Optimal. Leaf size=408 \[ -\frac {88 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {88 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {44 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Rubi [A]
time = 0.77, antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2043, 2046,
2049, 2057, 335, 311, 226, 1210} \begin {gather*} -\frac {44 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 )}{1105 a^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 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 )}{1105 a^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {88 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 a^{7/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{3315 a^3}-\frac {88 b^3 x \sqrt {a x+b \sqrt [3]{x}}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}+\frac {2}{7} x^2 \left (a x+b \sqrt [3]{x}\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2043
Rule 2046
Rule 2049
Rule 2057
Rubi steps
\begin {align*} \int x \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx &=3 \text {Subst}\left (\int x^5 \left (b x+a x^3\right )^{3/2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{7} (6 b) \text {Subst}\left (\int x^6 \sqrt {b x+a x^3} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {1}{119} \left (12 b^2\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (132 b^3\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1547 a}\\ &=-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {\left (44 b^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 a^2}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (44 b^5\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1105 a^3}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (44 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 )}{1105 a^3 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (88 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 )}{1105 a^3 \sqrt {b \sqrt [3]{x}+a x}}\\ &=\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}-\frac {\left (88 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 )}{1105 a^{7/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (88 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 )}{1105 a^{7/2} \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {88 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{1105 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {88 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{3315 a^3}-\frac {88 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^2}+\frac {24 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a}+\frac {12}{119} b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}+\frac {2}{7} x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}+\frac {88 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 )}{1105 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {44 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 )}{1105 a^{15/4} \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 = 10.07, size = 123, normalized size = 0.30 \begin {gather*} \frac {2 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x} \left (\left (b+a x^{2/3}\right )^2 \sqrt {1+\frac {a x^{2/3}}{b}} \left (77 b^2-143 a b x^{2/3}+221 a^2 x^{4/3}\right )-77 b^4 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {a x^{2/3}}{b}\right )\right )}{1547 a^3 \sqrt {1+\frac {a x^{2/3}}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 261, normalized size = 0.64
method | result | size |
default | \(\frac {\frac {622 x^{\frac {8}{3}} a^{4} b^{2}}{1547}+\frac {80 x^{\frac {10}{3}} a^{5} b}{119}-\frac {16 a^{3} b^{3} x^{2}}{4641}-\frac {88 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {44 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{1105}+\frac {2 a^{6} x^{4}}{7}+\frac {88 x^{\frac {2}{3}} a \,b^{5}}{3315}+\frac {176 x^{\frac {4}{3}} a^{2} b^{4}}{23205}}{a^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(261\) |
derivativedivides | \(\frac {2 a \,x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7}+\frac {46 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119}+\frac {24 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a}-\frac {88 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{2}}+\frac {88 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3315 a^{3}}-\frac {44 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{1105 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(271\) |
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 x \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 x\,{\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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