∖int (A+Bx)(d+ex)3 ________________________________________∖left(a+bx+cx2 ∖right)5∕2 ∖,dx

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3.2480 \(\int \frac{(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=397 \[ \frac{2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-3 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-3 a e^3\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{B e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

(2*(d + e*x)^2*(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*(

4*b^2*c^2*d^2*(B*d + 2*A*e) + 16*a*c^2*e*(A*c*d^2 + 3*a*B*d*e + a*A*e^2) - b^3*B

*(c*d^2*e - 3*a*e^3) - 4*b*c*(5*a*B*e*(c*d^2 + a*e^2) + 2*A*c*d*(c*d^2 + 3*a*e^2

)) - (2*b^3*B*c*d*e^2 - 3*b^4*B*e^3 + 2*b^2*c*e*(3*B*c*d^2 + 4*A*c*d*e + 11*a*B*

e^2) - 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e + 4*a*B*d*e^2 + a*A*e^3) + 8*c^2*(3*a*B*e*

(c*d^2 - a*e^2) + 2*A*c*d*(c*d^2 + a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a +

b*x + c*x^2]) + (B*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c

^(5/2)

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Rubi [A] time = 0.946205, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-3 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) \left (c d^2 e-3 a e^3\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}+\frac{B e^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(2*(d + e*x)^2*(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*(

4*b^2*c^2*d^2*(B*d + 2*A*e) + 16*a*c^2*e*(A*c*d^2 + 3*a*B*d*e + a*A*e^2) - b^3*B

*(c*d^2*e - 3*a*e^3) - 4*b*c*(5*a*B*e*(c*d^2 + a*e^2) + 2*A*c*d*(c*d^2 + 3*a*e^2

)) - (2*b^3*B*c*d*e^2 - 3*b^4*B*e^3 + 2*b^2*c*e*(3*B*c*d^2 + 4*A*c*d*e + 11*a*B*

e^2) - 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e + 4*a*B*d*e^2 + a*A*e^3) + 8*c^2*(3*a*B*e*

(c*d^2 - a*e^2) + 2*A*c*d*(c*d^2 + a*e^2)))*x))/(3*c^2*(b^2 - 4*a*c)^2*Sqrt[a +

b*x + c*x^2]) + (B*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c

^(5/2)

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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)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Mathematica [A] time = 4.16624, size = 503, normalized size = 1.27 \[ \frac{B e^3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{5/2}}-\frac{2 \left (A c^2 \left (-12 b (d-e x) \left (2 a^2 e^2+a c (d-e x)^2+2 c^2 d^2 x^2\right )+8 \left (2 a^3 e^3+3 a^2 c e \left (d^2+e^2 x^2\right )-3 a c^2 d x \left (d^2+e^2 x^2\right )-2 c^3 d^3 x^3\right )+6 b^2 \left (d^2-6 d e x+e^2 x^2\right ) (a e-c d x)+b^3 \left (d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )\right )+B \left (4 a^3 c e^2 (6 c (2 d+e x)-5 b e)+a^2 \left (3 b^3 e^3-42 b^2 c e^3 x-24 b c^2 d e (d-3 e x)+8 c^3 \left (d^3+9 d e^2 x^2+4 e^3 x^3\right )\right )+2 a \left (3 b^4 e^3 x-9 b^3 c e^3 x^2+b^2 c^2 \left (d^3-18 d^2 e x+9 d e^2 x^2-14 e^3 x^3\right )+6 b c^3 d x \left (d^2-3 d e x+3 e^2 x^2\right )-12 c^4 d^2 e x^3\right )+b x \left (3 b^4 e^3 x+4 b^3 c e^3 x^2+3 b^2 c^2 d \left (d^2-3 d e x-e^2 x^2\right )+6 b c^3 d^2 x (2 d-e x)+8 c^4 d^3 x^2\right )\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(-2*(A*c^2*(6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(d^3 + 9*d^2*e*x

- 9*d*e^2*x^2 - e^3*x^3) - 12*b*(d - e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^2 + a*c*(d -

e*x)^2) + 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e^2*x^2) - 3*a*c^2*d*

x*(d^2 + e^2*x^2))) + B*(4*a^3*c*e^2*(-5*b*e + 6*c*(2*d + e*x)) + b*x*(3*b^4*e^3

*x + 8*c^4*d^3*x^2 + 4*b^3*c*e^3*x^2 + 6*b*c^3*d^2*x*(2*d - e*x) + 3*b^2*c^2*d*(

d^2 - 3*d*e*x - e^2*x^2)) + 2*a*(3*b^4*e^3*x - 9*b^3*c*e^3*x^2 - 12*c^4*d^2*e*x^

3 + 6*b*c^3*d*x*(d^2 - 3*d*e*x + 3*e^2*x^2) + b^2*c^2*(d^3 - 18*d^2*e*x + 9*d*e^

2*x^2 - 14*e^3*x^3)) + a^2*(3*b^3*e^3 - 42*b^2*c*e^3*x - 24*b*c^2*d*e*(d - 3*e*x

) + 8*c^3*(d^3 + 9*d*e^2*x^2 + 4*e^3*x^3)))))/(3*c^2*(b^2 - 4*a*c)^2*(a + x*(b +

c*x))^(3/2)) + (B*e^3*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2)

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Maple [B] time = 0.018, size = 2443, normalized size = 6.2 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^2*e+a/c/(4*a*c-b^2)/(c*x^2+b*x+a

)^(3/2)*b*A*d*e^2+a/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b*B*d^2*e+16*a*c/(4*a*c-b^

2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d*e^2+16*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*

d^2*e-1/48*B*e^3/c^4*b^5/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+1/2*B*e^3/c^3*b^3/(4*a*

c-b^2)/(c*x^2+b*x+a)^(1/2)+1/2*B*e^3/c^2*b*x^2/(c*x^2+b*x+a)^(3/2)+1/8*B*e^3/c^3

*b^2*x/(c*x^2+b*x+a)^(3/2)+1/8/c^3*b^2/(c*x^2+b*x+a)^(3/2)*B*d*e^2+1/24/c^3*b^4/

(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*e^3-3*x^2/c/(c*x^2+b*x+a)^(3/2)*B*d*e^2-1/4/c^

2*b*x/(c*x^2+b*x+a)^(3/2)*A*e^3+4/3*A*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+32

/3*A*d^3*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+16/3*A*d^3*c/(4*a*c-b^2)^2/(c*x

^2+b*x+a)^(1/2)*b-1/6*B*e^3/c^3*b^5/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+1/3*B*e^3/

c^3*b*a/(c*x^2+b*x+a)^(3/2)-8*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*d^2*e-1/3/

c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^3-2/3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2

)*x*B*d^3-3/2*x/c/(c*x^2+b*x+a)^(3/2)*A*d*e^2-3/2*x/c/(c*x^2+b*x+a)^(3/2)*B*d^2*

e+1/4/c^2*b/(c*x^2+b*x+a)^(3/2)*A*d*e^2+1/4/c^2*b/(c*x^2+b*x+a)^(3/2)*B*d^2*e+1/

3/c^2*b^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*e^3-2*a/c^2/(c*x^2+b*x+a)^(3/2)*B*

d*e^2-3/c*b*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d*e^2-1/3/c/(c*x^2+b*x+a)^(3/2

)*B*d^3+B*e^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-x^2/c/(c*x^2+b

*x+a)^(3/2)*A*e^3+1/24/c^3*b^2/(c*x^2+b*x+a)^(3/2)*A*e^3-2/3*a/c^2/(c*x^2+b*x+a)

^(3/2)*A*e^3+1/2*B*e^3/c^3*b/(c*x^2+b*x+a)^(1/2)+2/3*A*d^3/(4*a*c-b^2)/(c*x^2+b*

x+a)^(3/2)*b-1/3*B*e^3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/48*B*e^3/c^4*b^3/(c*x^2+b*x+a

)^(3/2)-B*e^3/c^2*x/(c*x^2+b*x+a)^(1/2)-8/3*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2

)*B*d^3-1/c/(c*x^2+b*x+a)^(3/2)*A*d^2*e+1/4/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3

/2)*A*d*e^2+1/4/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d^2*e+4*b^2/(4*a*c-b^2

)^2/(c*x^2+b*x+a)^(1/2)*x*A*d*e^2+4*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^

2*e-4/c*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*A*e^3+1/8/c^3*b^4/(4*a*c-b^2)/(c

*x^2+b*x+a)^(3/2)*B*d*e^2-1/24*B*e^3/c^3*b^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-1

/3*B*e^3/c^2*b^4/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/4*B*e^3/c^3*b^3*a/(4*a*c-

b^2)/(c*x^2+b*x+a)^(3/2)+2*B*e^3/c^2*b^3*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+B*e

^3/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/

2)*b*A*d*e^2+8*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b*B*d^2*e-2*b/(4*a*c-b^2)/(c*

x^2+b*x+a)^(3/2)*x*A*d^2*e-1/c*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*d^2*e-16/3*

c*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d^3+2/c*b^3/(4*a*c-b^2)^2/(c*x^2+b*x+a

)^(1/2)*A*d*e^2+2/c*b^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*B*d^2*e+2*a/(4*a*c-b^2

)/(c*x^2+b*x+a)^(3/2)*x*A*d*e^2+2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d^2*e+2/

3/c*b^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^3+1/c^2*b^4/(4*a*c-b^2)^2/(c*x^2

+b*x+a)^(1/2)*B*d*e^2-1/2/c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*A*e^3-8*b*a/

(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*e^3-3/4/c^2*b*x/(c*x^2+b*x+a)^(3/2)*B*d*e^

2+1/12/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*A*e^3+1/2/c*b^2/(4*a*c-b^2)/(c*

x^2+b*x+a)^(3/2)*x*A*d*e^2-16*c*b/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*A*d^2*e+1/

2*B*e^3/c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x+4*B*e^3/c*b^2*a/(4*a*c-b^2)^

2/(c*x^2+b*x+a)^(1/2)*x+1/4/c^2*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x*B*d*e^2+2/

c*b^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^2-1/c*b*a/(4*a*c-b^2)/(c*x^2+b*x

+a)^(3/2)*x*A*e^3-3/2/c^2*b^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*B*d*e^2-24*b*a/(

4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x*B*d*e^2-12/c*b^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a

)^(1/2)*B*d*e^2

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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)^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 2.10874, 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)^3/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(4*(24*(2*B*a^3 - A*a^2*b)*c^2*d*e^2 + (4*(2*B*a^2 - 3*A*a*b)*c^3 + (2*B*a

*b^2 + A*b^3)*c^2)*d^3 + 6*(4*A*a^2*c^3 - (4*B*a^2*b - A*a*b^2)*c^2)*d^2*e + (3*

B*a^2*b^3 - 20*B*a^3*b*c + 16*A*a^3*c^2)*e^3 + (8*(B*b*c^4 - 2*A*c^5)*d^3 - 6*(B

*b^2*c^3 + 4*(B*a - A*b)*c^4)*d^2*e - 3*(B*b^3*c^2 + 8*A*a*c^4 - 2*(6*B*a*b - A*

b^2)*c^3)*d*e^2 + (4*B*b^4*c + 4*(8*B*a^2 + 3*A*a*b)*c^3 - (28*B*a*b^2 + A*b^3)*

c^2)*e^3)*x^3 + 3*(4*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 3*(B*b^3*c^2 + 4*(B*a*b - A*b

^2)*c^3)*d^2*e + 3*(4*(2*B*a^2 - A*a*b)*c^3 + (2*B*a*b^2 - A*b^3)*c^2)*d*e^2 + (

B*b^5 - 6*B*a*b^3*c + 2*A*a*b^2*c^2 + 8*A*a^2*c^3)*e^3)*x^2 + 3*(12*(2*B*a^2*b -

A*a*b^2)*c^2*d*e^2 + (B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - A*b^2)*c^3)*d^3 + 3*

(4*A*a*b*c^3 - (4*B*a*b^2 - A*b^3)*c^2)*d^2*e + 2*(B*a*b^4 - 7*B*a^2*b^2*c + 4*(

B*a^3 + A*a^2*b)*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*((B*b^4*c^2 - 8*

B*a*b^2*c^3 + 16*B*a^2*c^4)*e^3*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^

3)*e^3*x^3 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*e^3*x^2 + 2*(B*a*b^5 - 8*B*a^2

*b^3*c + 16*B*a^3*b*c^2)*e^3*x + (B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*e^3)

*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^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^

5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 -

6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x)*

sqrt(c)), -1/3*(2*(24*(2*B*a^3 - A*a^2*b)*c^2*d*e^2 + (4*(2*B*a^2 - 3*A*a*b)*c^3

+ (2*B*a*b^2 + A*b^3)*c^2)*d^3 + 6*(4*A*a^2*c^3 - (4*B*a^2*b - A*a*b^2)*c^2)*d^

2*e + (3*B*a^2*b^3 - 20*B*a^3*b*c + 16*A*a^3*c^2)*e^3 + (8*(B*b*c^4 - 2*A*c^5)*d

^3 - 6*(B*b^2*c^3 + 4*(B*a - A*b)*c^4)*d^2*e - 3*(B*b^3*c^2 + 8*A*a*c^4 - 2*(6*B

*a*b - A*b^2)*c^3)*d*e^2 + (4*B*b^4*c + 4*(8*B*a^2 + 3*A*a*b)*c^3 - (28*B*a*b^2

+ A*b^3)*c^2)*e^3)*x^3 + 3*(4*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 3*(B*b^3*c^2 + 4*(B*

a*b - A*b^2)*c^3)*d^2*e + 3*(4*(2*B*a^2 - A*a*b)*c^3 + (2*B*a*b^2 - A*b^3)*c^2)*

d*e^2 + (B*b^5 - 6*B*a*b^3*c + 2*A*a*b^2*c^2 + 8*A*a^2*c^3)*e^3)*x^2 + 3*(12*(2*

B*a^2*b - A*a*b^2)*c^2*d*e^2 + (B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - A*b^2)*c^3)

*d^3 + 3*(4*A*a*b*c^3 - (4*B*a*b^2 - A*b^3)*c^2)*d^2*e + 2*(B*a*b^4 - 7*B*a^2*b^

2*c + 4*(B*a^3 + A*a^2*b)*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*((B*b^

4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*e^3*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*

B*a^2*b*c^3)*e^3*x^3 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*e^3*x^2 + 2*(B*a*b^5

- 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*e^3*x + (B*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4

*c^2)*e^3)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a^2*b^4

*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2

*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^

5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*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((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.291863, size = 1067, normalized size = 2.69 \[ -\frac{B e^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} - \frac{2 \,{\left ({\left ({\left (\frac{{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e - 24 \, B a c^{4} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} + 36 \, B a b c^{3} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} - 24 \, A a c^{4} d e^{2} + 4 \, B b^{4} c e^{3} - 28 \, B a b^{2} c^{2} e^{3} - A b^{3} c^{2} e^{3} + 32 \, B a^{2} c^{3} e^{3} + 12 \, A a b c^{3} e^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e - 12 \, B a b c^{3} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e + 6 \, B a b^{2} c^{2} d e^{2} - 3 \, A b^{3} c^{2} d e^{2} + 24 \, B a^{2} c^{3} d e^{2} - 12 \, A a b c^{3} d e^{2} + B b^{5} e^{3} - 6 \, B a b^{3} c e^{3} + 2 \, A a b^{2} c^{2} e^{3} + 8 \, A a^{2} c^{3} e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (B b^{3} c^{2} d^{3} + 4 \, B a b c^{3} d^{3} - 2 \, A b^{2} c^{3} d^{3} - 8 \, A a c^{4} d^{3} - 12 \, B a b^{2} c^{2} d^{2} e + 3 \, A b^{3} c^{2} d^{2} e + 12 \, A a b c^{3} d^{2} e + 24 \, B a^{2} b c^{2} d e^{2} - 12 \, A a b^{2} c^{2} d e^{2} + 2 \, B a b^{4} e^{3} - 14 \, B a^{2} b^{2} c e^{3} + 8 \, B a^{3} c^{2} e^{3} + 8 \, A a^{2} b c^{2} e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{2 \, B a b^{2} c^{2} d^{3} + A b^{3} c^{2} d^{3} + 8 \, B a^{2} c^{3} d^{3} - 12 \, A a b c^{3} d^{3} - 24 \, B a^{2} b c^{2} d^{2} e + 6 \, A a b^{2} c^{2} d^{2} e + 24 \, A a^{2} c^{3} d^{2} e + 48 \, B a^{3} c^{2} d e^{2} - 24 \, A a^{2} b c^{2} d e^{2} + 3 \, B a^{2} b^{3} e^{3} - 20 \, B a^{3} b c e^{3} + 16 \, A a^{3} c^{2} e^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-B*e^3*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2) - 2/3

*((((8*B*b*c^4*d^3 - 16*A*c^5*d^3 - 6*B*b^2*c^3*d^2*e - 24*B*a*c^4*d^2*e + 24*A*

b*c^4*d^2*e - 3*B*b^3*c^2*d*e^2 + 36*B*a*b*c^3*d*e^2 - 6*A*b^2*c^3*d*e^2 - 24*A*

a*c^4*d*e^2 + 4*B*b^4*c*e^3 - 28*B*a*b^2*c^2*e^3 - A*b^3*c^2*e^3 + 32*B*a^2*c^3*

e^3 + 12*A*a*b*c^3*e^3)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(4*B*b^2*c^3*

d^3 - 8*A*b*c^4*d^3 - 3*B*b^3*c^2*d^2*e - 12*B*a*b*c^3*d^2*e + 12*A*b^2*c^3*d^2*

e + 6*B*a*b^2*c^2*d*e^2 - 3*A*b^3*c^2*d*e^2 + 24*B*a^2*c^3*d*e^2 - 12*A*a*b*c^3*

d*e^2 + B*b^5*e^3 - 6*B*a*b^3*c*e^3 + 2*A*a*b^2*c^2*e^3 + 8*A*a^2*c^3*e^3)/(b^4*

c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 3*(B*b^3*c^2*d^3 + 4*B*a*b*c^3*d^3 - 2*A*b^

2*c^3*d^3 - 8*A*a*c^4*d^3 - 12*B*a*b^2*c^2*d^2*e + 3*A*b^3*c^2*d^2*e + 12*A*a*b*

c^3*d^2*e + 24*B*a^2*b*c^2*d*e^2 - 12*A*a*b^2*c^2*d*e^2 + 2*B*a*b^4*e^3 - 14*B*a

^2*b^2*c*e^3 + 8*B*a^3*c^2*e^3 + 8*A*a^2*b*c^2*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*

a^2*c^4))*x + (2*B*a*b^2*c^2*d^3 + A*b^3*c^2*d^3 + 8*B*a^2*c^3*d^3 - 12*A*a*b*c^

3*d^3 - 24*B*a^2*b*c^2*d^2*e + 6*A*a*b^2*c^2*d^2*e + 24*A*a^2*c^3*d^2*e + 48*B*a

^3*c^2*d*e^2 - 24*A*a^2*b*c^2*d*e^2 + 3*B*a^2*b^3*e^3 - 20*B*a^3*b*c*e^3 + 16*A*

a^3*c^2*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)

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