3.675 \(\int \frac{1}{x^3 \sqrt{d x^2} (a+b x^2)} \, dx\)

Optimal. Leaf size=68 \[ \frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}} \]

[Out]

b/(a^2*Sqrt[d*x^2]) - 1/(3*a*x^2*Sqrt[d*x^2]) + (b^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[d*x^2])

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Rubi [A]  time = 0.0240175, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {15, 325, 205} \[ \frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}+\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

b/(a^2*Sqrt[d*x^2]) - 1/(3*a*x^2*Sqrt[d*x^2]) + (b^(3/2)*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[d*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, 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{1}{x^3 \sqrt{d x^2} \left (a+b x^2\right )} \, dx &=\frac{x \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{\sqrt{d x^2}}\\ &=-\frac{1}{3 a x^2 \sqrt{d x^2}}-\frac{(b x) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{a \sqrt{d x^2}}\\ &=\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}}+\frac{\left (b^2 x\right ) \int \frac{1}{a+b x^2} \, dx}{a^2 \sqrt{d x^2}}\\ &=\frac{b}{a^2 \sqrt{d x^2}}-\frac{1}{3 a x^2 \sqrt{d x^2}}+\frac{b^{3/2} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0215465, size = 58, normalized size = 0.85 \[ \frac{d \left (3 b^{3/2} x^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-\sqrt{a} \left (a-3 b x^2\right )\right )}{3 a^{5/2} \left (d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 58, normalized size = 0.9 \begin{align*}{\frac{1}{3\,{a}^{2}{x}^{2}} \left ( 3\,{b}^{2}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}+3\,b{x}^{2}\sqrt{ab}-a\sqrt{ab} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36295, size = 331, normalized size = 4.87 \begin{align*} \left [\frac{3 \, b d x^{4} \sqrt{-\frac{b}{a d}} \log \left (\frac{b x^{2} + 2 \, \sqrt{d x^{2}} a \sqrt{-\frac{b}{a d}} - a}{b x^{2} + a}\right ) + 2 \,{\left (3 \, b x^{2} - a\right )} \sqrt{d x^{2}}}{6 \, a^{2} d x^{4}}, \frac{3 \, b d x^{4} \sqrt{\frac{b}{a d}} \arctan \left (\sqrt{d x^{2}} \sqrt{\frac{b}{a d}}\right ) +{\left (3 \, b x^{2} - a\right )} \sqrt{d x^{2}}}{3 \, a^{2} d x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*b*d*x^4*sqrt(-b/(a*d))*log((b*x^2 + 2*sqrt(d*x^2)*a*sqrt(-b/(a*d)) - a)/(b*x^2 + a)) + 2*(3*b*x^2 - a)
*sqrt(d*x^2))/(a^2*d*x^4), 1/3*(3*b*d*x^4*sqrt(b/(a*d))*arctan(sqrt(d*x^2)*sqrt(b/(a*d))) + (3*b*x^2 - a)*sqrt
(d*x^2))/(a^2*d*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x**3*sqrt(d*x**2)*(a + b*x**2)), x)

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Giac [A]  time = 1.11755, size = 92, normalized size = 1.35 \begin{align*} \frac{1}{3} \, d^{2}{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} a^{2} d^{2}} + \frac{3 \, b d x^{2} - a d}{\sqrt{d x^{2}} a^{2} d^{3} x^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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