Optimal. Leaf size=401 \[ \frac {8388608 b^{12} \sqrt {b x^{2/3}+a x}}{11700675 a^{13}}-\frac {16777216 b^{13} \sqrt {b x^{2/3}+a x}}{11700675 a^{14} \sqrt [3]{x}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a} \]
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Rubi [A]
time = 0.49, antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027,
2039} \begin {gather*} -\frac {16777216 b^{13} \sqrt {a x+b x^{2/3}}}{11700675 a^{14} \sqrt [3]{x}}+\frac {8388608 b^{12} \sqrt {a x+b x^{2/3}}}{11700675 a^{13}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {a x+b x^{2/3}}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {a x+b x^{2/3}}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {a x+b x^{2/3}}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {a x+b x^{2/3}}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {a x+b x^{2/3}}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {a x+b x^{2/3}}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {a x+b x^{2/3}}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {a x+b x^{2/3}}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {a x+b x^{2/3}}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {a x+b x^{2/3}}}{225 a^2}+\frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2027
Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx &=\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {(26 b) \int \frac {x^{11/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{27 a}\\ &=-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}+\frac {\left (208 b^2\right ) \int \frac {x^{10/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{225 a^2}\\ &=\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {\left (4576 b^3\right ) \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx}{5175 a^3}\\ &=-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}+\frac {\left (18304 b^4\right ) \int \frac {x^{8/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{21735 a^4}\\ &=\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {\left (36608 b^5\right ) \int \frac {x^{7/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{45885 a^5}\\ &=-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}+\frac {\left (585728 b^6\right ) \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx}{780045 a^6}\\ &=\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {\left (1171456 b^7\right ) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{1671525 a^7}\\ &=-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}+\frac {\left (360448 b^8\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{557175 a^8}\\ &=\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {\left (65536 b^9\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{111435 a^9}\\ &=-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}+\frac {\left (524288 b^{10}\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{1002915 a^{10}}\\ &=\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {\left (1048576 b^{11}\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{2340135 a^{11}}\\ &=-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}+\frac {\left (4194304 b^{12}\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{11700675 a^{12}}\\ &=\frac {8388608 b^{12} \sqrt {b x^{2/3}+a x}}{11700675 a^{13}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}-\frac {\left (8388608 b^{13}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{35102025 a^{13}}\\ &=\frac {8388608 b^{12} \sqrt {b x^{2/3}+a x}}{11700675 a^{13}}-\frac {16777216 b^{13} \sqrt {b x^{2/3}+a x}}{11700675 a^{14} \sqrt [3]{x}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 185, normalized size = 0.46 \begin {gather*} \frac {2 \sqrt {b x^{2/3}+a x} \left (-8388608 b^{13}+4194304 a b^{12} \sqrt [3]{x}-3145728 a^2 b^{11} x^{2/3}+2621440 a^3 b^{10} x-2293760 a^4 b^9 x^{4/3}+2064384 a^5 b^8 x^{5/3}-1892352 a^6 b^7 x^2+1757184 a^7 b^6 x^{7/3}-1647360 a^8 b^5 x^{8/3}+1555840 a^9 b^4 x^3-1478048 a^{10} b^3 x^{10/3}+1410864 a^{11} b^2 x^{11/3}-1352078 a^{12} b x^4+1300075 a^{13} x^{13/3}\right )}{11700675 a^{14} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 167, normalized size = 0.42
method | result | size |
derivativedivides | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right )}{11700675 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{14}}\) | \(167\) |
default | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right )}{11700675 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{14}}\) | \(167\) |
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. 1294 vs.
\(2 (299) = 598\).
time = 270.07, size = 1294, normalized size = 3.23 \begin {gather*} \frac {{\left (211106232532992 \, b^{19} + 43980465111040 \, b^{18} + 206158430208 \, {\left (64 \, a^{3} - 3\right )} b^{16} - 4123168604160 \, b^{17} - 1073741824 \, {\left (11264 \, a^{3} - 53\right )} b^{15} + 15143273600 \, a^{15} - 402653184 \, {\left (5504 \, a^{3} + 1\right )} b^{14} + 12582912 \, {\left (3194880 \, a^{6} - 114688 \, a^{3} - 3\right )} b^{13} + 469762048 \, {\left (18816 \, a^{6} + 103 \, a^{3}\right )} b^{12} - 50331648 \, {\left (48816 \, a^{6} + 23 \, a^{3}\right )} b^{11} - 786432 \, {\left (45731840 \, a^{9} - 495872 \, a^{6} - 15 \, a^{3}\right )} b^{10} - 7340032 \, {\left (1349120 \, a^{9} + 3439 \, a^{6}\right )} b^{9} + 250478592 \, {\left (5600 \, a^{9} + 3 \, a^{6}\right )} b^{8} + 12288 \, {\left (2616979456 \, a^{12} - 21542400 \, a^{9} - 693 \, a^{6}\right )} b^{7} + 212992 \, {\left (43743616 \, a^{12} + 89111 \, a^{9}\right )} b^{6} - 638976 \, {\left (1652476 \, a^{12} + 935 \, a^{9}\right )} b^{5} + 3264 \, {\left (3608543232 \, a^{15} + 64599808 \, a^{12} + 2145 \, a^{9}\right )} b^{4} + 578816 \, {\left (13049856 \, a^{15} - 27313 \, a^{12}\right )} b^{3} + 217056 \, {\left (6211584 \, a^{15} + 2353 \, a^{12}\right )} b^{2} - 156009 \, {\left (2547712 \, a^{15} + 39 \, a^{12}\right )} b\right )} x + 2 \, {\left (1300075 \, {\left (16777216 \, a^{13} b^{6} + 6291456 \, a^{13} b^{5} + 196608 \, a^{13} b^{4} - 262144 \, a^{16} - 114688 \, a^{13} b^{3} - 2304 \, a^{13} b^{2} + 864 \, a^{13} b - 27 \, a^{13}\right )} x^{5} - 1478048 \, {\left (16777216 \, a^{10} b^{9} + 6291456 \, a^{10} b^{8} + 196608 \, a^{10} b^{7} - 114688 \, a^{10} b^{6} - 2304 \, a^{10} b^{5} + 864 \, a^{10} b^{4} - {\left (262144 \, a^{13} + 27 \, a^{10}\right )} b^{3}\right )} x^{4} + 1757184 \, {\left (16777216 \, a^{7} b^{12} + 6291456 \, a^{7} b^{11} + 196608 \, a^{7} b^{10} - 114688 \, a^{7} b^{9} - 2304 \, a^{7} b^{8} + 864 \, a^{7} b^{7} - {\left (262144 \, a^{10} + 27 \, a^{7}\right )} b^{6}\right )} x^{3} - 2293760 \, {\left (16777216 \, a^{4} b^{15} + 6291456 \, a^{4} b^{14} + 196608 \, a^{4} b^{13} - 114688 \, a^{4} b^{12} - 2304 \, a^{4} b^{11} + 864 \, a^{4} b^{10} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{9}\right )} x^{2} + 4194304 \, {\left (16777216 \, a b^{18} + 6291456 \, a b^{17} + 196608 \, a b^{16} - 114688 \, a b^{15} - 2304 \, a b^{14} + 864 \, a b^{13} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{12}\right )} x - 2 \, {\left (70368744177664 \, b^{19} + 26388279066624 \, b^{18} + 824633720832 \, b^{17} - 481036337152 \, b^{16} - 9663676416 \, b^{15} - 4194304 \, {\left (262144 \, a^{3} + 27\right )} b^{13} + 3623878656 \, b^{14} + 676039 \, {\left (16777216 \, a^{12} b^{7} + 6291456 \, a^{12} b^{6} + 196608 \, a^{12} b^{5} - 114688 \, a^{12} b^{4} - 2304 \, a^{12} b^{3} + 864 \, a^{12} b^{2} - {\left (262144 \, a^{15} + 27 \, a^{12}\right )} b\right )} x^{4} - 777920 \, {\left (16777216 \, a^{9} b^{10} + 6291456 \, a^{9} b^{9} + 196608 \, a^{9} b^{8} - 114688 \, a^{9} b^{7} - 2304 \, a^{9} b^{6} + 864 \, a^{9} b^{5} - {\left (262144 \, a^{12} + 27 \, a^{9}\right )} b^{4}\right )} x^{3} + 946176 \, {\left (16777216 \, a^{6} b^{13} + 6291456 \, a^{6} b^{12} + 196608 \, a^{6} b^{11} - 114688 \, a^{6} b^{10} - 2304 \, a^{6} b^{9} + 864 \, a^{6} b^{8} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b^{7}\right )} x^{2} - 1310720 \, {\left (16777216 \, a^{3} b^{16} + 6291456 \, a^{3} b^{15} + 196608 \, a^{3} b^{14} - 114688 \, a^{3} b^{13} - 2304 \, a^{3} b^{12} + 864 \, a^{3} b^{11} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{10}\right )} x\right )} x^{\frac {2}{3}} + 48 \, {\left (29393 \, {\left (16777216 \, a^{11} b^{8} + 6291456 \, a^{11} b^{7} + 196608 \, a^{11} b^{6} - 114688 \, a^{11} b^{5} - 2304 \, a^{11} b^{4} + 864 \, a^{11} b^{3} - {\left (262144 \, a^{14} + 27 \, a^{11}\right )} b^{2}\right )} x^{4} - 34320 \, {\left (16777216 \, a^{8} b^{11} + 6291456 \, a^{8} b^{10} + 196608 \, a^{8} b^{9} - 114688 \, a^{8} b^{8} - 2304 \, a^{8} b^{7} + 864 \, a^{8} b^{6} - {\left (262144 \, a^{11} + 27 \, a^{8}\right )} b^{5}\right )} x^{3} + 43008 \, {\left (16777216 \, a^{5} b^{14} + 6291456 \, a^{5} b^{13} + 196608 \, a^{5} b^{12} - 114688 \, a^{5} b^{11} - 2304 \, a^{5} b^{10} + 864 \, a^{5} b^{9} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{8}\right )} x^{2} - 65536 \, {\left (16777216 \, a^{2} b^{17} + 6291456 \, a^{2} b^{16} + 196608 \, a^{2} b^{15} - 114688 \, a^{2} b^{14} - 2304 \, a^{2} b^{13} + 864 \, a^{2} b^{12} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{11}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}}{11700675 \, {\left (16777216 \, a^{14} b^{6} + 6291456 \, a^{14} b^{5} + 196608 \, a^{14} b^{4} - 262144 \, a^{17} - 114688 \, a^{14} b^{3} - 2304 \, a^{14} b^{2} + 864 \, a^{14} b - 27 \, a^{14}\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 \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 206, normalized size = 0.51 \begin {gather*} \frac {16777216 \, b^{\frac {27}{2}}}{11700675 \, a^{14}} + \frac {2 \, {\left (1300075 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {27}{2}} - 18253053 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} b + 119041650 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b^{2} - 478056150 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{3} + 1320944625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{4} - 2657429775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{5} + 4015671660 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{6} - 4633467300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{7} + 4106936925 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{8} - 2788660875 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{9} + 1434168450 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{10} - 547591590 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{11} + 152108775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{12} - 35102025 \, \sqrt {a x^{\frac {1}{3}} + b} b^{13}\right )}}{11700675 \, a^{14}} \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 \frac {x^4}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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