3.3.4 \(\int \frac {x^3}{\sqrt {a x^2+b x^5}} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 \sqrt {a x^2+b x^5}}{3 b x} \]

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1588} \begin {gather*} \frac {2 \sqrt {a x^2+b x^5}}{3 b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[a*x^2 + b*x^5])/(3*b*x)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {a x^2+b x^5}} \, dx &=\frac {2 \sqrt {a x^2+b x^5}}{3 b x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^2 \left (a+b x^3\right )}}{3 b x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[x^2*(a + b*x^3)])/(3*b*x)

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IntegrateAlgebraic [A]  time = 0.04, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a x^2+b x^5}}{3 b x} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*Sqrt[a*x^2 + b*x^5])/(3*b*x)

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fricas [A]  time = 0.39, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, \sqrt {b x^{5} + a x^{2}}}{3 \, b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(b*x^5 + a*x^2)/(b*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt {b x^{5} + a x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^3/sqrt(b*x^5 + a*x^2), x)

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maple [A]  time = 0.05, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 \left (b \,x^{3}+a \right ) x}{3 \sqrt {b \,x^{5}+a \,x^{2}}\, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^5+a*x^2)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.40, size = 14, normalized size = 0.56 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(b*x^3 + a)/b

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mupad [B]  time = 5.19, size = 21, normalized size = 0.84 \begin {gather*} \frac {2\,\sqrt {b\,x^5+a\,x^2}}{3\,b\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x^2 + b*x^5)^(1/2),x)

[Out]

(2*(a*x^2 + b*x^5)^(1/2))/(3*b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**5+a*x**2)**(1/2),x)

[Out]

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

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