Optimal. Leaf size=304 \[ \frac {11050 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {5525 b^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{14421 a^{29/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.35, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2043, 2049,
2036, 335, 226} \begin {gather*} -\frac {5525 b^{27/4} \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 )}{14421 a^{29/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {11050 b^6 \sqrt {a x+b \sqrt [3]{x}}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {a x+b \sqrt [3]{x}}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{207 a^2}+\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{9 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2036
Rule 2043
Rule 2049
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \text {Subst}\left (\int \frac {x^{14}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {(25 b) \text {Subst}\left (\int \frac {x^{12}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{9 a}\\ &=-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}+\frac {\left (175 b^2\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{69 a^2}\\ &=\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {\left (2975 b^3\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1311 a^3}\\ &=-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}+\frac {\left (7735 b^4\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{3933 a^4}\\ &=\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {\left (7735 b^5\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^5}\\ &=-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}+\frac {\left (5525 b^6\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^6}\\ &=\frac {11050 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {\left (5525 b^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^7}\\ &=\frac {11050 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {\left (5525 b^7 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^7 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {11050 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {\left (11050 b^7 \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 )}{14421 a^7 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {11050 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {5525 b^{27/4} \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 )}{14421 a^{29/4} \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 = 161, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (16575 b^7+6630 a b^6 x^{2/3}-2210 a^2 b^5 x^{4/3}+1190 a^3 b^4 x^2-770 a^4 b^3 x^{8/3}+550 a^5 b^2 x^{10/3}-418 a^6 b x^4+4807 a^7 x^{14/3}-16575 b^7 \sqrt {1+\frac {a x^{2/3}}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {a x^{2/3}}{b}\right )\right )}{43263 a^7 \left (b+a x^{2/3}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 196, normalized size = 0.64
method | result | size |
default | \(-\frac {-1100 x^{\frac {11}{3}} a^{6} b^{2}+836 x^{\frac {13}{3}} a^{7} b +1540 a^{5} b^{3} x^{3}+4420 x^{\frac {5}{3}} a^{3} b^{5}-2380 x^{\frac {7}{3}} a^{4} b^{4}-9614 a^{8} x^{5}+16575 b^{7} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-13260 a^{2} b^{6} x -33150 x^{\frac {1}{3}} a \,b^{7}}{43263 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{8}}\) | \(196\) |
derivativedivides | \(\frac {2 x^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{9 a}-\frac {50 b \,x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{207 a^{2}}+\frac {350 b^{2} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 a^{3}}-\frac {1190 b^{3} x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3933 a^{4}}+\frac {15470 b^{4} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{43263 a^{5}}-\frac {2210 b^{5} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{4807 a^{6}}+\frac {11050 b^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{14421 a^{7}}-\frac {5525 b^{7} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14421 a^{8} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(267\) |
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}}{\sqrt {a x + b \sqrt [3]{x}}}\, 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}{\sqrt {a\,x+b\,x^{1/3}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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