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

Optimal. Leaf size=137 \[ \frac{256 b^4 \sqrt{a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac{128 b^3 \sqrt{a x+b x^{2/3}}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^2}+\frac{2 x \sqrt{a x+b x^{2/3}}}{3 a} \]

[Out]

(-128*b^3*Sqrt[b*x^(2/3) + a*x])/(105*a^4) + (256*b^4*Sqrt[b*x^(2/3) + a*x])/(105*a^5*x^(1/3)) + (32*b^2*x^(1/
3)*Sqrt[b*x^(2/3) + a*x])/(35*a^3) - (16*b*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(21*a^2) + (2*x*Sqrt[b*x^(2/3) + a*x
])/(3*a)

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Rubi [A]  time = 0.179859, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2016, 2002, 2014} \[ \frac{256 b^4 \sqrt{a x+b x^{2/3}}}{105 a^5 \sqrt [3]{x}}-\frac{128 b^3 \sqrt{a x+b x^{2/3}}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{a x+b x^{2/3}}}{21 a^2}+\frac{2 x \sqrt{a x+b x^{2/3}}}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(-128*b^3*Sqrt[b*x^(2/3) + a*x])/(105*a^4) + (256*b^4*Sqrt[b*x^(2/3) + a*x])/(105*a^5*x^(1/3)) + (32*b^2*x^(1/
3)*Sqrt[b*x^(2/3) + a*x])/(35*a^3) - (16*b*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(21*a^2) + (2*x*Sqrt[b*x^(2/3) + a*x
])/(3*a)

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 2002

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

Rule 2014

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] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx &=\frac{2 x \sqrt{b x^{2/3}+a x}}{3 a}-\frac{(8 b) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{9 a}\\ &=-\frac{16 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^2}+\frac{2 x \sqrt{b x^{2/3}+a x}}{3 a}+\frac{\left (16 b^2\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{21 a^2}\\ &=\frac{32 b^2 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^2}+\frac{2 x \sqrt{b x^{2/3}+a x}}{3 a}-\frac{\left (64 b^3\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{105 a^3}\\ &=-\frac{128 b^3 \sqrt{b x^{2/3}+a x}}{105 a^4}+\frac{32 b^2 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^2}+\frac{2 x \sqrt{b x^{2/3}+a x}}{3 a}+\frac{\left (128 b^4\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{315 a^4}\\ &=-\frac{128 b^3 \sqrt{b x^{2/3}+a x}}{105 a^4}+\frac{256 b^4 \sqrt{b x^{2/3}+a x}}{105 a^5 \sqrt [3]{x}}+\frac{32 b^2 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{35 a^3}-\frac{16 b x^{2/3} \sqrt{b x^{2/3}+a x}}{21 a^2}+\frac{2 x \sqrt{b x^{2/3}+a x}}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.063737, size = 74, normalized size = 0.54 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (48 a^2 b^2 x^{2/3}-40 a^3 b x+35 a^4 x^{4/3}-64 a b^3 \sqrt [3]{x}+128 b^4\right )}{105 a^5 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(128*b^4 - 64*a*b^3*x^(1/3) + 48*a^2*b^2*x^(2/3) - 40*a^3*b*x + 35*a^4*x^(4/3)))/(105
*a^5*x^(1/3))

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Maple [A]  time = 0.003, size = 68, normalized size = 0.5 \begin{align*}{\frac{2}{105\,{a}^{5}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 35\,{x}^{4/3}{a}^{4}-40\,x{a}^{3}b+48\,{x}^{2/3}{a}^{2}{b}^{2}-64\,\sqrt [3]{x}a{b}^{3}+128\,{b}^{4} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*x^(1/3)*(b+a*x^(1/3))*(35*x^(4/3)*a^4-40*x*a^3*b+48*x^(2/3)*a^2*b^2-64*x^(1/3)*a*b^3+128*b^4)/(b*x^(2/3)
+a*x)^(1/2)/a^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/sqrt(a*x + b*x^(2/3)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(a*x + b*x**(2/3)), x)

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Giac [A]  time = 1.13309, size = 108, normalized size = 0.79 \begin{align*} -\frac{256 \, b^{\frac{9}{2}}}{105 \, a^{5}} + \frac{2 \,{\left (35 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} - 180 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b + 378 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{2} - 420 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{3} + 315 \, \sqrt{a x^{\frac{1}{3}} + b} b^{4}\right )}}{105 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-256/105*b^(9/2)/a^5 + 2/105*(35*(a*x^(1/3) + b)^(9/2) - 180*(a*x^(1/3) + b)^(7/2)*b + 378*(a*x^(1/3) + b)^(5/
2)*b^2 - 420*(a*x^(1/3) + b)^(3/2)*b^3 + 315*sqrt(a*x^(1/3) + b)*b^4)/a^5