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

Optimal. Leaf size=283 \[ -\frac{131072 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{1616615 a^{10} x}+\frac{196608 b^8 \left (a x+b x^{2/3}\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac{49152 b^7 \left (a x+b x^{2/3}\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}+\frac{8192 b^6 \left (a x+b x^{2/3}\right )^{3/2}}{46189 a^7}-\frac{9216 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (a x+b x^{2/3}\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a} \]

[Out]

(8192*b^6*(b*x^(2/3) + a*x)^(3/2))/(46189*a^7) - (131072*b^9*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^10*x) + (1966
08*b^8*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^9*x^(2/3)) - (49152*b^7*(b*x^(2/3) + a*x)^(3/2))/(323323*a^8*x^(1/3
)) - (9216*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(46189*a^6) + (4608*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(3/2))/(2099
5*a^5) - (384*b^3*x*(b*x^(2/3) + a*x)^(3/2))/(1615*a^4) + (576*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(2261*a^3)
- (36*b*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(133*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(3/2))/(7*a)

________________________________________________________________________________________

Rubi [A]  time = 0.441085, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \[ -\frac{131072 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{1616615 a^{10} x}+\frac{196608 b^8 \left (a x+b x^{2/3}\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac{49152 b^7 \left (a x+b x^{2/3}\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}+\frac{8192 b^6 \left (a x+b x^{2/3}\right )^{3/2}}{46189 a^7}-\frac{9216 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (a x+b x^{2/3}\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[b*x^(2/3) + a*x],x]

[Out]

(8192*b^6*(b*x^(2/3) + a*x)^(3/2))/(46189*a^7) - (131072*b^9*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^10*x) + (1966
08*b^8*(b*x^(2/3) + a*x)^(3/2))/(1616615*a^9*x^(2/3)) - (49152*b^7*(b*x^(2/3) + a*x)^(3/2))/(323323*a^8*x^(1/3
)) - (9216*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(46189*a^6) + (4608*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(3/2))/(2099
5*a^5) - (384*b^3*x*(b*x^(2/3) + a*x)^(3/2))/(1615*a^4) + (576*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(2261*a^3)
- (36*b*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(133*a^2) + (2*x^2*(b*x^(2/3) + a*x)^(3/2))/(7*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 x^2 \sqrt{b x^{2/3}+a x} \, dx &=\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac{(6 b) \int x^{5/3} \sqrt{b x^{2/3}+a x} \, dx}{7 a}\\ &=-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac{\left (96 b^2\right ) \int x^{4/3} \sqrt{b x^{2/3}+a x} \, dx}{133 a^2}\\ &=\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac{\left (192 b^3\right ) \int x \sqrt{b x^{2/3}+a x} \, dx}{323 a^3}\\ &=-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac{\left (768 b^4\right ) \int x^{2/3} \sqrt{b x^{2/3}+a x} \, dx}{1615 a^4}\\ &=\frac{4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac{\left (1536 b^5\right ) \int \sqrt [3]{x} \sqrt{b x^{2/3}+a x} \, dx}{4199 a^5}\\ &=-\frac{9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac{\left (12288 b^6\right ) \int \sqrt{b x^{2/3}+a x} \, dx}{46189 a^6}\\ &=\frac{8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac{9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac{\left (8192 b^7\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{\sqrt [3]{x}} \, dx}{46189 a^7}\\ &=\frac{8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac{49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac{9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}+\frac{\left (32768 b^8\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{x^{2/3}} \, dx}{323323 a^8}\\ &=\frac{8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}+\frac{196608 b^8 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac{49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac{9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}-\frac{\left (65536 b^9\right ) \int \frac{\sqrt{b x^{2/3}+a x}}{x} \, dx}{1616615 a^9}\\ &=\frac{8192 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^7}-\frac{131072 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^{10} x}+\frac{196608 b^8 \left (b x^{2/3}+a x\right )^{3/2}}{1616615 a^9 x^{2/3}}-\frac{49152 b^7 \left (b x^{2/3}+a x\right )^{3/2}}{323323 a^8 \sqrt [3]{x}}-\frac{9216 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{46189 a^6}+\frac{4608 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{20995 a^5}-\frac{384 b^3 x \left (b x^{2/3}+a x\right )^{3/2}}{1615 a^4}+\frac{576 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{2261 a^3}-\frac{36 b x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{133 a^2}+\frac{2 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0951391, size = 144, normalized size = 0.51 \[ \frac{2 \left (a \sqrt [3]{x}+b\right ) \sqrt{a x+b x^{2/3}} \left (205920 a^7 b^2 x^{7/3}-192192 a^6 b^3 x^2+177408 a^5 b^4 x^{5/3}-161280 a^4 b^5 x^{4/3}-122880 a^2 b^7 x^{2/3}+143360 a^3 b^6 x-218790 a^8 b x^{8/3}+230945 a^9 x^3+98304 a b^8 \sqrt [3]{x}-65536 b^9\right )}{1616615 a^{10} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^(1/3))*Sqrt[b*x^(2/3) + a*x]*(-65536*b^9 + 98304*a*b^8*x^(1/3) - 122880*a^2*b^7*x^(2/3) + 143360*a
^3*b^6*x - 161280*a^4*b^5*x^(4/3) + 177408*a^5*b^4*x^(5/3) - 192192*a^6*b^3*x^2 + 205920*a^7*b^2*x^(7/3) - 218
790*a^8*b*x^(8/3) + 230945*a^9*x^3))/(1616615*a^10*x^(1/3))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 123, normalized size = 0.4 \begin{align*} -{\frac{2}{1616615\,{a}^{10}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( b+a\sqrt [3]{x} \right ) \left ( 218790\,{x}^{8/3}{a}^{8}b-205920\,{x}^{7/3}{a}^{7}{b}^{2}-177408\,{x}^{5/3}{a}^{5}{b}^{4}+161280\,{x}^{4/3}{a}^{4}{b}^{5}-230945\,{x}^{3}{a}^{9}+122880\,{x}^{2/3}{a}^{2}{b}^{7}+192192\,{x}^{2}{a}^{6}{b}^{3}-98304\,\sqrt [3]{x}a{b}^{8}-143360\,x{a}^{3}{b}^{6}+65536\,{b}^{9} \right ){\frac{1}{\sqrt [3]{x}}}} \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]

-2/1616615*(b*x^(2/3)+a*x)^(1/2)*(b+a*x^(1/3))*(218790*x^(8/3)*a^8*b-205920*x^(7/3)*a^7*b^2-177408*x^(5/3)*a^5
*b^4+161280*x^(4/3)*a^4*b^5-230945*x^3*a^9+122880*x^(2/3)*a^2*b^7+192192*x^2*a^6*b^3-98304*x^(1/3)*a*b^8-14336
0*x*a^3*b^6+65536*b^9)/x^(1/3)/a^10

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 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)

________________________________________________________________________________________

Giac [A]  time = 1.12632, size = 203, normalized size = 0.72 \begin{align*} \frac{131072 \, b^{\frac{21}{2}}}{1616615 \, a^{10}} + \frac{2 \,{\left (230945 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} - 2297295 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b + 10270260 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{2} - 27159132 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{3} + 47006190 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{4} - 55552770 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{5} + 45265220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{6} - 24942060 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{7} + 8729721 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{8} - 1616615 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{9}\right )}}{1616615 \, a^{10}} \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]

131072/1616615*b^(21/2)/a^10 + 2/1616615*(230945*(a*x^(1/3) + b)^(21/2) - 2297295*(a*x^(1/3) + b)^(19/2)*b + 1
0270260*(a*x^(1/3) + b)^(17/2)*b^2 - 27159132*(a*x^(1/3) + b)^(15/2)*b^3 + 47006190*(a*x^(1/3) + b)^(13/2)*b^4
- 55552770*(a*x^(1/3) + b)^(11/2)*b^5 + 45265220*(a*x^(1/3) + b)^(9/2)*b^6 - 24942060*(a*x^(1/3) + b)^(7/2)*b
^7 + 8729721*(a*x^(1/3) + b)^(5/2)*b^8 - 1616615*(a*x^(1/3) + b)^(3/2)*b^9)/a^10