3.3.90 \(\int \frac {x^3}{\sqrt {a x^2+b x^4}} \, dx\) [290]

Optimal. Leaf size=58 \[ \frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^2+b x^4}}\right )}{2 b^{3/2}} \]

[Out]

-1/2*a*arctanh(x^2*b^(1/2)/(b*x^4+a*x^2)^(1/2))/b^(3/2)+1/2*(b*x^4+a*x^2)^(1/2)/b

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2043, 654, 634, 212} \begin {gather*} \frac {\sqrt {a x^2+b x^4}}{2 b}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a x^2+b x^4}}\right )}{2 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a*x^2 + b*x^4],x]

[Out]

Sqrt[a*x^2 + b*x^4]/(2*b) - (a*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a*x^2 + b*x^4]])/(2*b^(3/2))

Rule 212

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

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 654

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

Rule 2043

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 74, normalized size = 1.28 \begin {gather*} \frac {x \left (\sqrt {b} x \left (a+b x^2\right )+a \sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{2 b^{3/2} \sqrt {x^2 \left (a+b x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a*x^2 + b*x^4],x]

[Out]

(x*(Sqrt[b]*x*(a + b*x^2) + a*Sqrt[a + b*x^2]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]]))/(2*b^(3/2)*Sqrt[x^2*(a + b
*x^2)])

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 64, normalized size = 1.10

method	result	size
default	\(\frac {x \sqrt {b \,x^{2}+a}\, \left (x \sqrt {b \,x^{2}+a}\, b^{\frac {3}{2}}-a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) b \right )}{2 \sqrt {b \,x^{4}+a \,x^{2}}\, b^{\frac {5}{2}}}\)	\(64\)
risch	\(\frac {x^{2} \left (b \,x^{2}+a \right )}{2 b \sqrt {x^{2} \left (b \,x^{2}+a \right )}}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) x \sqrt {b \,x^{2}+a}}{2 b^{\frac {3}{2}} \sqrt {x^{2} \left (b \,x^{2}+a \right )}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^4+a*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*x*(b*x^2+a)^(1/2)*(x*(b*x^2+a)^(1/2)*b^(3/2)-a*ln(x*b^(1/2)+(b*x^2+a)^(1/2))*b)/(b*x^4+a*x^2)^(1/2)/b^(5/2
)

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 52, normalized size = 0.90 \begin {gather*} -\frac {a \log \left (2 \, b x^{2} + a + 2 \, \sqrt {b x^{4} + a x^{2}} \sqrt {b}\right )}{4 \, b^{\frac {3}{2}}} + \frac {\sqrt {b x^{4} + a x^{2}}}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

-1/4*a*log(2*b*x^2 + a + 2*sqrt(b*x^4 + a*x^2)*sqrt(b))/b^(3/2) + 1/2*sqrt(b*x^4 + a*x^2)/b

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 114, normalized size = 1.97 \begin {gather*} \left [\frac {a \sqrt {b} \log \left (-2 \, b x^{2} - a + 2 \, \sqrt {b x^{4} + a x^{2}} \sqrt {b}\right ) + 2 \, \sqrt {b x^{4} + a x^{2}} b}{4 \, b^{2}}, \frac {a \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{4} + a x^{2}} \sqrt {-b}}{b x^{2} + a}\right ) + \sqrt {b x^{4} + a x^{2}} b}{2 \, b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**4+a*x**2)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 3.85, size = 59, normalized size = 1.02 \begin {gather*} -\frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{4 \, b^{\frac {3}{2}}} + \frac {\sqrt {b x^{2} + a} x}{2 \, b \mathrm {sgn}\left (x\right )} + \frac {a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^4+a*x^2)^(1/2),x, algorithm="giac")

[Out]

-1/4*a*log(abs(a))*sgn(x)/b^(3/2) + 1/2*sqrt(b*x^2 + a)*x/(b*sgn(x)) + 1/2*a*log(abs(-sqrt(b)*x + sqrt(b*x^2 +
a)))/(b^(3/2)*sgn(x))

________________________________________________________________________________________

Mupad [B]
time = 4.71, size = 53, normalized size = 0.91 \begin {gather*} \frac {\sqrt {b\,x^4+a\,x^2}}{2\,b}-\frac {a\,\ln \left (\frac {b\,x^2+\frac {a}{2}}{\sqrt {b}}+\sqrt {b\,x^4+a\,x^2}\right )}{4\,b^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x^2 + b*x^4)^(1/2),x)

[Out]

(a*x^2 + b*x^4)^(1/2)/(2*b) - (a*log((a/2 + b*x^2)/b^(1/2) + (a*x^2 + b*x^4)^(1/2)))/(4*b^(3/2))

________________________________________________________________________________________