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

Optimal. Leaf size=52 \[ \frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0157799, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {15, 321, 205} \[ \frac{x^2}{b \sqrt{d x^2}}-\frac{\sqrt{a} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

x^2/(b*Sqrt[d*x^2]) - (Sqrt[a]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b^(3/2)*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 321

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

Mathematica [A]  time = 0.0156811, size = 44, normalized size = 0.85 \[ \frac{x \left (\sqrt{b} x-\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{b^{3/2} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 38, normalized size = 0.7 \begin{align*}{\frac{x}{b} \left ( x\sqrt{ab}-a\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) \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(x^3/(b*x^2+a)/(d*x^2)^(1/2),x)

[Out]

x*(x*(a*b)^(1/2)-a*arctan(b*x/(a*b)^(1/2)))/(d*x^2)^(1/2)/b/(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(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 = 1.24982, size = 257, normalized size = 4.94 \begin{align*} \left [\frac{d \sqrt{-\frac{a}{b d}} \log \left (\frac{b x^{2} - 2 \, \sqrt{d x^{2}} b \sqrt{-\frac{a}{b d}} - a}{b x^{2} + a}\right ) + 2 \, \sqrt{d x^{2}}}{2 \, b d}, -\frac{d \sqrt{\frac{a}{b d}} \arctan \left (\frac{\sqrt{d x^{2}} b \sqrt{\frac{a}{b d}}}{a}\right ) - \sqrt{d x^{2}}}{b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.1171, size = 62, normalized size = 1.19 \begin{align*} -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} b} - \frac{\sqrt{d x^{2}}}{b}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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