3.1723 \(\int \frac{(a+b x) (A+B x)}{(d+e x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0347482, 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} \[ -\frac{2 \sqrt{d+e x} (-a B e-A b e+2 b B d)}{e^3}-\frac{2 (b d-a e) (B d-A e)}{e^3 \sqrt{d+e x}}+\frac{2 b B (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0513176, size = 68, normalized size = 0.86 \[ \frac{6 a e (-A e+2 B d+B e x)+6 A b e (2 d+e x)+2 b B \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*A*b*e*(2*d + e*x) + 6*a*e*(2*B*d - A*e + B*e*x) + 2*b*B*(-8*d^2 - 4*d*e*x + e^2*x^2))/(3*e^3*Sqrt[d + e*x])

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Maple [A]  time = 0.005, size = 73, normalized size = 0.9 \begin{align*} -{\frac{-2\,bB{x}^{2}{e}^{2}-6\,Ab{e}^{2}x-6\,Ba{e}^{2}x+8\,Bbdex+6\,aA{e}^{2}-12\,Abde-12\,Bade+16\,bB{d}^{2}}{3\,{e}^{3}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10067, size = 111, normalized size = 1.41 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} B b - 3 \,{\left (2 \, B b d -{\left (B a + A b\right )} e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (B b d^{2} + A a e^{2} -{\left (B a + A b\right )} d e\right )}}{\sqrt{e x + d} 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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.59746, size = 174, normalized size = 2.2 \begin{align*} \frac{2 \,{\left (B b e^{2} x^{2} - 8 \, B b d^{2} - 3 \, A a e^{2} + 6 \,{\left (B a + A b\right )} d e -{\left (4 \, B b d e - 3 \,{\left (B a + A b\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.6787, size = 76, normalized size = 0.96 \begin{align*} \frac{2 B b \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{\sqrt{d + e x} \left (2 A b e + 2 B a e - 4 B b d\right )}{e^{3}} + \frac{2 \left (- A e + B d\right ) \left (a e - b d\right )}{e^{3} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.36433, size = 135, normalized size = 1.71 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b e^{6} - 6 \, \sqrt{x e + d} B b d e^{6} + 3 \, \sqrt{x e + d} B a e^{7} + 3 \, \sqrt{x e + d} A b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((x*e + d)^(3/2)*B*b*e^6 - 6*sqrt(x*e + d)*B*b*d*e^6 + 3*sqrt(x*e + d)*B*a*e^7 + 3*sqrt(x*e + d)*A*b*e^7)*
e^(-9) - 2*(B*b*d^2 - B*a*d*e - A*b*d*e + A*a*e^2)*e^(-3)/sqrt(x*e + d)