### 3.1681 $$\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^{5/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0365931, 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} $\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0232539, size = 47, normalized size = 0.5 $-\frac{2 \sqrt{(a+b x)^2} (a e+2 b d+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(5/2),x]

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x+a)^2)^(1/2)/(e*x+d)^(5/2),x)

[Out]

-2/3/(e*x+d)^(3/2)*(3*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 = 1.19077, size = 47, normalized size = 0.5 \begin{align*} -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.55682, size = 103, normalized size = 1.1 \begin{align*} -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)**2)**(1/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.13638, size = 65, normalized size = 0.69 \begin{align*} -\frac{2 \,{\left (3 \,{\left (x e + d\right )} b \mathrm{sgn}\left (b x + a\right ) - b d \mathrm{sgn}\left (b x + a\right ) + a e \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

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