Optimal. Leaf size=343 \[ -\frac {1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac {45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac {11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \]
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Rubi [A] time = 0.62, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 12, 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 {1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac {45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac {11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x^2 \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {(22 b) \int x^{5/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{27 a}\\ &=-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (88 b^2\right ) \int x^{4/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{135 a^2}\\ &=\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (176 b^3\right ) \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx}{345 a^3}\\ &=-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (2816 b^4\right ) \int x^{2/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{7245 a^4}\\ &=\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (5632 b^5\right ) \int \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{19665 a^5}\\ &=-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (22528 b^6\right ) \int \left (b x^{2/3}+a x\right )^{3/2} \, dx}{111435 a^6}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (45056 b^7\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{334305 a^7}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (360448 b^8\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{4345965 a^8}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (65536 b^9\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{1448655 a^9}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}+\frac {\left (262144 b^{10}\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{13037895 a^{10}}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}+\frac {524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}-\frac {\left (524288 b^{11}\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{91265265 a^{11}}\\ &=\frac {45056 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{557175 a^7}-\frac {1048576 b^{11} \left (b x^{2/3}+a x\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac {524288 b^{10} \left (b x^{2/3}+a x\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac {131072 b^9 \left (b x^{2/3}+a x\right )^{5/2}}{4345965 a^{10} x}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac {90112 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}-\frac {11264 b^5 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{111435 a^6}+\frac {5632 b^4 x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{45885 a^5}-\frac {352 b^3 x \left (b x^{2/3}+a x\right )^{5/2}}{2415 a^4}+\frac {176 b^2 x^{4/3} \left (b x^{2/3}+a x\right )^{5/2}}{1035 a^3}-\frac {44 b x^{5/3} \left (b x^{2/3}+a x\right )^{5/2}}{225 a^2}+\frac {2 x^2 \left (b x^{2/3}+a x\right )^{5/2}}{9 a}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 172, normalized size = 0.50 \begin {gather*} \frac {2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt {a x+b x^{2/3}} \left (16900975 a^{11} x^{11/3}-14872858 a^{10} b x^{10/3}+12932920 a^9 b^2 x^3-11085360 a^8 b^3 x^{8/3}+9335040 a^7 b^4 x^{7/3}-7687680 a^6 b^5 x^2+6150144 a^5 b^6 x^{5/3}-4730880 a^4 b^7 x^{4/3}+3440640 a^3 b^8 x-2293760 a^2 b^9 x^{2/3}+1310720 a b^{10} \sqrt [3]{x}-524288 b^{11}\right )}{152108775 a^{12} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.51, size = 198, normalized size = 0.58 \begin {gather*} \frac {2 \left (x^{2/3} \left (a \sqrt [3]{x}+b\right )\right )^{3/2} \left (16900975 a^{13} x^{13/3}+18929092 a^{12} b x^4+88179 a^{11} b^2 x^{11/3}-92378 a^{10} b^3 x^{10/3}+97240 a^9 b^4 x^3-102960 a^8 b^5 x^{8/3}+109824 a^7 b^6 x^{7/3}-118272 a^6 b^7 x^2+129024 a^5 b^8 x^{5/3}-143360 a^4 b^9 x^{4/3}+163840 a^3 b^{10} x-196608 a^2 b^{11} x^{2/3}+262144 a b^{12} \sqrt [3]{x}-524288 b^{13}\right )}{152108775 a^{12} x \left (a \sqrt [3]{x}+b\right )} \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [B] time = 0.31, size = 770, normalized size = 2.24
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 145, normalized size = 0.42 \begin {gather*} \frac {2 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (a \,x^{\frac {1}{3}}+b \right ) \left (16900975 a^{11} x^{\frac {11}{3}}-14872858 a^{10} b \,x^{\frac {10}{3}}+12932920 a^{9} b^{2} x^{3}-11085360 a^{8} b^{3} x^{\frac {8}{3}}+9335040 a^{7} b^{4} x^{\frac {7}{3}}-7687680 a^{6} b^{5} x^{2}+6150144 a^{5} b^{6} x^{\frac {5}{3}}-4730880 a^{4} b^{7} x^{\frac {4}{3}}+3440640 a^{3} b^{8} x -2293760 a^{2} b^{9} x^{\frac {2}{3}}+1310720 a \,b^{10} x^{\frac {1}{3}}-524288 b^{11}\right )}{152108775 a^{12} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{2}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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