3.1679 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=94 \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0359643, antiderivative size = 94, 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} \[ \frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^2 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0275974, size = 47, normalized size = 0.5 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} (3 a e-2 b d+b e x)}{3 e^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 42, normalized size = 0.5 \begin{align*}{\frac{2\,bxe+6\,ae-4\,bd}{3\, \left ( bx+a \right ){e}^{2}}\sqrt{ex+d}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07143, size = 62, normalized size = 0.66 \begin{align*} \frac{2 \,{\left (b e^{2} x^{2} - 2 \, b d^{2} + 3 \, a d e -{\left (b d e - 3 \, a e^{2}\right )} x\right )}}{3 \, \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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55486, size = 63, normalized size = 0.67 \begin{align*} \frac{2 \,{\left (b e x - 2 \, b d + 3 \, a e\right )} \sqrt{e x + d}}{3 \, 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)^(1/2),x, algorithm="fricas")

[Out]

2/3*(b*e*x - 2*b*d + 3*a*e)*sqrt(e*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}}}{\sqrt{d + e x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14869, size = 70, normalized size = 0.74 \begin{align*} \frac{2}{3} \,{\left ({\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} b e^{\left (-1\right )} \mathrm{sgn}\left (b x + a\right ) + 3 \, \sqrt{x e + d} a \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*b*e^(-1)*sgn(b*x + a) + 3*sqrt(x*e + d)*a*sgn(b*x + a))*e^(-1)