3.31 \(\int \frac{x (A+B x)}{(a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{A+B x}{b \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0201247, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {778, 217, 206} \[ \frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}-\frac{A+B x}{b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(A + B*x))/(a + b*x^2)^(3/2),x]

[Out]

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

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0536328, size = 64, normalized size = 1.33 \[ \frac{\sqrt{a} B \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{b} (A+B x)}{b^{3/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(A + B*x))/(a + b*x^2)^(3/2),x]

[Out]

(-(Sqrt[b]*(A + B*x)) + Sqrt[a]*B*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*Sqrt[a + b*x^2])

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Maple [A]  time = 0.005, size = 54, normalized size = 1.1 \begin{align*} -{\frac{Bx}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}}-{\frac{A}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(B*x+A)/(b*x^2+a)^(3/2),x)

[Out]

-B*x/b/(b*x^2+a)^(1/2)+B/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-A/b/(b*x^2+a)^(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(x*(B*x+A)/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58117, size = 336, normalized size = 7. \begin{align*} \left [\frac{{\left (B b x^{2} + B a\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (B b x + A b\right )} \sqrt{b x^{2} + a}}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}}, -\frac{{\left (B b x^{2} + B a\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (B b x + A b\right )} \sqrt{b x^{2} + a}}{b^{3} x^{2} + a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.69385, size = 66, normalized size = 1.38 \begin{align*} A \left (\begin{cases} - \frac{1}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + B \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x**2+a)**(3/2),x)

[Out]

A*Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True)) + B*(asinh(sqrt(b)*x/sqrt(a))/b**(
3/2) - x/(sqrt(a)*b*sqrt(1 + b*x**2/a)))

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Giac [A]  time = 1.19577, size = 65, normalized size = 1.35 \begin{align*} -\frac{\frac{B x}{b} + \frac{A}{b}}{\sqrt{b x^{2} + a}} - \frac{B \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

-(B*x/b + A/b)/sqrt(b*x^2 + a) - B*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2)