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

Optimal. Leaf size=121 \[ -\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]

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Rubi [A]  time = 0.07, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {804, 636} \begin {gather*} -\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 804

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(b*f - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(m
*(b*(e*f + d*g) - 2*(c*d*f + a*e*g)))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)
, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {(4 (b B d-2 A c d+A b e-2 a B e)) \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (b B d-2 A c d+A b e-2 a B e) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [B]  time = 0.95, size = 314, normalized size = 2.60 \begin {gather*} \frac {2 A \left (4 b \left (2 a^2 e^2+3 a c (d-e x)^2+2 c^2 d x^2 (3 d-2 e x)\right )+8 c \left (-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3\right )+b^2 \left (2 c x \left (3 d^2-12 d e x+e^2 x^2\right )-4 a e (d-3 e x)\right )-\left (b^3 \left (d^2+6 d e x-3 e^2 x^2\right )\right )\right )-2 B \left (16 a^3 e^2+8 a^2 \left (b e (3 e x-2 d)+c \left (d^2+3 e^2 x^2\right )\right )+2 a \left (b^2 \left (d^2-12 d e x+3 e^2 x^2\right )+6 b c x (d-e x)^2-8 c^2 d e x^3\right )+b x \left (b^2 \left (3 d^2-6 d e x-e^2 x^2\right )+4 b c d x (3 d-e x)+8 c^2 d^2 x^2\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*(-(b^3*(d^2 + 6*d*e*x - 3*e^2*x^2)) + 4*b*(2*a^2*e^2 + 2*c^2*d*x^2*(3*d - 2*e*x) + 3*a*c*(d - e*x)^2) + 8
*c*(-2*a^2*d*e + 2*c^2*d^2*x^3 + a*c*x*(3*d^2 + e^2*x^2)) + b^2*(-4*a*e*(d - 3*e*x) + 2*c*x*(3*d^2 - 12*d*e*x
+ e^2*x^2))) - 2*B*(16*a^3*e^2 + b*x*(8*c^2*d^2*x^2 + 4*b*c*d*x*(3*d - e*x) + b^2*(3*d^2 - 6*d*e*x - e^2*x^2))
+ 8*a^2*(b*e*(-2*d + 3*e*x) + c*(d^2 + 3*e^2*x^2)) + 2*a*(-8*c^2*d*e*x^3 + 6*b*c*x*(d - e*x)^2 + b^2*(d^2 - 1
2*d*e*x + 3*e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))

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IntegrateAlgebraic [B]  time = 3.06, size = 425, normalized size = 3.51 \begin {gather*} -\frac {2 \left (16 a^3 B e^2-8 a^2 A b e^2+16 a^2 A c d e-16 a^2 b B d e+24 a^2 b B e^2 x+8 a^2 B c d^2+24 a^2 B c e^2 x^2+4 a A b^2 d e-12 a A b^2 e^2 x-12 a A b c d^2+24 a A b c d e x-12 a A b c e^2 x^2-24 a A c^2 d^2 x-8 a A c^2 e^2 x^3+2 a b^2 B d^2-24 a b^2 B d e x+6 a b^2 B e^2 x^2+12 a b B c d^2 x-24 a b B c d e x^2+12 a b B c e^2 x^3-16 a B c^2 d e x^3+A b^3 d^2+6 A b^3 d e x-3 A b^3 e^2 x^2-6 A b^2 c d^2 x+24 A b^2 c d e x^2-2 A b^2 c e^2 x^3-24 A b c^2 d^2 x^2+16 A b c^2 d e x^3-16 A c^3 d^2 x^3+3 b^3 B d^2 x-6 b^3 B d e x^2-b^3 B e^2 x^3+12 b^2 B c d^2 x^2-4 b^2 B c d e x^3+8 b B c^2 d^2 x^3\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b^3*d^2 + 2*a*b^2*B*d^2 - 12*a*A*b*c*d^2 + 8*a^2*B*c*d^2 + 4*a*A*b^2*d*e - 16*a^2*b*B*d*e + 16*a^2*A*c*
d*e - 8*a^2*A*b*e^2 + 16*a^3*B*e^2 + 3*b^3*B*d^2*x - 6*A*b^2*c*d^2*x + 12*a*b*B*c*d^2*x - 24*a*A*c^2*d^2*x + 6
*A*b^3*d*e*x - 24*a*b^2*B*d*e*x + 24*a*A*b*c*d*e*x - 12*a*A*b^2*e^2*x + 24*a^2*b*B*e^2*x + 12*b^2*B*c*d^2*x^2
- 24*A*b*c^2*d^2*x^2 - 6*b^3*B*d*e*x^2 + 24*A*b^2*c*d*e*x^2 - 24*a*b*B*c*d*e*x^2 - 3*A*b^3*e^2*x^2 + 6*a*b^2*B
*e^2*x^2 - 12*a*A*b*c*e^2*x^2 + 24*a^2*B*c*e^2*x^2 + 8*b*B*c^2*d^2*x^3 - 16*A*c^3*d^2*x^3 - 4*b^2*B*c*d*e*x^3
+ 16*A*b*c^2*d*e*x^3 - 16*a*B*c^2*d*e*x^3 - b^3*B*e^2*x^3 - 2*A*b^2*c*e^2*x^3 + 12*a*b*B*c*e^2*x^3 - 8*a*A*c^2
*e^2*x^3))/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(3/2))

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fricas [B]  time = 9.12, size = 469, normalized size = 3.88 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \, {\left (B b^{2} c + 4 \, {\left (B a - A b\right )} c^{2}\right )} d e - {\left (B b^{3} + 8 \, A a c^{2} - 2 \, {\left (6 \, B a b - A b^{2}\right )} c\right )} e^{2}\right )} x^{3} + {\left (2 \, B a b^{2} + A b^{3} + 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} d^{2} - 4 \, {\left (4 \, B a^{2} b - A a b^{2} - 4 \, A a^{2} c\right )} d e + 8 \, {\left (2 \, B a^{3} - A a^{2} b\right )} e^{2} + 3 \, {\left (4 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} - 2 \, {\left (B b^{3} + 4 \, {\left (B a b - A b^{2}\right )} c\right )} d e + {\left (2 \, B a b^{2} - A b^{3} + 4 \, {\left (2 \, B a^{2} - A a b\right )} c\right )} e^{2}\right )} x^{2} + 3 \, {\left ({\left (B b^{3} - 8 \, A a c^{2} + 2 \, {\left (2 \, B a b - A b^{2}\right )} c\right )} d^{2} - 2 \, {\left (4 \, B a b^{2} - A b^{3} - 4 \, A a b c\right )} d e + 4 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*((8*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(B*b^2*c + 4*(B*a - A*b)*c^2)*d*e - (B*b^3 + 8*A*a*c^2 - 2*(6*B*a*b - A*b
^2)*c)*e^2)*x^3 + (2*B*a*b^2 + A*b^3 + 4*(2*B*a^2 - 3*A*a*b)*c)*d^2 - 4*(4*B*a^2*b - A*a*b^2 - 4*A*a^2*c)*d*e
+ 8*(2*B*a^3 - A*a^2*b)*e^2 + 3*(4*(B*b^2*c - 2*A*b*c^2)*d^2 - 2*(B*b^3 + 4*(B*a*b - A*b^2)*c)*d*e + (2*B*a*b^
2 - A*b^3 + 4*(2*B*a^2 - A*a*b)*c)*e^2)*x^2 + 3*((B*b^3 - 8*A*a*c^2 + 2*(2*B*a*b - A*b^2)*c)*d^2 - 2*(4*B*a*b^
2 - A*b^3 - 4*A*a*b*c)*d*e + 4*(2*B*a^2*b - A*a*b^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16
*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*
b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)

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giac [B]  time = 0.33, size = 448, normalized size = 3.70 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (\frac {{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 4 \, B b^{2} c d e - 16 \, B a c^{2} d e + 16 \, A b c^{2} d e - B b^{3} e^{2} + 12 \, B a b c e^{2} - 2 \, A b^{2} c e^{2} - 8 \, A a c^{2} e^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (4 \, B b^{2} c d^{2} - 8 \, A b c^{2} d^{2} - 2 \, B b^{3} d e - 8 \, B a b c d e + 8 \, A b^{2} c d e + 2 \, B a b^{2} e^{2} - A b^{3} e^{2} + 8 \, B a^{2} c e^{2} - 4 \, A a b c e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (B b^{3} d^{2} + 4 \, B a b c d^{2} - 2 \, A b^{2} c d^{2} - 8 \, A a c^{2} d^{2} - 8 \, B a b^{2} d e + 2 \, A b^{3} d e + 8 \, A a b c d e + 8 \, B a^{2} b e^{2} - 4 \, A a b^{2} e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, B a b^{2} d^{2} + A b^{3} d^{2} + 8 \, B a^{2} c d^{2} - 12 \, A a b c d^{2} - 16 \, B a^{2} b d e + 4 \, A a b^{2} d e + 16 \, A a^{2} c d e + 16 \, B a^{3} e^{2} - 8 \, A a^{2} b e^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 433, normalized size = 3.58 \begin {gather*} \frac {-8 B a b c \,d^{2} x -\frac {32}{3} A b \,c^{2} d e \,x^{3}-8 B a b c \,e^{2} x^{3}+\frac {32}{3} B a \,c^{2} d e \,x^{3}+\frac {8}{3} B \,b^{2} c d e \,x^{3}+8 A a b c \,e^{2} x^{2}-16 A \,b^{2} c d e \,x^{2}+16 B a \,b^{2} d e x +16 B a b c d e \,x^{2}-16 A a b c d e x +\frac {16}{3} A \,a^{2} b \,e^{2}-\frac {16}{3} B \,a^{2} c \,d^{2}-\frac {4}{3} B a \,b^{2} d^{2}+\frac {32}{3} A \,c^{3} d^{2} x^{3}+\frac {2}{3} B \,b^{3} e^{2} x^{3}+2 A \,b^{3} e^{2} x^{2}-2 B \,b^{3} d^{2} x +8 A a \,b^{2} e^{2} x +16 A a \,c^{2} d^{2} x -4 A \,b^{3} d e x +4 A \,b^{2} c \,d^{2} x -16 B \,a^{2} b \,e^{2} x +\frac {16}{3} A a \,c^{2} e^{2} x^{3}+\frac {4}{3} A \,b^{2} c \,e^{2} x^{3}-\frac {16}{3} B b \,c^{2} d^{2} x^{3}+16 A b \,c^{2} d^{2} x^{2}-16 B \,a^{2} c \,e^{2} x^{2}-4 B a \,b^{2} e^{2} x^{2}+4 B \,b^{3} d e \,x^{2}-8 B \,b^{2} c \,d^{2} x^{2}-\frac {32}{3} A \,a^{2} c d e -\frac {8}{3} A a \,b^{2} d e +8 A a b c \,d^{2}+\frac {32}{3} B \,a^{2} b d e -\frac {2}{3} A \,b^{3} d^{2}-\frac {32}{3} B \,a^{3} e^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/(c*x^2+b*x+a)^(3/2)*(8*A*a*c^2*e^2*x^3+2*A*b^2*c*e^2*x^3-16*A*b*c^2*d*e*x^3+16*A*c^3*d^2*x^3-12*B*a*b*c*e^
2*x^3+16*B*a*c^2*d*e*x^3+B*b^3*e^2*x^3+4*B*b^2*c*d*e*x^3-8*B*b*c^2*d^2*x^3+12*A*a*b*c*e^2*x^2+3*A*b^3*e^2*x^2-
24*A*b^2*c*d*e*x^2+24*A*b*c^2*d^2*x^2-24*B*a^2*c*e^2*x^2-6*B*a*b^2*e^2*x^2+24*B*a*b*c*d*e*x^2+6*B*b^3*d*e*x^2-
12*B*b^2*c*d^2*x^2+12*A*a*b^2*e^2*x-24*A*a*b*c*d*e*x+24*A*a*c^2*d^2*x-6*A*b^3*d*e*x+6*A*b^2*c*d^2*x-24*B*a^2*b
*e^2*x+24*B*a*b^2*d*e*x-12*B*a*b*c*d^2*x-3*B*b^3*d^2*x+8*A*a^2*b*e^2-16*A*a^2*c*d*e-4*A*a*b^2*d*e+12*A*a*b*c*d
^2-A*b^3*d^2-16*B*a^3*e^2+16*B*a^2*b*d*e-8*B*a^2*c*d^2-2*B*a*b^2*d^2)/(16*a^2*c^2-8*a*b^2*c+b^4)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 3.09, size = 423, normalized size = 3.50 \begin {gather*} -\frac {2\,\left (16\,B\,a^3\,e^2-16\,B\,a^2\,b\,d\,e+24\,B\,a^2\,b\,e^2\,x-8\,A\,a^2\,b\,e^2+8\,B\,a^2\,c\,d^2+16\,A\,a^2\,c\,d\,e+24\,B\,a^2\,c\,e^2\,x^2+2\,B\,a\,b^2\,d^2-24\,B\,a\,b^2\,d\,e\,x+4\,A\,a\,b^2\,d\,e+6\,B\,a\,b^2\,e^2\,x^2-12\,A\,a\,b^2\,e^2\,x+12\,B\,a\,b\,c\,d^2\,x-12\,A\,a\,b\,c\,d^2-24\,B\,a\,b\,c\,d\,e\,x^2+24\,A\,a\,b\,c\,d\,e\,x+12\,B\,a\,b\,c\,e^2\,x^3-12\,A\,a\,b\,c\,e^2\,x^2-24\,A\,a\,c^2\,d^2\,x-16\,B\,a\,c^2\,d\,e\,x^3-8\,A\,a\,c^2\,e^2\,x^3+3\,B\,b^3\,d^2\,x+A\,b^3\,d^2-6\,B\,b^3\,d\,e\,x^2+6\,A\,b^3\,d\,e\,x-B\,b^3\,e^2\,x^3-3\,A\,b^3\,e^2\,x^2+12\,B\,b^2\,c\,d^2\,x^2-6\,A\,b^2\,c\,d^2\,x-4\,B\,b^2\,c\,d\,e\,x^3+24\,A\,b^2\,c\,d\,e\,x^2-2\,A\,b^2\,c\,e^2\,x^3+8\,B\,b\,c^2\,d^2\,x^3-24\,A\,b\,c^2\,d^2\,x^2+16\,A\,b\,c^2\,d\,e\,x^3-16\,A\,c^3\,d^2\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(A*b^3*d^2 + 16*B*a^3*e^2 - 8*A*a^2*b*e^2 + 2*B*a*b^2*d^2 + 8*B*a^2*c*d^2 + 3*B*b^3*d^2*x - 3*A*b^3*e^2*x^
2 - 16*A*c^3*d^2*x^3 - B*b^3*e^2*x^3 - 6*A*b^2*c*d^2*x + 24*B*a^2*b*e^2*x - 6*B*b^3*d*e*x^2 - 24*A*b*c^2*d^2*x
^2 + 6*B*a*b^2*e^2*x^2 - 8*A*a*c^2*e^2*x^3 + 24*B*a^2*c*e^2*x^2 + 12*B*b^2*c*d^2*x^2 - 2*A*b^2*c*e^2*x^3 + 8*B
*b*c^2*d^2*x^3 - 12*A*a*b*c*d^2 + 4*A*a*b^2*d*e + 16*A*a^2*c*d*e - 16*B*a^2*b*d*e + 6*A*b^3*d*e*x - 12*A*a*b^2
*e^2*x - 24*A*a*c^2*d^2*x + 12*B*a*b*c*d^2*x - 24*B*a*b^2*d*e*x - 12*A*a*b*c*e^2*x^2 + 12*B*a*b*c*e^2*x^3 + 24
*A*b^2*c*d*e*x^2 + 16*A*b*c^2*d*e*x^3 - 16*B*a*c^2*d*e*x^3 - 4*B*b^2*c*d*e*x^3 + 24*A*a*b*c*d*e*x - 24*B*a*b*c
*d*e*x^2))/(3*(4*a*c - b^2)^2*(a + b*x + c*x^2)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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