3.2395 $$\int \frac{d+e x}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=91 $\frac{8 (b+2 c x) (2 c d-b e)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}$

[Out]

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (8*(2*c*d - b*e)*(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 = 0.0213004, antiderivative size = 91, 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} $\frac{8 (b+2 c x) (2 c d-b e)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.12243, size = 110, normalized size = 1.21 $-\frac{2 \left (8 c \left (a^2 e-3 a c d x-2 c^2 d x^3\right )+2 b^2 (a e+3 c x (2 e x-d))+4 b c \left (2 c x^2 (e x-3 d)-3 a (d-e x)\right )+b^3 (d+3 e x)\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 131, normalized size = 1.4 \begin{align*} -{\frac{16\,b{c}^{2}e{x}^{3}-32\,{c}^{3}d{x}^{3}+24\,{b}^{2}ce{x}^{2}-48\,b{c}^{2}d{x}^{2}+24\,abcex-48\,a{c}^{2}dx+6\,{b}^{3}ex-12\,{b}^{2}cdx+16\,{a}^{2}ce+4\,a{b}^{2}e-24\,cabd+2\,{b}^{3}d}{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)/(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/3/(c*x^2+b*x+a)^(3/2)*(8*b*c^2*e*x^3-16*c^3*d*x^3+12*b^2*c*e*x^2-24*b*c^2*d*x^2+12*a*b*c*e*x-24*a*c^2*d*x+3
*b^3*e*x-6*b^2*c*d*x+8*a^2*c*e+2*a*b^2*e-12*a*b*c*d+b^3*d)/(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((e*x+d)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.44065, size = 521, normalized size = 5.73 \begin{align*} \frac{2 \,{\left (8 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} + 12 \,{\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} -{\left (b^{3} - 12 \, a b c\right )} d - 2 \,{\left (a b^{2} + 4 \, a^{2} c\right )} e + 3 \,{\left (2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} d -{\left (b^{3} + 4 \, a b c\right )} 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)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [B]  time = 1.12858, size = 306, normalized size = 3.36 \begin{align*} \frac{{\left (4 \,{\left (\frac{2 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (2 \, b^{2} c d + 8 \, a c^{2} d - b^{3} e - 4 \, a b c e\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d - 12 \, a b c d + 2 \, a b^{2} e + 8 \, a^{2} c e}{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)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

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