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

Optimal. Leaf size=103 \[ -\frac{32 a^3 \sqrt{a x^2+b x^3}}{35 b^4 x}+\frac{16 a^2 \sqrt{a x^2+b x^3}}{35 b^3}-\frac{12 a x \sqrt{a x^2+b x^3}}{35 b^2}+\frac{2 x^2 \sqrt{a x^2+b x^3}}{7 b} \]

[Out]

(16*a^2*Sqrt[a*x^2 + b*x^3])/(35*b^3) - (32*a^3*Sqrt[a*x^2 + b*x^3])/(35*b^4*x) - (12*a*x*Sqrt[a*x^2 + b*x^3])
/(35*b^2) + (2*x^2*Sqrt[a*x^2 + b*x^3])/(7*b)

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Rubi [A]  time = 0.148466, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 1588} \[ -\frac{32 a^3 \sqrt{a x^2+b x^3}}{35 b^4 x}+\frac{16 a^2 \sqrt{a x^2+b x^3}}{35 b^3}-\frac{12 a x \sqrt{a x^2+b x^3}}{35 b^2}+\frac{2 x^2 \sqrt{a x^2+b x^3}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x^4/Sqrt[a*x^2 + b*x^3],x]

[Out]

(16*a^2*Sqrt[a*x^2 + b*x^3])/(35*b^3) - (32*a^3*Sqrt[a*x^2 + b*x^3])/(35*b^4*x) - (12*a*x*Sqrt[a*x^2 + b*x^3])
/(35*b^2) + (2*x^2*Sqrt[a*x^2 + b*x^3])/(7*b)

Rule 2016

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*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

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^4}{\sqrt{a x^2+b x^3}} \, dx &=\frac{2 x^2 \sqrt{a x^2+b x^3}}{7 b}-\frac{(6 a) \int \frac{x^3}{\sqrt{a x^2+b x^3}} \, dx}{7 b}\\ &=-\frac{12 a x \sqrt{a x^2+b x^3}}{35 b^2}+\frac{2 x^2 \sqrt{a x^2+b x^3}}{7 b}+\frac{\left (24 a^2\right ) \int \frac{x^2}{\sqrt{a x^2+b x^3}} \, dx}{35 b^2}\\ &=\frac{16 a^2 \sqrt{a x^2+b x^3}}{35 b^3}-\frac{12 a x \sqrt{a x^2+b x^3}}{35 b^2}+\frac{2 x^2 \sqrt{a x^2+b x^3}}{7 b}-\frac{\left (16 a^3\right ) \int \frac{x}{\sqrt{a x^2+b x^3}} \, dx}{35 b^3}\\ &=\frac{16 a^2 \sqrt{a x^2+b x^3}}{35 b^3}-\frac{32 a^3 \sqrt{a x^2+b x^3}}{35 b^4 x}-\frac{12 a x \sqrt{a x^2+b x^3}}{35 b^2}+\frac{2 x^2 \sqrt{a x^2+b x^3}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0324024, size = 53, normalized size = 0.51 \[ \frac{2 \sqrt{x^2 (a+b x)} \left (8 a^2 b x-16 a^3-6 a b^2 x^2+5 b^3 x^3\right )}{35 b^4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[x^2*(a + b*x)]*(-16*a^3 + 8*a^2*b*x - 6*a*b^2*x^2 + 5*b^3*x^3))/(35*b^4*x)

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Maple [A]  time = 0.004, size = 55, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -5\,{x}^{3}{b}^{3}+6\,a{b}^{2}{x}^{2}-8\,{a}^{2}xb+16\,{a}^{3} \right ) x}{35\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35*(b*x+a)*(-5*b^3*x^3+6*a*b^2*x^2-8*a^2*b*x+16*a^3)*x/b^4/(b*x^3+a*x^2)^(1/2)

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Maxima [A]  time = 1.06037, size = 72, normalized size = 0.7 \begin{align*} \frac{2 \,{\left (5 \, b^{4} x^{4} - a b^{3} x^{3} + 2 \, a^{2} b^{2} x^{2} - 8 \, a^{3} b x - 16 \, a^{4}\right )}}{35 \, \sqrt{b x + a} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*b^4*x^4 - a*b^3*x^3 + 2*a^2*b^2*x^2 - 8*a^3*b*x - 16*a^4)/(sqrt(b*x + a)*b^4)

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Fricas [A]  time = 0.805416, size = 109, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 8 \, a^{2} b x - 16 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{35 \, b^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*b^3*x^3 - 6*a*b^2*x^2 + 8*a^2*b*x - 16*a^3)*sqrt(b*x^3 + a*x^2)/(b^4*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\sqrt{x^{2} \left (a + b x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4/sqrt(b*x^3 + a*x^2), x)