3.1724 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=79 \[ \frac{2 (-a B e-A b e+2 b B d)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e) (B d-A e)}{3 e^3 (d+e x)^{3/2}}+\frac{2 b B \sqrt{d+e x}}{e^3} \]

[Out]

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

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Rubi [A]  time = 0.0335756, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{2 (-a B e-A b e+2 b B d)}{e^3 \sqrt{d+e x}}-\frac{2 (b d-a e) (B d-A e)}{3 e^3 (d+e x)^{3/2}}+\frac{2 b B \sqrt{d+e x}}{e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.045578, size = 68, normalized size = 0.86 \[ -\frac{2 \left (a e (A e+2 B d+3 B e x)+A b e (2 d+3 e x)-b B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )}{3 e^3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 72, normalized size = 0.9 \begin{align*} -{\frac{-6\,bB{x}^{2}{e}^{2}+6\,Ab{e}^{2}x+6\,Ba{e}^{2}x-24\,Bbdex+2\,aA{e}^{2}+4\,Abde+4\,Bade-16\,bB{d}^{2}}{3\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06058, size = 107, normalized size = 1.35 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{e x + d} B b}{e^{2}} - \frac{B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e - 3 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{2}}\right )}}{3 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50717, size = 196, normalized size = 2.48 \begin{align*} \frac{2 \,{\left (3 \, B b e^{2} x^{2} + 8 \, B b d^{2} - A a e^{2} - 2 \,{\left (B a + A b\right )} d e + 3 \,{\left (4 \, B b d e -{\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.29563, size = 355, normalized size = 4.49 \begin{align*} \begin{cases} - \frac{2 A a e^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{4 A b d e}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{6 A b e^{2} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{4 B a d e}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} - \frac{6 B a e^{2} x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{16 B b d^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{24 B b d e x}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} + \frac{6 B b e^{2} x^{2}}{3 d e^{3} \sqrt{d + e x} + 3 e^{4} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{A b x^{2}}{2} + \frac{B a x^{2}}{2} + \frac{B b x^{3}}{3}}{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)*(B*x+A)/(e*x+d)**(5/2),x)

[Out]

Piecewise((-2*A*a*e**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 4*A*b*d*e/(3*d*e**3*sqrt(d + e*x) +
3*e**4*x*sqrt(d + e*x)) - 6*A*b*e**2*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 4*B*a*d*e/(3*d*e**
3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) - 6*B*a*e**2*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 1
6*B*b*d**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)) + 24*B*b*d*e*x/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x
*sqrt(d + e*x)) + 6*B*b*e**2*x**2/(3*d*e**3*sqrt(d + e*x) + 3*e**4*x*sqrt(d + e*x)), Ne(e, 0)), ((A*a*x + A*b*
x**2/2 + B*a*x**2/2 + B*b*x**3/3)/d**(5/2), True))

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Giac [A]  time = 1.73002, size = 119, normalized size = 1.51 \begin{align*} 2 \, \sqrt{x e + d} B b e^{\left (-3\right )} + \frac{2 \,{\left (6 \,{\left (x e + d\right )} B b d - B b d^{2} - 3 \,{\left (x e + d\right )} B a e - 3 \,{\left (x e + d\right )} A b e + B a d e + A b d e - A a e^{2}\right )} e^{\left (-3\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)*(B*x+A)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

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