Optimal. Leaf size=437 \[ -\frac {4807 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 a^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{27/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {4807 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{13/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{663 a^6}-\frac {24035 b^3 x \sqrt {a x+b \sqrt [3]{x}}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{119 a^3}+\frac {23 x^3 \sqrt {a x+b \sqrt [3]{x}}}{7 a^2}-\frac {3 x^4}{a \sqrt {a x+b \sqrt [3]{x}}} \]
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Rubi [A] time = 0.67, antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2018, 2022, 2024, 2032, 329, 305, 220, 1196} \[ -\frac {4807 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{13/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {6555 b^2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{1547 a^4}-\frac {4807 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 a^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{27/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{663 a^6}-\frac {24035 b^3 x \sqrt {a x+b \sqrt [3]{x}}}{4641 a^5}-\frac {437 b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{119 a^3}+\frac {23 x^3 \sqrt {a x+b \sqrt [3]{x}}}{7 a^2}-\frac {3 x^4}{a \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2018
Rule 2022
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \frac {x^4}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^{14}}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {69 \operatorname {Subst}\left (\int \frac {x^{11}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}-\frac {(437 b) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14 a^2}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}+\frac {\left (6555 b^2\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{238 a^3}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}-\frac {\left (72105 b^3\right ) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3094 a^4}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}+\frac {\left (24035 b^4\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1326 a^5}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^6}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}-\frac {\left (4807 b^5\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{442 a^6}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^6}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}-\frac {\left (4807 b^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{442 a^6 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^6}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}-\frac {\left (4807 b^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^6 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^6}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}-\frac {\left (4807 b^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^{13/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (4807 b^{11/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{221 a^{13/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {4807 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{13/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^6}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}+\frac {4807 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{27/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 a^{27/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 131, normalized size = 0.30 \[ \frac {2 x^{2/3} \left (663 a^5 x^{10/3}-897 a^4 b x^{8/3}+1311 a^3 b^2 x^2-2185 a^2 b^3 x^{4/3}+33649 b^5 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )+4807 a b^4 x^{2/3}-33649 b^5\right )}{4641 a^6 \sqrt {a x+b \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 8.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} x^{6} + 3 \, a^{2} b^{2} x^{\frac {14}{3}} - 2 \, a b^{3} x^{4} - {\left (2 \, a^{3} b x^{5} - b^{4} x^{3}\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{6} x^{4} + 2 \, a^{3} b^{3} x^{2} + b^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 384, normalized size = 0.88 \[ -\frac {-2652 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{6} x^{4}+3588 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{5} b \,x^{\frac {10}{3}}-5244 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{4} b^{2} x^{\frac {8}{3}}+8740 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{3} b^{3} x^{2}-19228 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a^{2} b^{4} x^{\frac {4}{3}}+201894 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{6} \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-100947 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{6} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-39452 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, a \,b^{5} x^{\frac {2}{3}}-27846 \sqrt {a x +b \,x^{\frac {1}{3}}}\, a \,b^{5} x^{\frac {2}{3}}}{9282 \left (a \,x^{\frac {2}{3}}+b \right ) a^{7} x^{\frac {1}{3}}} \]
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 \[ \int \frac {x^{4}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^4}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{4}}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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