3.1731 \(\int \frac{(a+b x)^2 (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]

[Out]

(2*(b*d - a*e)^2*(B*d - A*e))/(e^4*Sqrt[d + e*x]) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[d + e*x])/
e^4 - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3/2))/(3*e^4) + (2*b^2*B*(d + e*x)^(5/2))/(5*e^4)

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Rubi [A]  time = 0.0516942, 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} \[ -\frac{2 b (d+e x)^{3/2} (-2 a B e-A b e+3 b B d)}{3 e^4}+\frac{2 \sqrt{d+e x} (b d-a e) (-a B e-2 A b e+3 b B d)}{e^4}+\frac{2 (b d-a e)^2 (B d-A e)}{e^4 \sqrt{d+e x}}+\frac{2 b^2 B (d+e x)^{5/2}}{5 e^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.085603, size = 107, normalized size = 0.86 \[ \frac{2 \left (-5 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+15 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)+15 (b d-a e)^2 (B d-A e)+3 b^2 B (d+e x)^3\right )}{15 e^4 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^2*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

(2*(15*(b*d - a*e)^2*(B*d - A*e) + 15*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 5*b*(3*b*B*d - A*b*e
- 2*a*B*e)*(d + e*x)^2 + 3*b^2*B*(d + e*x)^3))/(15*e^4*Sqrt[d + e*x])

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Maple [A]  time = 0.004, size = 169, normalized size = 1.4 \begin{align*} -{\frac{-6\,B{b}^{2}{x}^{3}{e}^{3}-10\,A{b}^{2}{e}^{3}{x}^{2}-20\,Bab{e}^{3}{x}^{2}+12\,B{b}^{2}d{e}^{2}{x}^{2}-60\,Aab{e}^{3}x+40\,A{b}^{2}d{e}^{2}x-30\,B{a}^{2}{e}^{3}x+80\,Babd{e}^{2}x-48\,B{b}^{2}{d}^{2}ex+30\,{a}^{2}A{e}^{3}-120\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-60\,B{a}^{2}d{e}^{2}+160\,Bab{d}^{2}e-96\,B{b}^{2}{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x)

[Out]

-2/15/(e*x+d)^(1/2)*(-3*B*b^2*e^3*x^3-5*A*b^2*e^3*x^2-10*B*a*b*e^3*x^2+6*B*b^2*d*e^2*x^2-30*A*a*b*e^3*x+20*A*b
^2*d*e^2*x-15*B*a^2*e^3*x+40*B*a*b*d*e^2*x-24*B*b^2*d^2*e*x+15*A*a^2*e^3-60*A*a*b*d*e^2+40*A*b^2*d^2*e-30*B*a^
2*d*e^2+80*B*a*b*d^2*e-48*B*b^2*d^3)/e^4

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Maxima [A]  time = 1.87326, size = 225, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B b^{2} - 5 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\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 )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (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}\right )}}{\sqrt{e x + d} e^{3}}\right )}}{15 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*B*b^2 - 5*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(3/2) + 15*(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)*sqrt(e*x + d))/e^3 + 15*(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)/(sqrt(e*x + d)*e^3))/e

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Fricas [A]  time = 1.89739, size = 360, normalized size = 2.9 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} e^{3} x^{3} + 48 \, B b^{2} d^{3} - 15 \, A a^{2} e^{3} - 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 30 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2} -{\left (6 \, B b^{2} d e^{2} - 5 \,{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} +{\left (24 \, B b^{2} d^{2} e - 20 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{5} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^2*e^3*x^3 + 48*B*b^2*d^3 - 15*A*a^2*e^3 - 40*(2*B*a*b + A*b^2)*d^2*e + 30*(B*a^2 + 2*A*a*b)*d*e^2
- (6*B*b^2*d*e^2 - 5*(2*B*a*b + A*b^2)*e^3)*x^2 + (24*B*b^2*d^2*e - 20*(2*B*a*b + A*b^2)*d*e^2 + 15*(B*a^2 + 2
*A*a*b)*e^3)*x)*sqrt(e*x + d)/(e^5*x + d*e^4)

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Sympy [A]  time = 23.6279, size = 150, normalized size = 1.21 \begin{align*} \frac{2 B b^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 A b^{2} e + 4 B a b e - 6 B b^{2} d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (4 A a b e^{2} - 4 A b^{2} d e + 2 B a^{2} e^{2} - 8 B a b d e + 6 B b^{2} d^{2}\right )}{e^{4}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )^{2}}{e^{4} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)/(e*x+d)**(3/2),x)

[Out]

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

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Giac [A]  time = 2.05699, size = 296, normalized size = 2.39 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{2} e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e^{16} + 45 \, \sqrt{x e + d} B b^{2} d^{2} e^{16} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{17} - 60 \, \sqrt{x e + d} B a b d e^{17} - 30 \, \sqrt{x e + d} A b^{2} d e^{17} + 15 \, \sqrt{x e + d} B a^{2} e^{18} + 30 \, \sqrt{x e + d} A a b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B b^{2} d^{3} - 2 \, B a b d^{2} e - A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} - A a^{2} e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*b^2*e^16 - 15*(x*e + d)^(3/2)*B*b^2*d*e^16 + 45*sqrt(x*e + d)*B*b^2*d^2*e^16 + 10*(x
*e + d)^(3/2)*B*a*b*e^17 + 5*(x*e + d)^(3/2)*A*b^2*e^17 - 60*sqrt(x*e + d)*B*a*b*d*e^17 - 30*sqrt(x*e + d)*A*b
^2*d*e^17 + 15*sqrt(x*e + d)*B*a^2*e^18 + 30*sqrt(x*e + d)*A*a*b*e^18)*e^(-20) + 2*(B*b^2*d^3 - 2*B*a*b*d^2*e
- A*b^2*d^2*e + B*a^2*d*e^2 + 2*A*a*b*d*e^2 - A*a^2*e^3)*e^(-4)/sqrt(x*e + d)