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

Optimal. Leaf size=90 \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0208019, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {638, 613} \[ -\frac{8 (b+2 c x) (b B-2 A c)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 638

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

Rule 613

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

Rubi steps

\begin{align*} \int \frac{A+B x}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{(4 (b B-2 A c)) \int \frac{1}{\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)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{8 (b B-2 A c) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.105536, size = 99, normalized size = 1.1 \[ -\frac{2 \left (B \left (8 a^2 c+2 a b (b+6 c x)+b x \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )+A (b+2 c x) \left (-4 c \left (3 a+2 c x^2\right )+b^2-8 b c x\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 132, normalized size = 1.5 \begin{align*}{\frac{32\,A{x}^{3}{c}^{3}-16\,B{x}^{3}b{c}^{2}+48\,A{x}^{2}b{c}^{2}-24\,B{x}^{2}{b}^{2}c+48\,aA{c}^{2}x+12\,A{b}^{2}cx-24\,abBcx-6\,{b}^{3}Bx+24\,Aabc-2\,A{b}^{3}-16\,B{a}^{2}c-4\,Ba{b}^{2}}{48\,{a}^{2}{c}^{2}-24\,a{b}^{2}c+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((B*x+A)/(c*x^2+b*x+a)^(5/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.25549, size = 524, normalized size = 5.82 \begin{align*} -\frac{2 \,{\left (2 \, B a b^{2} + A b^{3} + 8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 12 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c + 3 \,{\left (B b^{3} - 8 \, A a c^{2} + 2 \,{\left (2 \, B a b - A b^{2}\right )} c\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((B*x+A)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(2*B*a*b^2 + A*b^3 + 8*(B*b*c^2 - 2*A*c^3)*x^3 + 12*(B*b^2*c - 2*A*b*c^2)*x^2 + 4*(2*B*a^2 - 3*A*a*b)*c +
3*(B*b^3 - 8*A*a*c^2 + 2*(2*B*a*b - A*b^2)*c)*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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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((B*x+A)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.19913, size = 293, normalized size = 3.26 \begin{align*} -\frac{{\left (4 \,{\left (\frac{2 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (B b^{3} + 4 \, B a b c - 2 \, A b^{2} c - 8 \, A a c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{2 \, B a b^{2} + A b^{3} + 8 \, B a^{2} c - 12 \, A a b c}{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((B*x+A)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

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