/
2)*b-10/3*d^4*e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-5/3*d^4*e*b^2/c/(4*a*c-b^2)/
(c*x^2+b*x+a)^(3/2)+e^5*a/c^3*b*x/(c*x^2+b*x+a)^(3/2)+2*e^5*a^2/c^3*b^2/(4*a*c-b
^2)/(c*x^2+b*x+a)^(3/2)+16*e^5*a^2/c^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-19/
3*e^5*b^4/c^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-5/2*e^5*b^3/c^3/(4*a*c-b^2)/(c
*x^2+b*x+a)^(1/2)*x+5/48*e^5*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5/6*e^5*b
^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-19/24*e^5*b^4/c^4*a/(4*a*c-b^2)/(c*x^
2+b*x+a)^(3/2)-5/24*d*e^4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-38/3*e^5*b^3
/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+4*e^5*a^2/c^2*b/(4*a*c-b^2)/(c*x^2+b*
x+a)^(3/2)*x+32*e^5*a^2/c*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-5/3*d*e^4*b^4/c^
2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+5/4*d*e^4*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x
+a)^(3/2)+10*d*e^4*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+5*d*e^4/c^2*b^2/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+5/6*d^2*e^3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(
3/2)*x-80*d^2*e^3*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-10*d^2*e^3*b/c*a/(4*a*
c-b^2)/(c*x^2+b*x+a)^(3/2)*x+5/2*d*e^4*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)
*x-5/4*e^5*b^2/c^3*x^2/(c*x^2+b*x+a)^(3/2)+5/12*e^5*b^6/c^4/(4*a*c-b^2)^2/(c*x^2
+b*x+a)^(1/2)+4*e^5*a/c^2*x^2/(c*x^2+b*x+a)^(3/2)-5/4*e^5*b^4/c^4/(4*a*c-b^2)/(c
*x^2+b*x+a)^(1/2)-5*d^3*e^2*x/c/(c*x^2+b*x+a)^(3/2)+5/6*d^3*e^2*b/c^2/(c*x^2+b*x
+a)^(3/2)-e^5*b^2/c^4*a/(c*x^2+b*x+a)^(3/2)+5/96*e^5*b^4/c^5/(c*x^2+b*x+a)^(3/2)
-5/4*e^5*b^2/c^4/(c*x^2+b*x+a)^(1/2)+e^5*x^4/c/(c*x^2+b*x+a)^(3/2)+5*d*e^4/c^(5/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/3*d^4*e/c/(c*x^2+b*x+a)^(3/2)+8
/3*e^5*a^2/c^3/(c*x^2+b*x+a)^(3/2)-5/2*e^5*b/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))+2/3*d^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b
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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((e*x + d)^5/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.602785, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")
[Out]
[1/12*(4*(160*a^2*b*c^3*d^3*e^2 - 320*a^3*c^3*d^2*e^3 + 3*(b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4)*e^5*x^4 - 2*(b^3*c^3 - 12*a*b*c^4)*d^5 - 20*(a*b^2*c^3 + 4*a^2*c^
4)*d^4*e - 10*(3*a^2*b^3*c - 20*a^3*b*c^2)*d*e^4 + (15*a^2*b^4 - 100*a^3*b^2*c +
128*a^4*c^2)*e^5 + 4*(8*c^6*d^5 - 20*b*c^5*d^4*e + 10*(b^2*c^4 + 4*a*c^5)*d^3*e
^2 + 5*(b^3*c^3 - 12*a*b*c^4)*d^2*e^3 - 10*(b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*d
*e^4 + (5*b^5*c - 37*a*b^3*c^2 + 64*a^2*b*c^3)*e^5)*x^3 + 3*(16*b*c^5*d^5 - 40*b
^2*c^4*d^4*e + 20*(b^3*c^3 + 4*a*b*c^4)*d^3*e^2 - 40*(a*b^2*c^3 + 4*a^2*c^4)*d^2
*e^3 - 10*(b^5*c - 6*a*b^3*c^2)*d*e^4 + (5*b^6 - 30*a*b^4*c + 16*a^2*b^2*c^2 + 6
4*a^3*c^3)*e^5)*x^2 + 6*(40*a*b^2*c^3*d^3*e^2 - 80*a^2*b*c^3*d^2*e^3 + 2*(b^2*c^
4 + 4*a*c^5)*d^5 - 5*(b^3*c^3 + 4*a*b*c^4)*d^4*e - 10*(a*b^4*c - 7*a^2*b^2*c^2 +
4*a^3*c^3)*d*e^4 + (5*a*b^5 - 35*a^2*b^3*c + 52*a^3*b*c^2)*e^5)*x)*sqrt(c*x^2 +
b*x + a)*sqrt(c) - 15*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*
b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*
d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^
3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 +
(2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e
^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b
^4*c + 16*a^3*b^2*c^2)*e^5)*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)))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^
4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16
*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^
2*b^3*c^4 + 16*a^3*b*c^5)*x)*sqrt(c)), 1/6*(2*(160*a^2*b*c^3*d^3*e^2 - 320*a^3*c
^3*d^2*e^3 + 3*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^5*x^4 - 2*(b^3*c^3 - 12*a*
b*c^4)*d^5 - 20*(a*b^2*c^3 + 4*a^2*c^4)*d^4*e - 10*(3*a^2*b^3*c - 20*a^3*b*c^2)*
d*e^4 + (15*a^2*b^4 - 100*a^3*b^2*c + 128*a^4*c^2)*e^5 + 4*(8*c^6*d^5 - 20*b*c^5
*d^4*e + 10*(b^2*c^4 + 4*a*c^5)*d^3*e^2 + 5*(b^3*c^3 - 12*a*b*c^4)*d^2*e^3 - 10*
(b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*d*e^4 + (5*b^5*c - 37*a*b^3*c^2 + 64*a^2*b*c
^3)*e^5)*x^3 + 3*(16*b*c^5*d^5 - 40*b^2*c^4*d^4*e + 20*(b^3*c^3 + 4*a*b*c^4)*d^3
*e^2 - 40*(a*b^2*c^3 + 4*a^2*c^4)*d^2*e^3 - 10*(b^5*c - 6*a*b^3*c^2)*d*e^4 + (5*
b^6 - 30*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3)*e^5)*x^2 + 6*(40*a*b^2*c^3*d^3*e
^2 - 80*a^2*b*c^3*d^2*e^3 + 2*(b^2*c^4 + 4*a*c^5)*d^5 - 5*(b^3*c^3 + 4*a*b*c^4)*
d^4*e - 10*(a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*c^3)*d*e^4 + (5*a*b^5 - 35*a^2*b^3*c
+ 52*a^3*b*c^2)*e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 15*(2*(a^2*b^4*c - 8*a
^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2
*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*
c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a
*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^
4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 +
16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*x)*arctan(1/2*
(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 +
16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^
5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3
- 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)*sqrt(-c))]
_______________________________________________________________________________________
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((e*x+d)**5/(c*x**2+b*x+a)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.233123, size = 1064, normalized size = 2.74 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^5/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")
[Out]
1/3*((((3*(b^4*c^2*e^5 - 8*a*b^2*c^3*e^5 + 16*a^2*c^4*e^5)*x/(b^4*c^3 - 8*a*b^2*
c^4 + 16*a^2*c^5) + 4*(8*c^6*d^5 - 20*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 40*a*c^
5*d^3*e^2 + 5*b^3*c^3*d^2*e^3 - 60*a*b*c^4*d^2*e^3 - 10*b^4*c^2*d*e^4 + 70*a*b^2
*c^3*d*e^4 - 80*a^2*c^4*d*e^4 + 5*b^5*c*e^5 - 37*a*b^3*c^2*e^5 + 64*a^2*b*c^3*e^
5)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 3*(16*b*c^5*d^5 - 40*b^2*c^4*d^4*e
+ 20*b^3*c^3*d^3*e^2 + 80*a*b*c^4*d^3*e^2 - 40*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d
^2*e^3 - 10*b^5*c*d*e^4 + 60*a*b^3*c^2*d*e^4 + 5*b^6*e^5 - 30*a*b^4*c*e^5 + 16*a
^2*b^2*c^2*e^5 + 64*a^3*c^3*e^5)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))*x + 6*(2*
b^2*c^4*d^5 + 8*a*c^5*d^5 - 5*b^3*c^3*d^4*e - 20*a*b*c^4*d^4*e + 40*a*b^2*c^3*d^
3*e^2 - 80*a^2*b*c^3*d^2*e^3 - 10*a*b^4*c*d*e^4 + 70*a^2*b^2*c^2*d*e^4 - 40*a^3*
c^3*d*e^4 + 5*a*b^5*e^5 - 35*a^2*b^3*c*e^5 + 52*a^3*b*c^2*e^5)/(b^4*c^3 - 8*a*b^
2*c^4 + 16*a^2*c^5))*x - (2*b^3*c^3*d^5 - 24*a*b*c^4*d^5 + 20*a*b^2*c^3*d^4*e +
80*a^2*c^4*d^4*e - 160*a^2*b*c^3*d^3*e^2 + 320*a^3*c^3*d^2*e^3 + 30*a^2*b^3*c*d*
e^4 - 200*a^3*b*c^2*d*e^4 - 15*a^2*b^4*e^5 + 100*a^3*b^2*c*e^5 - 128*a^4*c^2*e^5
)/(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5))/(c*x^2 + b*x + a)^(3/2) - 5/2*(2*c*d*e^4
- b*e^5)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)