3.312 \(\int \frac{x^2}{\sqrt{a x^3+b x^4}} \, dx\)

Optimal. Leaf size=56 \[ \frac{\sqrt{a x^3+b x^4}}{b x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0823131, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2024, 2029, 206} \[ \frac{\sqrt{a x^3+b x^4}}{b x}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a x^3+b x^4}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

Rule 2024

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n +
1)*(a*x^j + b*x^n)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^(n - j)*(m + j*p - n + j + 1))/(b*(m + n*p + 1)
), Int[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j
, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2029

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

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

Mathematica [A]  time = 0.0405259, size = 75, normalized size = 1.34 \[ \frac{\sqrt{b} x^2 (a+b x)-a^{3/2} x^{3/2} \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2} \sqrt{x^3 (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a*x^3 + b*x^4],x]

[Out]

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

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Maple [A]  time = 0.005, size = 78, normalized size = 1.4 \begin{align*}{\frac{x}{2}\sqrt{x \left ( bx+a \right ) } \left ( 2\,\sqrt{b{x}^{2}+ax}{b}^{3/2}-a\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{b{x}^{2}+ax}\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) b \right ){\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}{b}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{b x^{4} + a x^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/sqrt(b*x^4 + a*x^3), x)

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Fricas [A]  time = 0.809629, size = 277, normalized size = 4.95 \begin{align*} \left [\frac{a \sqrt{b} x \log \left (\frac{2 \, b x^{2} + a x - 2 \, \sqrt{b x^{4} + a x^{3}} \sqrt{b}}{x}\right ) + 2 \, \sqrt{b x^{4} + a x^{3}} b}{2 \, b^{2} x}, \frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{4} + a x^{3}} \sqrt{-b}}{b x^{2}}\right ) + \sqrt{b x^{4} + a x^{3}} b}{b^{2} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21226, size = 55, normalized size = 0.98 \begin{align*} \frac{\sqrt{b + \frac{a}{x}} x}{b} + \frac{a \arctan \left (\frac{\sqrt{b + \frac{a}{x}}}{\sqrt{-b}}\right )}{\sqrt{-b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b + a/x)*x/b + a*arctan(sqrt(b + a/x)/sqrt(-b))/(sqrt(-b)*b)