3.1680 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=92 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^2 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.0357957, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x}}{e^2 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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

Mathematica [A]  time = 0.0238114, size = 45, normalized size = 0.49 \[ \frac{2 \sqrt{(a+b x)^2} (-a e+2 b d+b e x)}{e^2 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.041, size = 42, normalized size = 0.5 \begin{align*} -2\,{\frac{ \left ( -bxe+ae-2\,bd \right ) \sqrt{ \left ( bx+a \right ) ^{2}}}{\sqrt{ex+d}{e}^{2} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x)

[Out]

-2/(e*x+d)^(1/2)*(-b*e*x+a*e-2*b*d)*((b*x+a)^2)^(1/2)/e^2/(b*x+a)

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Maxima [A]  time = 1.09141, size = 34, normalized size = 0.37 \begin{align*} \frac{2 \,{\left (b e x + 2 \, b d - a e\right )}}{\sqrt{e x + d} e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42545, size = 74, normalized size = 0.8 \begin{align*} \frac{2 \,{\left (b e x + 2 \, b d - a e\right )} \sqrt{e x + d}}{e^{3} x + d e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\left (a + b x\right )^{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)**2)**(1/2)/(e*x+d)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.15545, size = 72, normalized size = 0.78 \begin{align*} 2 \, \sqrt{x e + d} b e^{\left (-2\right )} \mathrm{sgn}\left (b x + a\right ) + \frac{2 \,{\left (b d \mathrm{sgn}\left (b x + a\right ) - a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{\sqrt{x e + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

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