3.1732 \(\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{2 b \sqrt{d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 b^2 B (d+e x)^{3/2}}{3 e^4} \]

[Out]

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

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Rubi [A]  time = 0.0520287, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ -\frac{2 b \sqrt{d+e x} (-2 a B e-A b e+3 b B d)}{e^4}-\frac{2 (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 \sqrt{d+e x}}+\frac{2 (b d-a e)^2 (B d-A e)}{3 e^4 (d+e x)^{3/2}}+\frac{2 b^2 B (d+e x)^{3/2}}{3 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0945331, size = 105, normalized size = 0.85 \[ \frac{2 \left (-3 b (d+e x)^2 (-2 a B e-A b e+3 b B d)-3 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)+(b d-a e)^2 (B d-A e)+b^2 B (d+e x)^3\right )}{3 e^4 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((b*d - a*e)^2*(B*d - A*e) - 3*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 3*b*(3*b*B*d - A*b*e - 2
*a*B*e)*(d + e*x)^2 + b^2*B*(d + e*x)^3))/(3*e^4*(d + e*x)^(3/2))

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Maple [A]  time = 0.005, size = 168, normalized size = 1.4 \begin{align*} -{\frac{-2\,B{b}^{2}{x}^{3}{e}^{3}-6\,A{b}^{2}{e}^{3}{x}^{2}-12\,Bab{e}^{3}{x}^{2}+12\,B{b}^{2}d{e}^{2}{x}^{2}+12\,Aab{e}^{3}x-24\,A{b}^{2}d{e}^{2}x+6\,B{a}^{2}{e}^{3}x-48\,Babd{e}^{2}x+48\,B{b}^{2}{d}^{2}ex+2\,{a}^{2}A{e}^{3}+8\,Aabd{e}^{2}-16\,A{b}^{2}{d}^{2}e+4\,B{a}^{2}d{e}^{2}-32\,Bab{d}^{2}e+32\,B{b}^{2}{d}^{3}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/(e*x+d)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.11416, size = 220, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B b^{2} - 3 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )} \sqrt{e x + d}}{e^{3}} + \frac{B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((e*x + d)^(3/2)*B*b^2 - 3*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*sqrt(e*x + d))/e^3 + (B*b^2*d^3 - A*a^2*e^3
- (2*B*a*b + A*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2 - 3*(3*B*b^2*d^2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*
a*b)*e^2)*(e*x + d))/((e*x + d)^(3/2)*e^3))/e

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Fricas [A]  time = 1.68479, size = 367, normalized size = 2.96 \begin{align*} \frac{2 \,{\left (B b^{2} e^{3} x^{3} - 16 \, B b^{2} d^{3} - A a^{2} e^{3} + 8 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e - 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 3 \,{\left (2 \, B b^{2} d e^{2} -{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} - 3 \,{\left (8 \, B b^{2} d^{2} e - 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(B*b^2*e^3*x^3 - 16*B*b^2*d^3 - A*a^2*e^3 + 8*(2*B*a*b + A*b^2)*d^2*e - 2*(B*a^2 + 2*A*a*b)*d*e^2 - 3*(2*B
*b^2*d*e^2 - (2*B*a*b + A*b^2)*e^3)*x^2 - 3*(8*B*b^2*d^2*e - 4*(2*B*a*b + A*b^2)*d*e^2 + (B*a^2 + 2*A*a*b)*e^3
)*x)*sqrt(e*x + d)/(e^6*x^2 + 2*d*e^5*x + d^2*e^4)

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Sympy [A]  time = 1.64116, size = 709, normalized size = 5.72 \begin{align*} \begin{cases} - \frac{2 A a^{2} e^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{8 A a b d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 A a b e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{16 A b^{2} d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{24 A b^{2} d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{6 A b^{2} e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{4 B a^{2} d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 B a^{2} e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{32 B a b d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 B a b d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{12 B a b e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 B b^{2} d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 B b^{2} d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 B b^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 B b^{2} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{A a^{2} x + A a b x^{2} + \frac{A b^{2} x^{3}}{3} + \frac{B a^{2} x^{2}}{2} + \frac{2 B a b x^{3}}{3} + \frac{B b^{2} x^{4}}{4}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*A*a**2*e**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 8*A*a*b*d*e**2/(3*d*e**4*sqrt(d
+ e*x) + 3*e**5*x*sqrt(d + e*x)) - 12*A*a*b*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 16*A*b*
*2*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 24*A*b**2*d*e**2*x/(3*d*e**4*sqrt(d + e*x) + 3*e
**5*x*sqrt(d + e*x)) + 6*A*b**2*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 4*B*a**2*d*e**2/
(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 6*B*a**2*e**3*x/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d
+ e*x)) + 32*B*a*b*d**2*e/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 48*B*a*b*d*e**2*x/(3*d*e**4*sqrt
(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 12*B*a*b*e**3*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 3
2*B*b**2*d**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) - 48*B*b**2*d**2*e*x/(3*d*e**4*sqrt(d + e*x) +
3*e**5*x*sqrt(d + e*x)) - 12*B*b**2*d*e**2*x**2/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)) + 2*B*b**2*
e**3*x**3/(3*d*e**4*sqrt(d + e*x) + 3*e**5*x*sqrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + A*a*b*x**2 + A*b**2*x**3/
3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4)/d**(5/2), True))

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Giac [A]  time = 1.53422, size = 278, normalized size = 2.24 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{8} - 9 \, \sqrt{x e + d} B b^{2} d e^{8} + 6 \, \sqrt{x e + d} B a b e^{9} + 3 \, \sqrt{x e + d} A b^{2} e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} B b^{2} d^{2} - B b^{2} d^{3} - 12 \,{\left (x e + d\right )} B a b d e - 6 \,{\left (x e + d\right )} A b^{2} d e + 2 \, B a b d^{2} e + A b^{2} d^{2} e + 3 \,{\left (x e + d\right )} B a^{2} e^{2} + 6 \,{\left (x e + d\right )} A a b e^{2} - B a^{2} d e^{2} - 2 \, A a b d e^{2} + A a^{2} e^{3}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((x*e + d)^(3/2)*B*b^2*e^8 - 9*sqrt(x*e + d)*B*b^2*d*e^8 + 6*sqrt(x*e + d)*B*a*b*e^9 + 3*sqrt(x*e + d)*A*b
^2*e^9)*e^(-12) - 2/3*(9*(x*e + d)*B*b^2*d^2 - B*b^2*d^3 - 12*(x*e + d)*B*a*b*d*e - 6*(x*e + d)*A*b^2*d*e + 2*
B*a*b*d^2*e + A*b^2*d^2*e + 3*(x*e + d)*B*a^2*e^2 + 6*(x*e + d)*A*a*b*e^2 - B*a^2*d*e^2 - 2*A*a*b*d*e^2 + A*a^
2*e^3)*e^(-4)/(x*e + d)^(3/2)