3.2479 \(\int \frac{(A+B x) (d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=608 \[ \frac{2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{2 (d+e x) \left (-2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )-x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )+4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )-8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 B e \left (c d^2-5 a e^2\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{e \sqrt{a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{3 c^3 \left (b^2-4 a c\right )^2}+\frac{e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{2 c^{7/2}} \]

[Out]

(2*(d + e*x)^3*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*
e) + 2*c*(A*c*d - a*B*e))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (2*(
d + e*x)*(b^3*B*e*(c*d^2 - 5*a*e^2) - 8*a*c^2*e*(A*c*d^2 + 8*a*B*d*e + 3*a*A*e^2
) - 2*b^2*c*(2*B*c*d^3 + 5*A*c*d^2*e - 2*a*B*d*e^2 - a*A*e^3) + 4*b*c*(2*A*c*d*(
c*d^2 + 3*a*e^2) + a*B*e*(5*c*d^2 + 7*a*e^2)) - (5*b^4*B*e^3 - 2*b^3*c*e^2*(3*B*
d + A*e) - 4*b^2*c*e*(B*c*d^2 + A*c*d*e + 8*a*B*e^2) + 8*b*c^2*(B*c*d^3 + 3*A*c*
d^2*e + 6*a*B*d*e^2 + 2*a*A*e^3) - 16*c^2*(2*a*B*e*(c*d^2 - a*e^2) + A*c*d*(c*d^
2 + 2*a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) + (e*(15*b^4*B*
e^3 - 2*b^3*c*e^2*(7*B*d + 3*A*e) - 4*b^2*c*e*(2*B*c*d^2 + A*c*d*e + 25*a*B*e^2)
+ 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 13*a*B*d*e^2 + 5*a*A*e^3) - 16*c^2*(4*a*B*
e*(c*d^2 - 2*a*e^2) + A*c*d*(2*c*d^2 + 5*a*e^2)))*Sqrt[a + b*x + c*x^2])/(3*c^3*
(b^2 - 4*a*c)^2) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqr
t[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(7/2))

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Rubi [A]  time = 2.23532, antiderivative size = 608, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 (d+e x)^3 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{2 (d+e x) \left (-2 b^2 c \left (-a A e^3-2 a B d e^2+5 A c d^2 e+2 B c d^3\right )-x \left (-4 b^2 c e \left (8 a B e^2+A c d e+B c d^2\right )+8 b c^2 \left (2 a A e^3+6 a B d e^2+3 A c d^2 e+B c d^3\right )-16 c^2 \left (A c d \left (2 a e^2+c d^2\right )+2 a B e \left (c d^2-a e^2\right )\right )-2 b^3 c e^2 (A e+3 B d)+5 b^4 B e^3\right )+4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+a B e \left (7 a e^2+5 c d^2\right )\right )-8 a c^2 e \left (3 a A e^2+8 a B d e+A c d^2\right )+b^3 B e \left (c d^2-5 a e^2\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{e \sqrt{a+b x+c x^2} \left (-4 b^2 c e \left (25 a B e^2+A c d e+2 B c d^2\right )+8 b c^2 \left (5 a A e^3+13 a B d e^2+6 A c d^2 e+2 B c d^3\right )-16 c^2 \left (A c d \left (5 a e^2+2 c d^2\right )+4 a B e \left (c d^2-2 a e^2\right )\right )-2 b^3 c e^2 (3 A e+7 B d)+15 b^4 B e^3\right )}{3 c^3 \left (b^2-4 a c\right )^2}+\frac{e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{2 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*(d + e*x)^3*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*
e) + 2*c*(A*c*d - a*B*e))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (2*(
d + e*x)*(b^3*B*e*(c*d^2 - 5*a*e^2) - 8*a*c^2*e*(A*c*d^2 + 8*a*B*d*e + 3*a*A*e^2
) - 2*b^2*c*(2*B*c*d^3 + 5*A*c*d^2*e - 2*a*B*d*e^2 - a*A*e^3) + 4*b*c*(2*A*c*d*(
c*d^2 + 3*a*e^2) + a*B*e*(5*c*d^2 + 7*a*e^2)) - (5*b^4*B*e^3 - 2*b^3*c*e^2*(3*B*
d + A*e) - 4*b^2*c*e*(B*c*d^2 + A*c*d*e + 8*a*B*e^2) + 8*b*c^2*(B*c*d^3 + 3*A*c*
d^2*e + 6*a*B*d*e^2 + 2*a*A*e^3) - 16*c^2*(2*a*B*e*(c*d^2 - a*e^2) + A*c*d*(c*d^
2 + 2*a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2]) + (e*(15*b^4*B*
e^3 - 2*b^3*c*e^2*(7*B*d + 3*A*e) - 4*b^2*c*e*(2*B*c*d^2 + A*c*d*e + 25*a*B*e^2)
+ 8*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e + 13*a*B*d*e^2 + 5*a*A*e^3) - 16*c^2*(4*a*B*
e*(c*d^2 - 2*a*e^2) + A*c*d*(2*c*d^2 + 5*a*e^2)))*Sqrt[a + b*x + c*x^2])/(3*c^3*
(b^2 - 4*a*c)^2) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(b + 2*c*x)/(2*Sqr
t[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(7/2))

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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((B*x+A)*(e*x+d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 6.43254, size = 847, normalized size = 1.39 \[ \frac{(8 B c d-5 b B e+2 A c e) \log \left (b+2 c x+2 \sqrt{c} \sqrt{a+x (b+c x)}\right ) e^3}{2 c^{7/2}}+\frac{B \left (128 a^4 c^2 e^4+4 a^3 c \left (-48 \left (d^2+e x d-e^2 x^2\right ) c^2+2 b e (20 d+39 e x) c-25 b^2 e^2\right ) e^2+b x \left (15 e^4 x b^5+4 c e^3 x (5 e x-6 d) b^4+c^2 e^3 x^2 (3 e x-32 d) b^3-6 c^3 d^2 \left (d^2-4 e x d-2 e^2 x^2\right ) b^2+8 c^4 d^3 x (2 e x-3 d) b-16 c^5 d^4 x^2\right )+a^2 \left (-16 \left (d^4+18 e^2 x^2 d^2+16 e^3 x^3 d-3 e^4 x^4\right ) c^4+32 b e \left (2 d^3-9 e x d^2+8 e^3 x^3\right ) c^3+48 b^2 e^3 x (7 d+e x) c^2-6 b^3 e^3 (4 d+35 e x) c+15 b^4 e^4\right )+2 a \left (15 e^4 x b^5-3 c e^3 x (8 d+15 e x) b^4+2 c^2 e^3 x^2 (36 d-37 e x) b^3-2 c^3 \left (d^4-24 e x d^3+18 e^2 x^2 d^2-56 e^3 x^3 d+6 e^4 x^4\right ) b^2-12 c^4 d^2 x \left (d^2-4 e x d+6 e^2 x^2\right ) b+32 c^5 d^3 e x^3\right )\right )-2 A c \left (3 e^4 x^2 b^5+2 e^4 x \left (2 c x^2+3 a\right ) b^4+\left (3 a^2 e^4-18 a c x^2 e^4+c^2 d \left (d^3+12 e x d^2-18 e^2 x^2 d-4 e^3 x^3\right )\right ) b^3-2 c \left (21 a^2 x e^4+2 a c \left (-2 d^3+18 e x d^2-6 e^2 x^2 d+7 e^3 x^3\right ) e+3 c^2 d^2 x \left (d^2-8 e x d+2 e^2 x^2\right )\right ) b^2-4 c \left (5 a^3 e^4+12 a^2 c d (d-2 e x) e^2+2 c^3 d^3 x^2 (3 d-4 e x)+3 a c^2 d \left (d^3-4 e x d^2+6 e^2 x^2 d-4 e^3 x^3\right )\right ) b+8 c^2 \left (-2 c^3 x^3 d^4-3 a c^2 x \left (d^2+2 e^2 x^2\right ) d^2+a^3 e^3 (8 d+3 e x)+4 a^2 c e \left (d^3+3 e^2 x^2 d+e^3 x^3\right )\right )\right )}{3 c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*A*c*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18*a*c*e
^4*x^2 + c^2*d*(d^3 + 12*d^2*e*x - 18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b*c*(5*a^3*e^4
+ 2*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x) + 3*a*c^2*d*(d^3 - 4
*d^2*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4*x^3 + a^3*e^3*(8*d + 3*
e*x) - 3*a*c^2*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3)
) - 2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*(d^2 - 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-
2*d^3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e^3*x^3))) + B*(128*a^4*c^2*e^4 + b*x*(15*b
^5*e^4*x - 16*c^5*d^4*x^2 + 8*b*c^4*d^3*x*(-3*d + 2*e*x) + b^3*c^2*e^3*x^2*(-32*
d + 3*e*x) + 4*b^4*c*e^3*x*(-6*d + 5*e*x) - 6*b^2*c^3*d^2*(d^2 - 4*d*e*x - 2*e^2
*x^2)) + 4*a^3*c*e^2*(-25*b^2*e^2 + 2*b*c*e*(20*d + 39*e*x) - 48*c^2*(d^2 + d*e*
x - e^2*x^2)) + a^2*(15*b^4*e^4 + 48*b^2*c^2*e^3*x*(7*d + e*x) - 6*b^3*c*e^3*(4*
d + 35*e*x) + 32*b*c^3*e*(2*d^3 - 9*d^2*e*x + 8*e^3*x^3) - 16*c^4*(d^4 + 18*d^2*
e^2*x^2 + 16*d*e^3*x^3 - 3*e^4*x^4)) + 2*a*(15*b^5*e^4*x + 32*c^5*d^3*e*x^3 + 2*
b^3*c^2*e^3*x^2*(36*d - 37*e*x) - 3*b^4*c*e^3*x*(8*d + 15*e*x) - 12*b*c^4*d^2*x*
(d^2 - 4*d*e*x + 6*e^2*x^2) - 2*b^2*c^3*(d^4 - 24*d^3*e*x + 18*d^2*e^2*x^2 - 56*
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)) + (e^3
*(8*B*c*d - 5*b*B*e + 2*A*c*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])
/(2*c^(7/2))

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Maple [B]  time = 0.032, size = 3912, normalized size = 6.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(c*x^2+b*x+a)^(5/2),x)

[Out]

16/c*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^3+2/c^2*b^2*a/(4*a*c-b^2)/(
c*x^2+b*x+a)^(3/2)*x*B*d*e^3-3*x/c/(c*x^2+b*x+a)^(3/2)*A*d^2*e^2-2*x/c/(c*x^2+b*
x+a)^(3/2)*B*d^3*e+1/2/c^2*b/(c*x^2+b*x+a)^(3/2)*A*d^2*e^2-32/3*b^2/(4*a*c-b^2)^
2/(c*x^2+b*x+a)^(1/2)*A*d^3*e-2/3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^4-1/3/
c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^4+5/96*B*e^4/c^5*b^6/(4*a*c-b^2)/(c*x^
2+b*x+a)^(3/2)+5/12*B*e^4/c^4*b^6/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)-B*e^4/c^4*b^
2*a/(c*x^2+b*x+a)^(3/2)+5/2*B*e^4/c^3*b*x/(c*x^2+b*x+a)^(1/2)-5/4*B*e^4/c^4*b^4/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+4*B*e^4*a/c^2*x^2/(c*x^2+b*x+a)^(3/2)+5/6*B*e^4/
c^2*b*x^3/(c*x^2+b*x+a)^(3/2)-5/4*B*e^4/c^3*b^2*x^2/(c*x^2+b*x+a)^(3/2)-5/16*B*e
^4/c^4*b^3*x/(c*x^2+b*x+a)^(3/2)-4/c^2*x/(c*x^2+b*x+a)^(1/2)*B*d*e^3+2/c^3*b/(c*
x^2+b*x+a)^(1/2)*B*d*e^3+1/2/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*A*e^4-4/3*x
^3/c/(c*x^2+b*x+a)^(3/2)*B*d*e^3-1/48/c^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*
e^4-1/6/c^3*b^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*e^4+1/3/c^3*b*a/(c*x^2+b*x+a
)^(3/2)*A*e^4+1/8/c^3*b^2*x/(c*x^2+b*x+a)^(3/2)*A*e^4-1/12/c^4*b^3/(c*x^2+b*x+a)
^(3/2)*B*d*e^3+32/3*A*d^4*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*A*d^4*c/(
4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b+4/3*A*d^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x
+1/2/c^2*b*x^2/(c*x^2+b*x+a)^(3/2)*A*e^4-4*a/c^2/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2+1
/4/c^3*b^2/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2-8/3*a/c^2/(c*x^2+b*x+a)^(3/2)*A*d*e^3-4
*x^2/c/(c*x^2+b*x+a)^(3/2)*A*d*e^3-6*x^2/c/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2+1/6/c^3
*b^2/(c*x^2+b*x+a)^(3/2)*A*d*e^3+1/3/c^2*b/(c*x^2+b*x+a)^(3/2)*B*d^3*e-4/c*b*a/(
4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d*e^3-6/c*b*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)
*x*B*d^2*e^2-1/3/c/(c*x^2+b*x+a)^(3/2)*B*d^4+1/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*A*e^4+4/c*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^4+1/3
/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^3*e+8*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+
a)^(1/2)*x*A*d^2*e^2+16/3*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^3*e+4/c*b^
3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d^2*e^2+8/3/c*b^3/(4*a*c-b^2)^2/(c*x^2+b*x
+a)^(1/2)*B*d^3*e+4*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*d^2*e^2+8/3*a/(4*a*c-b
^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^3*e+16*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*A*d^2
*e^2+32/3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*B*d^3*e-8/3*b/(4*a*c-b^2)/(c*x^2
+b*x+a)^(3/2)*x*A*d^3*e-4/3/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d^3*e-16/3*c
*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^4+B*e^4*a/c^3*b*x/(c*x^2+b*x+a)^(3/2)
+2*B*e^4*a^2/c^3*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16*B*e^4*a^2/c^2*b^2/(4*a*c
-b^2)^2/(c*x^2+b*x+a)^(1/2)+5/48*B*e^4/c^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x
+5/6*B*e^4/c^3*b^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-19/24*B*e^4/c^4*b^4*a/(4*
a*c-b^2)/(c*x^2+b*x+a)^(3/2)-19/3*B*e^4/c^3*b^4*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)-5/2*B*e^4/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+4/3/c^3*b*a/(c*x^2+b*x+a
)^(3/2)*B*d*e^3+1/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*A*e^4+2/c^3*b^3/(4*a
*c-b^2)/(c*x^2+b*x+a)^(1/2)*B*d*e^3+2/c^2*b*x^2/(c*x^2+b*x+a)^(3/2)*B*d*e^3+1/2/
c^3*b^2*x/(c*x^2+b*x+a)^(3/2)*B*d*e^3-1/24/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)*x*A*e^4-1/12/c^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d*e^3-1/3/c^2*b^4/(4*a
*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^4-2/3/c^3*b^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)*B*d*e^3+1/4/c^3*b^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*e^4+2/c^2*b^3*a/(4*a
*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*e^4-1/c^2*b*x/(c*x^2+b*x+a)^(3/2)*A*d*e^3-3/2/c^
2*b*x/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2+1/6/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*
A*d*e^3+1/4/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2+4/3/c^2*b^4/(4*a*c
-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d*e^3+2/c^2*b^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*
B*d^2*e^2+1/2/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d^2*e^2-8/3*b^2/(4*a*c-b
^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^4+5/96*B*e^4/c^5*b^4/(c*x^2+b*x+a)^(3/2)-5/4*B*e^4
/c^4*b^2/(c*x^2+b*x+a)^(1/2)-5/2*B*e^4/c^(7/2)*b*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))+B*e^4*x^4/c/(c*x^2+b*x+a)^(3/2)+8/3*B*e^4*a^2/c^3/(c*x^2+b*x+a)^(3/
2)+1/2/c^3*b/(c*x^2+b*x+a)^(1/2)*A*e^4+4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b
*x+a)^(1/2))*B*d*e^3-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)*A*e^4-1/48/c^4*b^3/(c*x^2+b*x
+a)^(3/2)*A*e^4-1/c^2*x/(c*x^2+b*x+a)^(1/2)*A*e^4-4/3/c/(c*x^2+b*x+a)^(3/2)*A*d^
3*e+2/3*A*d^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b+1/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a
)^(3/2)*x*A*d^2*e^2+2/3/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^3*e+2*a/c/(4
*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*A*d^2*e^2+4/3*a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2
)*b*B*d^3*e+32*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d^2*e^2+64/3*a*c/(4*a*c
-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^3*e+1/3/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2
)*x*A*d*e^3+1/2/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^2*e^2+8/3/c*b^3/(4
*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d*e^3+4/c*b^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1
/2)*x*B*d^2*e^2-2/c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d*e^3+4*B*e^4*a^2/
c^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+32*B*e^4*a^2/c*b/(4*a*c-b^2)^2/(c*x^2+b*
x+a)^(1/2)*x-38/3*B*e^4/c^2*b^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-19/12*B*e^
4/c^3*b^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-1/6/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x
+a)^(3/2)*x*B*d*e^3-4/3/c^2*b^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^3+1/2/
c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*e^4+1/c^3*b^3*a/(4*a*c-b^2)/(c*x^2
+b*x+a)^(3/2)*B*d*e^3-24/c*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^2*e^2+8/c
^2*b^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d*e^3+4/c^2*b^2/(4*a*c-b^2)/(c*x^2+
b*x+a)^(1/2)*x*B*d*e^3-3/c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^2*e^2-32*
b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d*e^3-48*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+
a)^(1/2)*x*B*d^2*e^2-16/c*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d*e^3-64/3*c
*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d^3*e

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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((B*x + A)*(e*x + d)^4/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.14146, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(4*(3*(B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*e^4*x^4 - 96*(2*B*a^3 - A
*a^2*b)*c^3*d^2*e^2 - 2*(4*(2*B*a^2 - 3*A*a*b)*c^4 + (2*B*a*b^2 + A*b^3)*c^3)*d^
4 - 16*(4*A*a^2*c^4 - (4*B*a^2*b - A*a*b^2)*c^3)*d^3*e - 8*(3*B*a^2*b^3*c - 20*B
*a^3*b*c^2 + 16*A*a^3*c^3)*d*e^3 + (15*B*a^2*b^4 + 8*(16*B*a^4 + 5*A*a^3*b)*c^2
- 2*(50*B*a^3*b^2 + 3*A*a^2*b^3)*c)*e^4 - 4*(4*(B*b*c^5 - 2*A*c^6)*d^4 - 4*(B*b^
2*c^4 + 4*(B*a - A*b)*c^5)*d^3*e - 3*(B*b^3*c^3 + 8*A*a*c^5 - 2*(6*B*a*b - A*b^2
)*c^4)*d^2*e^2 + 2*(4*B*b^4*c^2 + 4*(8*B*a^2 + 3*A*a*b)*c^4 - (28*B*a*b^2 + A*b^
3)*c^3)*d*e^3 - (5*B*b^5*c - 16*A*a^2*c^4 + 2*(32*B*a^2*b + 7*A*a*b^2)*c^3 - (37
*B*a*b^3 + 2*A*b^4)*c^2)*e^4)*x^3 - 3*(8*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 8*(B*b^3*
c^3 + 4*(B*a*b - A*b^2)*c^4)*d^3*e + 12*(4*(2*B*a^2 - A*a*b)*c^4 + (2*B*a*b^2 -
A*b^3)*c^3)*d^2*e^2 + 8*(B*b^5*c - 6*B*a*b^3*c^2 + 2*A*a*b^2*c^3 + 8*A*a^2*c^4)*
d*e^3 - (5*B*b^6 + 64*B*a^3*c^3 + 4*(4*B*a^2*b^2 + 3*A*a*b^3)*c^2 - 2*(15*B*a*b^
4 + A*b^5)*c)*e^4)*x^2 - 6*(24*(2*B*a^2*b - A*a*b^2)*c^3*d^2*e^2 + (B*b^3*c^3 -
8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^4 + 4*(4*A*a*b*c^4 - (4*B*a*b^2 - A*b^3)*
c^3)*d^3*e + 8*(B*a*b^4*c - 7*B*a^2*b^2*c^2 + 4*(B*a^3 + A*a^2*b)*c^3)*d*e^3 - (
5*B*a*b^5 - 8*A*a^3*c^3 + 2*(26*B*a^3*b + 7*A*a^2*b^2)*c^2 - (35*B*a^2*b^3 + 2*A
*a*b^4)*c)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(8*(B*a^2*b^4*c - 8*B*a^3*b
^2*c^2 + 16*B*a^4*c^3)*d*e^3 - (5*B*a^2*b^5 - 32*A*a^4*c^3 + 16*(5*B*a^4*b + A*a
^3*b^2)*c^2 - 2*(20*B*a^3*b^3 + A*a^2*b^4)*c)*e^4 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^
4 + 16*B*a^2*c^5)*d*e^3 - (5*B*b^5*c^2 - 32*A*a^2*c^5 + 16*(5*B*a^2*b + A*a*b^2)
*c^4 - 2*(20*B*a*b^3 + A*b^4)*c^3)*e^4)*x^4 + 2*(8*(B*b^5*c^2 - 8*B*a*b^3*c^3 +
16*B*a^2*b*c^4)*d*e^3 - (5*B*b^6*c - 32*A*a^2*b*c^4 + 16*(5*B*a^2*b^2 + A*a*b^3)
*c^3 - 2*(20*B*a*b^4 + A*b^5)*c^2)*e^4)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B
*a^3*c^4)*d*e^3 - (5*B*b^7 + 12*A*a*b^4*c^2 + 160*B*a^3*b*c^3 - 64*A*a^3*c^4 - 2
*(15*B*a*b^5 + A*b^6)*c)*e^4)*x^2 + 2*(8*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3
*b*c^3)*d*e^3 - (5*B*a*b^6 - 32*A*a^3*b*c^3 + 16*(5*B*a^3*b^2 + A*a^2*b^3)*c^2 -
2*(20*B*a^2*b^4 + A*a*b^5)*c)*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)))/((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*(3*(B*b^4*c^2 - 8*B*a*b^2
*c^3 + 16*B*a^2*c^4)*e^4*x^4 - 96*(2*B*a^3 - A*a^2*b)*c^3*d^2*e^2 - 2*(4*(2*B*a^
2 - 3*A*a*b)*c^4 + (2*B*a*b^2 + A*b^3)*c^3)*d^4 - 16*(4*A*a^2*c^4 - (4*B*a^2*b -
A*a*b^2)*c^3)*d^3*e - 8*(3*B*a^2*b^3*c - 20*B*a^3*b*c^2 + 16*A*a^3*c^3)*d*e^3 +
(15*B*a^2*b^4 + 8*(16*B*a^4 + 5*A*a^3*b)*c^2 - 2*(50*B*a^3*b^2 + 3*A*a^2*b^3)*c
)*e^4 - 4*(4*(B*b*c^5 - 2*A*c^6)*d^4 - 4*(B*b^2*c^4 + 4*(B*a - A*b)*c^5)*d^3*e -
3*(B*b^3*c^3 + 8*A*a*c^5 - 2*(6*B*a*b - A*b^2)*c^4)*d^2*e^2 + 2*(4*B*b^4*c^2 +
4*(8*B*a^2 + 3*A*a*b)*c^4 - (28*B*a*b^2 + A*b^3)*c^3)*d*e^3 - (5*B*b^5*c - 16*A*
a^2*c^4 + 2*(32*B*a^2*b + 7*A*a*b^2)*c^3 - (37*B*a*b^3 + 2*A*b^4)*c^2)*e^4)*x^3
- 3*(8*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 8*(B*b^3*c^3 + 4*(B*a*b - A*b^2)*c^4)*d^3*e
+ 12*(4*(2*B*a^2 - A*a*b)*c^4 + (2*B*a*b^2 - A*b^3)*c^3)*d^2*e^2 + 8*(B*b^5*c -
6*B*a*b^3*c^2 + 2*A*a*b^2*c^3 + 8*A*a^2*c^4)*d*e^3 - (5*B*b^6 + 64*B*a^3*c^3 +
4*(4*B*a^2*b^2 + 3*A*a*b^3)*c^2 - 2*(15*B*a*b^4 + A*b^5)*c)*e^4)*x^2 - 6*(24*(2*
B*a^2*b - A*a*b^2)*c^3*d^2*e^2 + (B*b^3*c^3 - 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^
4)*d^4 + 4*(4*A*a*b*c^4 - (4*B*a*b^2 - A*b^3)*c^3)*d^3*e + 8*(B*a*b^4*c - 7*B*a^
2*b^2*c^2 + 4*(B*a^3 + A*a^2*b)*c^3)*d*e^3 - (5*B*a*b^5 - 8*A*a^3*c^3 + 2*(26*B*
a^3*b + 7*A*a^2*b^2)*c^2 - (35*B*a^2*b^3 + 2*A*a*b^4)*c)*e^4)*x)*sqrt(c*x^2 + b*
x + a)*sqrt(-c) + 3*(8*(B*a^2*b^4*c - 8*B*a^3*b^2*c^2 + 16*B*a^4*c^3)*d*e^3 - (5
*B*a^2*b^5 - 32*A*a^4*c^3 + 16*(5*B*a^4*b + A*a^3*b^2)*c^2 - 2*(20*B*a^3*b^3 + A
*a^2*b^4)*c)*e^4 + (8*(B*b^4*c^3 - 8*B*a*b^2*c^4 + 16*B*a^2*c^5)*d*e^3 - (5*B*b^
5*c^2 - 32*A*a^2*c^5 + 16*(5*B*a^2*b + A*a*b^2)*c^4 - 2*(20*B*a*b^3 + A*b^4)*c^3
)*e^4)*x^4 + 2*(8*(B*b^5*c^2 - 8*B*a*b^3*c^3 + 16*B*a^2*b*c^4)*d*e^3 - (5*B*b^6*
c - 32*A*a^2*b*c^4 + 16*(5*B*a^2*b^2 + A*a*b^3)*c^3 - 2*(20*B*a*b^4 + A*b^5)*c^2
)*e^4)*x^3 + (8*(B*b^6*c - 6*B*a*b^4*c^2 + 32*B*a^3*c^4)*d*e^3 - (5*B*b^7 + 12*A
*a*b^4*c^2 + 160*B*a^3*b*c^3 - 64*A*a^3*c^4 - 2*(15*B*a*b^5 + A*b^6)*c)*e^4)*x^2
+ 2*(8*(B*a*b^5*c - 8*B*a^2*b^3*c^2 + 16*B*a^3*b*c^3)*d*e^3 - (5*B*a*b^6 - 32*A
*a^3*b*c^3 + 16*(5*B*a^3*b^2 + A*a^2*b^3)*c^2 - 2*(20*B*a^2*b^4 + A*a*b^5)*c)*e^
4)*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))]

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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((B*x+A)*(e*x+d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284612, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

Done