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

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

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Rubi [A]  time = 0.44, antiderivative size = 283, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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])

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])

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*}

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Mathematica [A]  time = 0.13, size = 144, normalized size = 0.51 \begin {gather*} \frac {2 \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \left (230945 a^9 x^3-218790 a^8 b x^{8/3}+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}+143360 a^3 b^6 x-122880 a^2 b^7 x^{2/3}+98304 a b^8 \sqrt [3]{x}-65536 b^9\right )}{1616615 a^{10} \sqrt [3]{x}} \end {gather*}

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))

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IntegrateAlgebraic [A]  time = 0.10, size = 133, normalized size = 0.47 \begin {gather*} \frac {2 \left (a x+b x^{2/3}\right )^{3/2} \left (230945 a^9 x^3-218790 a^8 b x^{8/3}+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}+143360 a^3 b^6 x-122880 a^2 b^7 x^{2/3}+98304 a b^8 \sqrt [3]{x}-65536 b^9\right )}{1616615 a^{10} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*x^(2/3) + a*x)^(3/2)*(-65536*b^9 + 98304*a*b^8*x^(1/3) - 122880*a^2*b^7*x^(2/3) + 143360*a^3*b^6*x - 161
280*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) - 218790*a^8*b*x^(8
/3) + 230945*a^9*x^3))/(1616615*a^10*x)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [A]  time = 0.22, size = 312, normalized size = 1.10 \begin {gather*} \frac {131072 \, b^{\frac {21}{2}}}{1616615 \, a^{10}} + \frac {2 \, {\left (\frac {21 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )} b}{a^{9}} + \frac {5 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\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} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{a^{9}}\right )}}{1616615 \, a} \end {gather*}

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*(21*(12155*(a*x^(1/3) + b)^(19/2) - 122265*(a*x^(1/3) + b)^(17/2)*b +
554268*(a*x^(1/3) + b)^(15/2)*b^2 - 1492260*(a*x^(1/3) + b)^(13/2)*b^3 + 2645370*(a*x^(1/3) + b)^(11/2)*b^4 -
3233230*(a*x^(1/3) + b)^(9/2)*b^5 + 2771340*(a*x^(1/3) + b)^(7/2)*b^6 - 1662804*(a*x^(1/3) + b)^(5/2)*b^7 + 6
92835*(a*x^(1/3) + b)^(3/2)*b^8 - 230945*sqrt(a*x^(1/3) + b)*b^9)*b/a^9 + 5*(46189*(a*x^(1/3) + b)^(21/2) - 51
0510*(a*x^(1/3) + b)^(19/2)*b + 2567565*(a*x^(1/3) + b)^(17/2)*b^2 - 7759752*(a*x^(1/3) + b)^(15/2)*b^3 + 1566
8730*(a*x^(1/3) + b)^(13/2)*b^4 - 22221108*(a*x^(1/3) + b)^(11/2)*b^5 + 22632610*(a*x^(1/3) + b)^(9/2)*b^6 - 1
6628040*(a*x^(1/3) + b)^(7/2)*b^7 + 8729721*(a*x^(1/3) + b)^(5/2)*b^8 - 3233230*(a*x^(1/3) + b)^(3/2)*b^9 + 96
9969*sqrt(a*x^(1/3) + b)*b^10)/a^9)/a

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maple [A]  time = 0.05, size = 123, normalized size = 0.43 \begin {gather*} -\frac {2 \sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (a \,x^{\frac {1}{3}}+b \right ) \left (-230945 a^{9} x^{3}+218790 a^{8} b \,x^{\frac {8}{3}}-205920 a^{7} b^{2} x^{\frac {7}{3}}+192192 a^{6} b^{3} x^{2}-177408 a^{5} b^{4} x^{\frac {5}{3}}+161280 a^{4} b^{5} x^{\frac {4}{3}}-143360 a^{3} b^{6} x +122880 a^{2} b^{7} x^{\frac {2}{3}}-98304 a \,b^{8} x^{\frac {1}{3}}+65536 b^{9}\right )}{1616615 a^{10} x^{\frac {1}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1616615*(a*x+b*x^(2/3))^(1/2)*(a*x^(1/3)+b)*(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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a x + b x^{\frac {2}{3}}} x^{2}\,{d x} \end {gather*}

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\sqrt {a\,x+b\,x^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x + b*x^(2/3))^(1/2),x)

[Out]

int(x^2*(a*x + b*x^(2/3))^(1/2), x)

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

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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