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

Optimal. Leaf size=225 \[ -\frac{4096 b^7 \sqrt{a x+b x^{2/3}}}{2145 a^8 \sqrt [3]{x}}+\frac{2048 b^6 \sqrt{a x+b x^{2/3}}}{2145 a^7}-\frac{512 b^5 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{715 a^6}+\frac{256 b^4 x^{2/3} \sqrt{a x+b x^{2/3}}}{429 a^5}-\frac{224 b^3 x \sqrt{a x+b x^{2/3}}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{a x+b x^{2/3}}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{a x+b x^{2/3}}}{65 a^2}+\frac{2 x^2 \sqrt{a x+b x^{2/3}}}{5 a} \]

[Out]

(2048*b^6*Sqrt[b*x^(2/3) + a*x])/(2145*a^7) - (4096*b^7*Sqrt[b*x^(2/3) + a*x])/(2145*a^8*x^(1/3)) - (512*b^5*x
^(1/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^6) + (256*b^4*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(429*a^5) - (224*b^3*x*Sqrt[
b*x^(2/3) + a*x])/(429*a^4) + (336*b^2*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^3) - (28*b*x^(5/3)*Sqrt[b*x^(2/3)
+ a*x])/(65*a^2) + (2*x^2*Sqrt[b*x^(2/3) + a*x])/(5*a)

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Rubi [A]  time = 0.34578, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ -\frac{4096 b^7 \sqrt{a x+b x^{2/3}}}{2145 a^8 \sqrt [3]{x}}+\frac{2048 b^6 \sqrt{a x+b x^{2/3}}}{2145 a^7}-\frac{512 b^5 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{715 a^6}+\frac{256 b^4 x^{2/3} \sqrt{a x+b x^{2/3}}}{429 a^5}-\frac{224 b^3 x \sqrt{a x+b x^{2/3}}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{a x+b x^{2/3}}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{a x+b x^{2/3}}}{65 a^2}+\frac{2 x^2 \sqrt{a x+b x^{2/3}}}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(2048*b^6*Sqrt[b*x^(2/3) + a*x])/(2145*a^7) - (4096*b^7*Sqrt[b*x^(2/3) + a*x])/(2145*a^8*x^(1/3)) - (512*b^5*x
^(1/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^6) + (256*b^4*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(429*a^5) - (224*b^3*x*Sqrt[
b*x^(2/3) + a*x])/(429*a^4) + (336*b^2*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^3) - (28*b*x^(5/3)*Sqrt[b*x^(2/3)
+ a*x])/(65*a^2) + (2*x^2*Sqrt[b*x^(2/3) + a*x])/(5*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^2}{\sqrt{b x^{2/3}+a x}} \, dx &=\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}-\frac{(14 b) \int \frac{x^{5/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{15 a}\\ &=-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}+\frac{\left (56 b^2\right ) \int \frac{x^{4/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{65 a^2}\\ &=\frac{336 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}-\frac{\left (112 b^3\right ) \int \frac{x}{\sqrt{b x^{2/3}+a x}} \, dx}{143 a^3}\\ &=-\frac{224 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}+\frac{\left (896 b^4\right ) \int \frac{x^{2/3}}{\sqrt{b x^{2/3}+a x}} \, dx}{1287 a^4}\\ &=\frac{256 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^5}-\frac{224 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}-\frac{\left (256 b^5\right ) \int \frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}} \, dx}{429 a^5}\\ &=-\frac{512 b^5 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{715 a^6}+\frac{256 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^5}-\frac{224 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}+\frac{\left (1024 b^6\right ) \int \frac{1}{\sqrt{b x^{2/3}+a x}} \, dx}{2145 a^6}\\ &=\frac{2048 b^6 \sqrt{b x^{2/3}+a x}}{2145 a^7}-\frac{512 b^5 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{715 a^6}+\frac{256 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^5}-\frac{224 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}-\frac{\left (2048 b^7\right ) \int \frac{1}{\sqrt [3]{x} \sqrt{b x^{2/3}+a x}} \, dx}{6435 a^7}\\ &=\frac{2048 b^6 \sqrt{b x^{2/3}+a x}}{2145 a^7}-\frac{4096 b^7 \sqrt{b x^{2/3}+a x}}{2145 a^8 \sqrt [3]{x}}-\frac{512 b^5 \sqrt [3]{x} \sqrt{b x^{2/3}+a x}}{715 a^6}+\frac{256 b^4 x^{2/3} \sqrt{b x^{2/3}+a x}}{429 a^5}-\frac{224 b^3 x \sqrt{b x^{2/3}+a x}}{429 a^4}+\frac{336 b^2 x^{4/3} \sqrt{b x^{2/3}+a x}}{715 a^3}-\frac{28 b x^{5/3} \sqrt{b x^{2/3}+a x}}{65 a^2}+\frac{2 x^2 \sqrt{b x^{2/3}+a x}}{5 a}\\ \end{align*}

Mathematica [A]  time = 0.0879218, size = 111, normalized size = 0.49 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (504 a^5 b^2 x^{5/3}-560 a^4 b^3 x^{4/3}-768 a^2 b^5 x^{2/3}+640 a^3 b^4 x-462 a^6 b x^2+429 a^7 x^{7/3}+1024 a b^6 \sqrt [3]{x}-2048 b^7\right )}{2145 a^8 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(-2048*b^7 + 1024*a*b^6*x^(1/3) - 768*a^2*b^5*x^(2/3) + 640*a^3*b^4*x - 560*a^4*b^3*x
^(4/3) + 504*a^5*b^2*x^(5/3) - 462*a^6*b*x^2 + 429*a^7*x^(7/3)))/(2145*a^8*x^(1/3))

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Maple [A]  time = 0.005, size = 101, normalized size = 0.5 \begin{align*}{\frac{2}{2145\,{a}^{8}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 429\,{x}^{7/3}{a}^{7}-462\,{x}^{2}{a}^{6}b+504\,{x}^{5/3}{a}^{5}{b}^{2}-560\,{x}^{4/3}{a}^{4}{b}^{3}+640\,x{a}^{3}{b}^{4}-768\,{x}^{2/3}{a}^{2}{b}^{5}+1024\,\sqrt [3]{x}a{b}^{6}-2048\,{b}^{7} \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^2/(b*x^(2/3)+a*x)^(1/2),x)

[Out]

2/2145*x^(1/3)*(b+a*x^(1/3))*(429*x^(7/3)*a^7-462*x^2*a^6*b+504*x^(5/3)*a^5*b^2-560*x^(4/3)*a^4*b^3+640*x*a^3*
b^4-768*x^(2/3)*a^2*b^5+1024*x^(1/3)*a*b^6-2048*b^7)/(b*x^(2/3)+a*x)^(1/2)/a^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\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^2/(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/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^2/(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^{2}}{\sqrt{a x + b x^{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16623, size = 165, normalized size = 0.73 \begin{align*} \frac{4096 \, b^{\frac{15}{2}}}{2145 \, a^{8}} + \frac{2 \,{\left (429 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} - 3465 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b + 12285 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{2} - 25025 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{3} + 32175 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{4} - 27027 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{5} + 15015 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{6} - 6435 \, \sqrt{a x^{\frac{1}{3}} + b} b^{7}\right )}}{2145 \, a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4096/2145*b^(15/2)/a^8 + 2/2145*(429*(a*x^(1/3) + b)^(15/2) - 3465*(a*x^(1/3) + b)^(13/2)*b + 12285*(a*x^(1/3)
+ b)^(11/2)*b^2 - 25025*(a*x^(1/3) + b)^(9/2)*b^3 + 32175*(a*x^(1/3) + b)^(7/2)*b^4 - 27027*(a*x^(1/3) + b)^(
5/2)*b^5 + 15015*(a*x^(1/3) + b)^(3/2)*b^6 - 6435*sqrt(a*x^(1/3) + b)*b^7)/a^8