Optimal. Leaf size=73 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (x \sqrt [3]{\frac {b}{a}}+1\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \]
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Rubi [A] time = 0.20, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {2140, 206} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {a} \left (x \sqrt [3]{\frac {b}{a}}+1\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt {a} \sqrt [3]{\frac {b}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2140
Rubi steps
\begin {align*} \int \frac {1+\sqrt {3}+\sqrt [3]{\frac {b}{a}} x}{\left (1-\sqrt {3}+\sqrt [3]{\frac {b}{a}} x\right ) \sqrt {a+b x^3}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1+\sqrt [3]{\frac {b}{a}} x}{\sqrt {a+b x^3}}\right )}{\sqrt [3]{\frac {b}{a}}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {a} \left (1+\sqrt [3]{\frac {b}{a}} x\right )}{\sqrt {a+b x^3}}\right )}{\sqrt {-3+2 \sqrt {3}} \sqrt {a} \sqrt [3]{\frac {b}{a}}}\\ \end {align*}
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Mathematica [C] time = 1.28, size = 663, normalized size = 9.08 \begin {gather*} \frac {x \left (-\frac {3 \left (10496 \sqrt {3} a^3 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )-18176 a^3 F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )+b x^3 \left (2 \left (3 \sqrt {3}-5\right ) a-b x^3\right ) \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right ) \left (3 b x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+\left (5-3 \sqrt {3}\right ) F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )+8 \left (3 \sqrt {3}-5\right ) a F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )\right )}{a \left (2 \left (3 \sqrt {3}-5\right ) a-b x^3\right ) \left (3 b x^3 \left (F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )+\left (5-3 \sqrt {3}\right ) F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )+8 \left (3 \sqrt {3}-5\right ) a F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )\right )}+12 \left (\sqrt {3}-3\right ) x \sqrt [3]{\frac {b}{a}} \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )-8 x^2 \left (\frac {b}{a}\right )^{2/3} \sqrt {\frac {3 b x^3}{a}+3} F_1\left (1;\frac {1}{2},1;2;-\frac {b x^3}{a},\frac {b x^3}{6 \sqrt {3} a-10 a}\right )\right )}{24 \left (3 \sqrt {3}-5\right ) \sqrt {a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 12.79, size = 90, normalized size = 1.23 \begin {gather*} -\frac {2 \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \sqrt [6]{\frac {b}{a}} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {2}{\sqrt {3}}} \sqrt {b} \sqrt [6]{\frac {b}{a}} \sqrt {a+b x^3}}{a \left (\frac {b}{a}\right )^{2/3}+b x}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.66, size = 1273, normalized size = 17.44
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\frac {b}{a}\right )^{\frac {1}{3}} x +1+\sqrt {3}}{\left (\left (\frac {b}{a}\right )^{\frac {1}{3}} x +1-\sqrt {3}\right ) \sqrt {b \,x^{3}+a}}\, dx \end {gather*}
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*} \int \frac {x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \sqrt {3} + 1}{\sqrt {b x^{3} + a} {\left (x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \sqrt {3} + 1\right )}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {3}+x\,{\left (\frac {b}{a}\right )}^{1/3}+1}{\sqrt {b\,x^3+a}\,\left (x\,{\left (\frac {b}{a}\right )}^{1/3}-\sqrt {3}+1\right )} \,d x \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 \sqrt [3]{\frac {b}{a}} + 1 + \sqrt {3}}{\sqrt {a + b x^{3}} \left (x \sqrt [3]{\frac {b}{a}} - \sqrt {3} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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