3.1626 \(\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4 b \sqrt{d+e x} (b d-a e)}{e^3}-\frac{2 (b d-a e)^2}{e^3 \sqrt{d+e x}}+\frac{2 b^2 (d+e x)^{3/2}}{3 e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0221356, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac{4 b \sqrt{d+e x} (b d-a e)}{e^3}-\frac{2 (b d-a e)^2}{e^3 \sqrt{d+e x}}+\frac{2 b^2 (d+e x)^{3/2}}{3 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0305628, size = 59, normalized size = 0.88 \[ \frac{2 \left (-3 a^2 e^2+6 a b e (2 d+e x)+b^2 \left (-8 d^2-4 d e x+e^2 x^2\right )\right )}{3 e^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 63, normalized size = 0.9 \begin{align*} -{\frac{-2\,{b}^{2}{x}^{2}{e}^{2}-12\,xab{e}^{2}+8\,x{b}^{2}de+6\,{a}^{2}{e}^{2}-24\,abde+16\,{b}^{2}{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^2*x^2+2*a*b*x+a^2)/(e*x+d)^(3/2),x)

[Out]

-2/3*(-b^2*e^2*x^2-6*a*b*e^2*x+4*b^2*d*e*x+3*a^2*e^2-12*a*b*d*e+8*b^2*d^2)/(e*x+d)^(1/2)/e^3

________________________________________________________________________________________

Maxima [A]  time = 1.10379, size = 101, normalized size = 1.51 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} b^{2} - 6 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{e^{2}} - \frac{3 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\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^2*x^2+2*a*b*x+a^2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.57028, size = 157, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (b^{2} e^{2} x^{2} - 8 \, b^{2} d^{2} + 12 \, a b d e - 3 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 3 \, a b 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^2*x^2+2*a*b*x+a^2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 10.2983, size = 65, normalized size = 0.97 \begin{align*} \frac{2 b^{2} \left (d + e x\right )^{\frac{3}{2}}}{3 e^{3}} + \frac{\sqrt{d + e x} \left (4 a b e - 4 b^{2} d\right )}{e^{3}} - \frac{2 \left (a e - b d\right )^{2}}{e^{3} \sqrt{d + e x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21209, size = 112, normalized size = 1.67 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{6} - 6 \, \sqrt{x e + d} b^{2} d e^{6} + 6 \, \sqrt{x e + d} a b e^{7}\right )} e^{\left (-9\right )} - \frac{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} 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^2*x^2+2*a*b*x+a^2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

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