Optimal. Leaf size=169 \[ \frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {512 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}} \]
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Rubi [A]
time = 0.17, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2027, 2041,
2039} \begin {gather*} -\frac {512 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (a x+b x^{2/3}\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2027
Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {(2 b) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{3 a}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {\left (16 b^2\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{39 a^2}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}-\frac {\left (32 b^3\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{143 a^3}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}+\frac {\left (128 b^4\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{1287 a^4}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}-\frac {\left (256 b^5\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{9009 a^5}\\ &=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {512 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}}\\ \end {align*}
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Mathematica [A]
time = 4.76, size = 94, normalized size = 0.56 \begin {gather*} \frac {2 \left (b+a \sqrt [3]{x}\right ) \left (b x^{2/3}+a x\right )^{3/2} \left (-256 b^5+640 a b^4 \sqrt [3]{x}-1120 a^2 b^3 x^{2/3}+1680 a^3 b^2 x-2310 a^4 b x^{4/3}+3003 a^5 x^{5/3}\right )}{15015 a^6 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 79, normalized size = 0.47
method | result | size |
derivativedivides | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 a^{4} b \,x^{\frac {4}{3}}+1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 a \,b^{4} x^{\frac {1}{3}}-256 b^{5}\right )}{15015 x \,a^{6}}\) | \(79\) |
default | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 a^{4} b \,x^{\frac {4}{3}}+1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 a \,b^{4} x^{\frac {1}{3}}-256 b^{5}\right )}{15015 x \,a^{6}}\) | \(79\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 768 vs.
\(2 (125) = 250\).
time = 287.48, size = 768, normalized size = 4.54 \begin {gather*} \frac {2 \, {\left (4 \, {\left (805306368 \, b^{13} + 167772160 \, b^{12} + 786432 \, {\left (64 \, a^{3} - 3\right )} b^{10} - 15728640 \, b^{11} - 4096 \, {\left (11264 \, a^{3} - 53\right )} b^{9} + 4372368 \, a^{9} - 1536 \, {\left (5504 \, a^{3} + 1\right )} b^{8} - 48 \, {\left (242810880 \, a^{6} + 114688 \, a^{3} + 3\right )} b^{7} - 1792 \, {\left (1353984 \, a^{6} - 103 \, a^{3}\right )} b^{6} + 192 \, {\left (1152384 \, a^{6} - 23 \, a^{3}\right )} b^{5} - 3 \, {\left (3633315840 \, a^{9} - 12027392 \, a^{6} - 15 \, a^{3}\right )} b^{4} - 112 \, {\left (35389440 \, a^{9} + 29281 \, a^{6}\right )} b^{3} - 819 \, {\left (368640 \, a^{9} - 31 \, a^{6}\right )} b^{2} + 693 \, {\left (40960 \, a^{9} + 3 \, a^{6}\right )} b\right )} x + {\left (3003 \, {\left (16777216 \, a^{7} b^{6} + 6291456 \, a^{7} b^{5} + 196608 \, a^{7} b^{4} - 262144 \, a^{10} - 114688 \, a^{7} b^{3} - 2304 \, a^{7} b^{2} + 864 \, a^{7} b - 27 \, a^{7}\right )} x^{3} - 70 \, {\left (16777216 \, a^{4} b^{9} + 6291456 \, a^{4} b^{8} + 196608 \, a^{4} b^{7} - 114688 \, a^{4} b^{6} - 2304 \, a^{4} b^{5} + 864 \, a^{4} b^{4} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{3}\right )} x^{2} + 128 \, {\left (16777216 \, a b^{12} + 6291456 \, a b^{11} + 196608 \, a b^{10} - 114688 \, a b^{9} - 2304 \, a b^{8} + 864 \, a b^{7} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{6}\right )} x - 16 \, {\left (268435456 \, b^{13} + 100663296 \, b^{12} + 3145728 \, b^{11} - 1835008 \, b^{10} - 36864 \, b^{9} - 16 \, {\left (262144 \, a^{3} + 27\right )} b^{7} + 13824 \, b^{8} - 231 \, {\left (16777216 \, a^{6} b^{7} + 6291456 \, a^{6} b^{6} + 196608 \, a^{6} b^{5} - 114688 \, a^{6} b^{4} - 2304 \, a^{6} b^{3} + 864 \, a^{6} b^{2} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b\right )} x^{2} - 5 \, {\left (16777216 \, a^{3} b^{10} + 6291456 \, a^{3} b^{9} + 196608 \, a^{3} b^{8} - 114688 \, a^{3} b^{7} - 2304 \, a^{3} b^{6} + 864 \, a^{3} b^{5} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{4}\right )} x\right )} x^{\frac {2}{3}} + 3 \, {\left (21 \, {\left (16777216 \, a^{5} b^{8} + 6291456 \, a^{5} b^{7} + 196608 \, a^{5} b^{6} - 114688 \, a^{5} b^{5} - 2304 \, a^{5} b^{4} + 864 \, a^{5} b^{3} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{2}\right )} x^{2} - 32 \, {\left (16777216 \, a^{2} b^{11} + 6291456 \, a^{2} b^{10} + 196608 \, a^{2} b^{9} - 114688 \, a^{2} b^{8} - 2304 \, a^{2} b^{7} + 864 \, a^{2} b^{6} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{5}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}\right )}}{15015 \, {\left (16777216 \, a^{6} b^{6} + 6291456 \, a^{6} b^{5} + 196608 \, a^{6} b^{4} - 262144 \, a^{9} - 114688 \, a^{6} b^{3} - 2304 \, a^{6} b^{2} + 864 \, a^{6} b - 27 \, a^{6}\right )} 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 \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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs.
\(2 (125) = 250\).
time = 2.26, size = 434, normalized size = 2.57 \begin {gather*} \frac {2}{3003} \, b {\left (\frac {256 \, b^{\frac {13}{2}}}{a^{6}} + \frac {\frac {13 \, {\left (63 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} - 385 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b + 990 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{2} - 1386 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{4} - 693 \, \sqrt {a x^{\frac {1}{3}} + b} b^{5}\right )} b}{a^{5}} + \frac {3 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )}}{a^{5}}}{a}\right )} - \frac {2}{15015} \, a {\left (\frac {1024 \, b^{\frac {15}{2}}}{a^{7}} - \frac {\frac {15 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )} b}{a^{6}} + \frac {7 \, {\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 )}}{a^{6}}}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.14, size = 40, normalized size = 0.24 \begin {gather*} \frac {x\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},6;\ 7;\ -\frac {a\,x^{1/3}}{b}\right )}{2\,{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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