3.1625 \(\int \frac{a^2+2 a b x+b^2 x^2}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.022423, antiderivative size = 69, 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} \[ -\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0325503, size = 60, normalized size = 0.87 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15*a^2*e^2 + 10*a*b*e*(-2*d + e*x) + b^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

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Maple [A]  time = 0.047, size = 63, normalized size = 0.9 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}{e}^{2}+20\,xab{e}^{2}-8\,x{b}^{2}de+30\,{a}^{2}{e}^{2}-40\,abde+16\,{b}^{2}{d}^{2}}{15\,{e}^{3}}\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)^(1/2),x)

[Out]

2/15*(3*b^2*e^2*x^2+10*a*b*e^2*x-4*b^2*d*e*x+15*a^2*e^2-20*a*b*d*e+8*b^2*d^2)*(e*x+d)^(1/2)/e^3

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Maxima [A]  time = 1.05642, size = 111, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{e x + d} a^{2} + \frac{10 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} a b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )}}{15 \, 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)^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(e*x + d)*a^2 + 10*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*b/e + (3*(e*x + d)^(5/2) - 10*(e*x + d
)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2/e^2)/e

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Fricas [A]  time = 1.5299, size = 146, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 20 \, a b d e + 15 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \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)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*b^2*e^2*x^2 + 8*b^2*d^2 - 20*a*b*d*e + 15*a^2*e^2 - 2*(2*b^2*d*e - 5*a*b*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 9.33131, size = 236, normalized size = 3.42 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d}{\sqrt{d + e x}} + 2 a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \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)**(1/2),x)

[Out]

Piecewise((-(2*a**2*d/sqrt(d + e*x) + 2*a**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 4*a*b*d*(-d/sqrt(d + e*x) -
sqrt(d + e*x))/e + 4*a*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*b**2*d*(d**2/sqrt
(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*b**2*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x)
+ d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2)/e, Ne(e, 0)), ((a**2*x + a*b*x**2 + b**2*x**3/3)/sqrt(d), Tru
e))

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Giac [A]  time = 1.15855, size = 115, normalized size = 1.67 \begin{align*} \frac{2}{15} \,{\left (10 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )} +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )} + 15 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\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)^(1/2),x, algorithm="giac")

[Out]

2/15*(10*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqr
t(x*e + d)*d^2)*b^2*e^(-2) + 15*sqrt(x*e + d)*a^2)*e^(-1)