Optimal. Leaf size=401 \[ -\frac {16777216 b^{13} \sqrt {a x+b x^{2/3}}}{11700675 a^{14} \sqrt [3]{x}}+\frac {8388608 b^{12} \sqrt {a x+b x^{2/3}}}{11700675 a^{13}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {a x+b x^{2/3}}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {a x+b x^{2/3}}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {a x+b x^{2/3}}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {a x+b x^{2/3}}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {a x+b x^{2/3}}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {a x+b x^{2/3}}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {a x+b x^{2/3}}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {a x+b x^{2/3}}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {a x+b x^{2/3}}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {a x+b x^{2/3}}}{225 a^2}+\frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a} \]
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Rubi [A] time = 0.73, 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 = {2016, 2002, 2014} \[ -\frac {16777216 b^{13} \sqrt {a x+b x^{2/3}}}{11700675 a^{14} \sqrt [3]{x}}+\frac {8388608 b^{12} \sqrt {a x+b x^{2/3}}}{11700675 a^{13}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {a x+b x^{2/3}}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {a x+b x^{2/3}}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {a x+b x^{2/3}}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {a x+b x^{2/3}}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {a x+b x^{2/3}}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {a x+b x^{2/3}}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {a x+b x^{2/3}}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {a x+b x^{2/3}}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {a x+b x^{2/3}}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {a x+b x^{2/3}}}{225 a^2}+\frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
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 = 0.23, size = 185, normalized size = 0.46 \[ \frac {2 \sqrt {a x+b x^{2/3}} \left (1300075 a^{13} x^{13/3}-1352078 a^{12} b x^4+1410864 a^{11} b^2 x^{11/3}-1478048 a^{10} b^3 x^{10/3}+1555840 a^9 b^4 x^3-1647360 a^8 b^5 x^{8/3}+1757184 a^7 b^6 x^{7/3}-1892352 a^6 b^7 x^2+2064384 a^5 b^8 x^{5/3}-2293760 a^4 b^9 x^{4/3}+2621440 a^3 b^{10} x-3145728 a^2 b^{11} x^{2/3}+4194304 a b^{12} \sqrt [3]{x}-8388608 b^{13}\right )}{11700675 a^{14} \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 206, normalized size = 0.51 \[ \frac {16777216 \, b^{\frac {27}{2}}}{11700675 \, a^{14}} + \frac {2 \, {\left (1300075 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {27}{2}} - 18253053 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} b + 119041650 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b^{2} - 478056150 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{3} + 1320944625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{4} - 2657429775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{5} + 4015671660 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{6} - 4633467300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{7} + 4106936925 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{8} - 2788660875 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{9} + 1434168450 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{10} - 547591590 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{11} + 152108775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{12} - 35102025 \, \sqrt {a x^{\frac {1}{3}} + b} b^{13}\right )}}{11700675 \, a^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 167, normalized size = 0.42 \[ \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right ) x^{\frac {1}{3}}}{11700675 \sqrt {a x +b \,x^{\frac {2}{3}}}\, a^{14}} \]
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 \[ \int \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}}\,{d x} \]
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 \[ \int \frac {x^4}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]
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 \[ \int \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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