Optimal. Leaf size=437 \[ -\frac {4807 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{13/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {3 x^4}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {4807 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^6}-\frac {24035 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^5}+\frac {6555 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^4}-\frac {437 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^3}+\frac {23 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a^2}+\frac {4807 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{27/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{442 a^{27/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.46, 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 = {2043, 2047,
2049, 2057, 335, 311, 226, 1210} \begin {gather*} -\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 \text {ArcTan}\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 \text {ArcTan}\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}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2043
Rule 2047
Rule 2049
Rule 2057
Rubi steps
\begin {align*} \int \frac {x^4}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx &=3 \text {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 \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.08, size = 131, normalized size = 0.30 \begin {gather*} \frac {2 x^{2/3} \left (-33649 b^5+4807 a b^4 x^{2/3}-2185 a^2 b^3 x^{4/3}+1311 a^3 b^2 x^2-897 a^4 b x^{8/3}+663 a^5 x^{10/3}+33649 b^5 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )\right )}{4641 a^6 \sqrt {b \sqrt [3]{x}+a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 384, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {3 x^{\frac {2}{3}} b^{5}}{a^{6} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {2 x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 a^{2}}-\frac {80 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119 a^{3}}+\frac {1914 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a^{4}}-\frac {10112 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{5}}+\frac {2818 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 a^{6}}-\frac {4807 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{442 a^{7} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(303\) |
default | \(\frac {5244 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{4} b^{2}-3588 x^{\frac {10}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5} b -8740 x^{2} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b^{3}-201894 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+100947 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+2652 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{6} x^{4}+27846 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {2}{3}} a \,b^{5}+39452 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{5}+19228 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{4}}{9282 a^{7} x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\) | \(384\) |
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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}}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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