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

Optimal. Leaf size=78 \[ \frac{8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0326602, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {728, 636} \[ \frac{8 (2 c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0399624, size = 95, normalized size = 1.22 \[ \frac{4 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )-2 b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+16 b c^2 d x^2 (3 d-2 e x)+32 c^3 d^2 x^3}{3 b^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.047, size = 117, normalized size = 1.5 \begin{align*} -{\frac{2\,x \left ( cx+b \right ) \left ( -2\,{b}^{2}c{e}^{2}{x}^{3}+16\,b{c}^{2}de{x}^{3}-16\,{c}^{3}{d}^{2}{x}^{3}-3\,{b}^{3}{e}^{2}{x}^{2}+24\,{b}^{2}cde{x}^{2}-24\,b{c}^{2}{d}^{2}{x}^{2}+6\,{b}^{3}dex-6\,{b}^{2}c{d}^{2}x+{d}^{2}{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x*(c*x+b)*(-2*b^2*c*e^2*x^3+16*b*c^2*d*e*x^3-16*c^3*d^2*x^3-3*b^3*e^2*x^2+24*b^2*c*d*e*x^2-24*b*c^2*d^2*x
^2+6*b^3*d*e*x-6*b^2*c*d^2*x+b^3*d^2)/b^4/(c*x^2+b*x)^(5/2)

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Maxima [B]  time = 1.07666, size = 274, normalized size = 3.51 \begin{align*} -\frac{4 \, c d^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, c^{2} d^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} + \frac{4 \, d e x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{32 \, c d e x}{3 \, \sqrt{c x^{2} + b x} b^{3}} + \frac{4 \, e^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, e^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{2 \, d^{2}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, c d^{2}}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{16 \, d e}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, e^{2}}{3 \, \sqrt{c x^{2} + b x} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*c*d^2*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*c^2*d^2*x/(sqrt(c*x^2 + b*x)*b^4) + 4/3*d*e*x/((c*x^2 + b*x)^(3/
2)*b) - 32/3*c*d*e*x/(sqrt(c*x^2 + b*x)*b^3) + 4/3*e^2*x/(sqrt(c*x^2 + b*x)*b^2) - 2/3*e^2*x/((c*x^2 + b*x)^(3
/2)*c) - 2/3*d^2/((c*x^2 + b*x)^(3/2)*b) + 16/3*c*d^2/(sqrt(c*x^2 + b*x)*b^3) - 16/3*d*e/(sqrt(c*x^2 + b*x)*b^
2) + 2/3*e^2/(sqrt(c*x^2 + b*x)*b*c)

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Fricas [A]  time = 2.00924, size = 259, normalized size = 3.32 \begin{align*} -\frac{2 \,{\left (b^{3} d^{2} - 2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x^{3} - 3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )} x^{2} - 6 \,{\left (b^{2} c d^{2} - b^{3} d e\right )} x\right )} \sqrt{c x^{2} + b x}}{3 \,{\left (b^{4} c^{2} x^{4} + 2 \, b^{5} c x^{3} + b^{6} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b^3*d^2 - 2*(8*c^3*d^2 - 8*b*c^2*d*e + b^2*c*e^2)*x^3 - 3*(8*b*c^2*d^2 - 8*b^2*c*d*e + b^3*e^2)*x^2 - 6*
(b^2*c*d^2 - b^3*d*e)*x)*sqrt(c*x^2 + b*x)/(b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.3299, size = 166, normalized size = 2.13 \begin{align*} \frac{{\left (x{\left (\frac{2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{6 \,{\left (b^{2} c d^{2} - b^{3} d e\right )}}{b^{4} c^{2}}\right )} x - \frac{d^{2}}{b c^{2}}}{3 \,{\left (c x^{2} + b x\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)^(5/2),x, algorithm="giac")

[Out]

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