### 3.2394 $$\int \frac{(d+e x)^2}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=98 $\frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0374289, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {728, 636} $\frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(b + 2*c*x)*(d + e*x)^2)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (8*(2*c*d - 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 728

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

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]

Rubi steps

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

Mathematica [A]  time = 0.730552, size = 167, normalized size = 1.7 $\frac{2 \left (4 b \left (2 a^2 e^2+3 a c (d-e x)^2+2 c^2 d x^2 (3 d-2 e x)\right )+8 c \left (-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3\right )+b^2 \left (2 c x \left (3 d^2-12 d e x+e^2 x^2\right )-4 a e (d-3 e x)\right )+b^3 \left (-\left (d^2+6 d e x-3 e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.047, size = 215, normalized size = 2.2 \begin{align*}{\frac{16\,a{c}^{2}{e}^{2}{x}^{3}+4\,{b}^{2}c{e}^{2}{x}^{3}-32\,b{c}^{2}de{x}^{3}+32\,{c}^{3}{d}^{2}{x}^{3}+24\,abc{e}^{2}{x}^{2}+6\,{b}^{3}{e}^{2}{x}^{2}-48\,{b}^{2}cde{x}^{2}+48\,b{c}^{2}{d}^{2}{x}^{2}+24\,a{b}^{2}{e}^{2}x-48\,abcdex+48\,a{c}^{2}{d}^{2}x-12\,{b}^{3}dex+12\,{b}^{2}c{d}^{2}x+16\,{a}^{2}b{e}^{2}-32\,{a}^{2}cde-8\,a{b}^{2}de+24\,abc{d}^{2}-2\,{b}^{3}{d}^{2}}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 8.82085, size = 644, normalized size = 6.57 \begin{align*} \frac{2 \,{\left (8 \, a^{2} b e^{2} + 2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e +{\left (b^{2} c + 4 \, a c^{2}\right )} e^{2}\right )} x^{3} -{\left (b^{3} - 12 \, a b c\right )} d^{2} - 4 \,{\left (a b^{2} + 4 \, a^{2} c\right )} d e + 3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e +{\left (b^{3} + 4 \, a b c\right )} e^{2}\right )} x^{2} + 6 \,{\left (2 \, a b^{2} e^{2} +{\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} + 4 \, a b c\right )} d e\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{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.11447, size = 387, normalized size = 3.95 \begin{align*} \frac{{\left ({\left (\frac{2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2} + 4 \, a c^{2} e^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2} + 4 \, a b c e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (b^{2} c d^{2} + 4 \, a c^{2} d^{2} - b^{3} d e - 4 \, a b c d e + 2 \, a b^{2} e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d^{2} - 12 \, a b c d^{2} + 4 \, a b^{2} d e + 16 \, a^{2} c d e - 8 \, a^{2} b e^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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