3.812 \(\int \frac{\sqrt{x} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=144 \[ \frac{2 \sqrt{x} (a+b x) (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{a} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(2*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(3/2)*(a + b*x))/(3*b*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (2*Sqrt[a]*(A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(5/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.0703647, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \[ \frac{2 \sqrt{x} (a+b x) (A b-a B)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{a} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x]*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(A*b - a*B)*Sqrt[x]*(a + b*x))/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*B*x^(3/2)*(a + b*x))/(3*b*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (2*Sqrt[a]*(A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(b^(5/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sqrt{x} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{\sqrt{x} (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (\frac{3 A b^2}{2}-\frac{3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) \sqrt{x} (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a \left (\frac{3 A b^2}{2}-\frac{3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) \sqrt{x} (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (4 a \left (\frac{3 A b^2}{2}-\frac{3 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{3 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) \sqrt{x} (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{3/2} (a+b x)}{3 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 \sqrt{a} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0440493, size = 82, normalized size = 0.57 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} (-3 a B+3 A b+b B x)+3 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{3 b^{5/2} \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x]*(A + B*x))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(2*(a + b*x)*(Sqrt[b]*Sqrt[x]*(3*A*b - 3*a*B + b*B*x) + 3*Sqrt[a]*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt
[a]]))/(3*b^(5/2)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.008, size = 94, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{3\,{b}^{2}} \left ( B\sqrt{ab}{x}^{{\frac{3}{2}}}b+3\,A\sqrt{ab}\sqrt{x}b-3\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) ab-3\,B\sqrt{ab}\sqrt{x}a+3\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{2} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*x^(1/2)/((b*x+a)^2)^(1/2),x)

[Out]

2/3*(b*x+a)*(B*(a*b)^(1/2)*x^(3/2)*b+3*A*(a*b)^(1/2)*x^(1/2)*b-3*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a*b-3*B*(a*b)
^(1/2)*x^(1/2)*a+3*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^2)/((b*x+a)^2)^(1/2)/b^2/(a*b)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.39636, size = 305, normalized size = 2.12 \begin{align*} \left [-\frac{3 \,{\left (B a - A b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (B b x - 3 \, B a + 3 \, A b\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/3*(3*(B*a - A*b)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(B*b*x - 3*B*a + 3*A*b)*
sqrt(x))/b^2, 2/3*(3*(B*a - A*b)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (B*b*x - 3*B*a + 3*A*b)*sqrt(x))/b^
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)*x**(1/2)/((b*x+a)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16416, size = 127, normalized size = 0.88 \begin{align*} \frac{2 \,{\left (B a^{2} \mathrm{sgn}\left (b x + a\right ) - A a b \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (B b^{2} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) - 3 \, B a b \sqrt{x} \mathrm{sgn}\left (b x + a\right ) + 3 \, A b^{2} \sqrt{x} \mathrm{sgn}\left (b x + a\right )\right )}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(B*a^2*sgn(b*x + a) - A*a*b*sgn(b*x + a))*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^2) + 2/3*(B*b^2*x^(3/2)*s
gn(b*x + a) - 3*B*a*b*sqrt(x)*sgn(b*x + a) + 3*A*b^2*sqrt(x)*sgn(b*x + a))/b^3