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

Optimal. Leaf size=47 \[ \frac {4 \sqrt {a x^2+b x^3}}{b^2 x}-\frac {2 x^2}{b \sqrt {a x^2+b x^3}} \]

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Rubi [A]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 1588} \begin {gather*} \frac {4 \sqrt {a x^2+b x^3}}{b^2 x}-\frac {2 x^2}{b \sqrt {a x^2+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a*x^2 + b*x^3)^(3/2),x]

[Out]

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

Rule 2015

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.55 \begin {gather*} \frac {2 x (2 a+b x)}{b^2 \sqrt {x^2 (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a*x^2 + b*x^3)^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.41, size = 37, normalized size = 0.79 \begin {gather*} \frac {2 (2 a+b x) \sqrt {a x^2+b x^3}}{b^2 x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4/(a*x^2 + b*x^3)^(3/2),x]

[Out]

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

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fricas [A]  time = 0.39, size = 38, normalized size = 0.81 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )}}{b^{3} x^{2} + a b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^3 + a*x^2)*(b*x + 2*a)/(b^3*x^2 + a*b^2*x)

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giac [A]  time = 0.28, size = 28, normalized size = 0.60 \begin {gather*} \frac {2 \, {\left (\frac {1}{b} + \frac {2 \, a}{b^{2} x}\right )}}{\sqrt {\frac {b}{x} + \frac {a}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1/b + 2*a/(b^2*x))/sqrt(b/x + a/x^2)

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maple [A]  time = 0.05, size = 34, normalized size = 0.72 \begin {gather*} \frac {2 \left (b x +a \right ) \left (b x +2 a \right ) x^{3}}{\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.54, size = 19, normalized size = 0.40 \begin {gather*} \frac {2 \, {\left (b x + 2 \, a\right )}}{\sqrt {b x + a} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x + 2*a)/(sqrt(b*x + a)*b^2)

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mupad [B]  time = 5.17, size = 35, normalized size = 0.74 \begin {gather*} \frac {2\,\left (2\,a+b\,x\right )\,\sqrt {b\,x^3+a\,x^2}}{b^2\,x\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a*x^2 + b*x^3)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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