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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.02, 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])

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 126, normalized size = 2.42 \[ \left [\frac {d \sqrt {-\frac {a}{b d}} \log \left (\frac {b x^{2} - 2 \, \sqrt {d x^{2}} b \sqrt {-\frac {a}{b d}} - a}{b x^{2} + a}\right ) + 2 \, \sqrt {d x^{2}}}{2 \, b d}, -\frac {d \sqrt {\frac {a}{b d}} \arctan \left (\frac {\sqrt {d x^{2}} b \sqrt {\frac {a}{b d}}}{a}\right ) - \sqrt {d x^{2}}}{b d}\right ] \]

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)]

________________________________________________________________________________________

giac [A]  time = 0.43, size = 46, normalized size = 0.88 \[ -\frac {\frac {a d \arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d} b} - \frac {\sqrt {d x^{2}}}{b}}{d} \]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 38, normalized size = 0.73 \[ \frac {\left (-a \arctan \left (\frac {b x}{\sqrt {a b}}\right )+\sqrt {a b}\, x \right ) x}{\sqrt {d \,x^{2}}\, \sqrt {a b}\, b} \]

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(1/(a*b)^(1/2)*b*x))/(d*x^2)^(1/2)/b/(a*b)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.79, size = 49, normalized size = 0.94 \[ -\frac {\frac {a d^{2} \arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d} b} - \frac {\sqrt {d x^{2}} d}{b}}{d^{2}} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 0.62, size = 37, normalized size = 0.71 \[ \frac {\sqrt {x^2}}{b\,\sqrt {d}}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{b^{3/2}\,\sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2)^(1/2)/(b*d^(1/2)) - (a^(1/2)*atan((b^(1/2)*(x^2)^(1/2))/a^(1/2)))/(b^(3/2)*d^(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \]

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)

________________________________________________________________________________________