3.1208 \(\int \frac{(A+B x) (d+e x)^2}{(b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 (d+e x)^2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{8 (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{3 b^4 \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0518084, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {804, 636} \[ -\frac{2 (d+e x)^2 (A b-x (b B-2 A c))}{3 b^2 \left (b x+c x^2\right )^{3/2}}-\frac{8 (x (2 c d-b e)+b d) (A b e-2 A c d+b B d)}{3 b^4 \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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

Mathematica [A]  time = 0.0854175, size = 149, normalized size = 1.62 \[ \frac{2 \left (A \left (2 b^2 c x \left (3 d^2-12 d e x+e^2 x^2\right )+b^3 \left (-\left (d^2+6 d e x-3 e^2 x^2\right )\right )+8 b c^2 d x^2 (3 d-2 e x)+16 c^3 d^2 x^3\right )+b B x \left (b^2 \left (-3 d^2+6 d e x+e^2 x^2\right )+4 b c d x (e x-3 d)-8 c^2 d^2 x^2\right )\right )}{3 b^4 (x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 197, normalized size = 2.1 \begin{align*} -{\frac{2\,x \left ( cx+b \right ) \left ( -2\,A{b}^{2}c{e}^{2}{x}^{3}+16\,Ab{c}^{2}de{x}^{3}-16\,A{c}^{3}{d}^{2}{x}^{3}-B{b}^{3}{e}^{2}{x}^{3}-4\,B{b}^{2}cde{x}^{3}+8\,Bb{c}^{2}{d}^{2}{x}^{3}-3\,A{b}^{3}{e}^{2}{x}^{2}+24\,A{b}^{2}cde{x}^{2}-24\,Ab{c}^{2}{d}^{2}{x}^{2}-6\,B{b}^{3}de{x}^{2}+12\,B{b}^{2}c{d}^{2}{x}^{2}+6\,A{b}^{3}dex-6\,A{b}^{2}c{d}^{2}x+3\,B{b}^{3}{d}^{2}x+A{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((B*x+A)*(e*x+d)^2/(c*x^2+b*x)^(5/2),x)

[Out]

-2/3*x*(c*x+b)*(-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-B*b^3*e^2*x^3-4*B*b^2*c*d*e*x^3+8*B*b*c
^2*d^2*x^3-3*A*b^3*e^2*x^2+24*A*b^2*c*d*e*x^2-24*A*b*c^2*d^2*x^2-6*B*b^3*d*e*x^2+12*B*b^2*c*d^2*x^2+6*A*b^3*d*
e*x-6*A*b^2*c*d^2*x+3*B*b^3*d^2*x+A*b^3*d^2)/b^4/(c*x^2+b*x)^(5/2)

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Maxima [B]  time = 1.0911, size = 468, normalized size = 5.09 \begin{align*} -\frac{B e^{2} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{4 \, A c d^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, A c^{2} d^{2} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} - \frac{B b e^{2} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c^{2}} + \frac{2 \, B e^{2} x}{3 \, \sqrt{c x^{2} + b x} b c} - \frac{2 \, A d^{2}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, A c d^{2}}{3 \, \sqrt{c x^{2} + b x} b^{3}} + \frac{B e^{2}}{3 \, \sqrt{c x^{2} + b x} c^{2}} + \frac{4 \,{\left (2 \, B d e + A e^{2}\right )} x}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \,{\left (B d^{2} + 2 \, A d e\right )} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{2 \,{\left (2 \, B d e + A e^{2}\right )} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{16 \,{\left (B d^{2} + 2 \, A d e\right )} c x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \,{\left (B d^{2} + 2 \, A d e\right )}}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \,{\left (2 \, B d e + A e^{2}\right )}}{3 \, \sqrt{c x^{2} + b x} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*e^2*x^2/((c*x^2 + b*x)^(3/2)*c) - 4/3*A*c*d^2*x/((c*x^2 + b*x)^(3/2)*b^2) + 32/3*A*c^2*d^2*x/(sqrt(c*x^2 +
b*x)*b^4) - 1/3*B*b*e^2*x/((c*x^2 + b*x)^(3/2)*c^2) + 2/3*B*e^2*x/(sqrt(c*x^2 + b*x)*b*c) - 2/3*A*d^2/((c*x^2
+ b*x)^(3/2)*b) + 16/3*A*c*d^2/(sqrt(c*x^2 + b*x)*b^3) + 1/3*B*e^2/(sqrt(c*x^2 + b*x)*c^2) + 4/3*(2*B*d*e + A*
e^2)*x/(sqrt(c*x^2 + b*x)*b^2) + 2/3*(B*d^2 + 2*A*d*e)*x/((c*x^2 + b*x)^(3/2)*b) - 2/3*(2*B*d*e + A*e^2)*x/((c
*x^2 + b*x)^(3/2)*c) - 16/3*(B*d^2 + 2*A*d*e)*c*x/(sqrt(c*x^2 + b*x)*b^3) - 8/3*(B*d^2 + 2*A*d*e)/(sqrt(c*x^2
+ b*x)*b^2) + 2/3*(2*B*d*e + A*e^2)/(sqrt(c*x^2 + b*x)*b*c)

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Fricas [B]  time = 1.84641, size = 389, normalized size = 4.23 \begin{align*} -\frac{2 \,{\left (A b^{3} d^{2} +{\left (8 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \,{\left (B b^{2} c - 4 \, A b c^{2}\right )} d e -{\left (B b^{3} + 2 \, A b^{2} c\right )} e^{2}\right )} x^{3} - 3 \,{\left (A b^{3} e^{2} - 4 \,{\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} + 2 \,{\left (B b^{3} - 4 \, A b^{2} c\right )} d e\right )} x^{2} + 3 \,{\left (2 \, A b^{3} d e +{\left (B b^{3} - 2 \, A b^{2} c\right )} d^{2}\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((B*x+A)*(e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(A*b^3*d^2 + (8*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(B*b^2*c - 4*A*b*c^2)*d*e - (B*b^3 + 2*A*b^2*c)*e^2)*x^3 - 3*
(A*b^3*e^2 - 4*(B*b^2*c - 2*A*b*c^2)*d^2 + 2*(B*b^3 - 4*A*b^2*c)*d*e)*x^2 + 3*(2*A*b^3*d*e + (B*b^3 - 2*A*b^2*
c)*d^2)*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 (A + B x\right ) \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((B*x+A)*(e*x+d)**2/(c*x**2+b*x)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.29368, size = 254, normalized size = 2.76 \begin{align*} -\frac{{\left (x{\left (\frac{{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 16 \, A b c^{2} d e - B b^{3} e^{2} - 2 \, A b^{2} c e^{2}\right )} x}{b^{4} 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 \, A b^{2} c d e - A b^{3} e^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (B b^{3} d^{2} - 2 \, A b^{2} c d^{2} + 2 \, A b^{3} d e\right )}}{b^{4} c^{2}}\right )} x + \frac{A 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((B*x+A)*(e*x+d)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

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