Optimal. Leaf size=345 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{a^2 x^3+b^2 x}}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{a^2 x^3+b^2 x}-2^{3/4} a x+2^{3/4} b}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{a^2 x^3+b^2 x}}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{a^2 x^3+b^2 x}+2^{3/4} a x-2^{3/4} b}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {-2 \sqrt [4]{2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}-\frac {a^2 x^2}{\sqrt [4]{2}}+2^{3/4} a b x-\frac {b^2}{\sqrt [4]{2}}}{2 a^{5/4} \sqrt [4]{b} x \sqrt [4]{a^2 x^3+b^2 x}-2 \sqrt [4]{a} b^{5/4} \sqrt [4]{a^2 x^3+b^2 x}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}} \]
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Rubi [C] time = 0.30, antiderivative size = 150, normalized size of antiderivative = 0.43, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 959, 466, 511, 510} \begin {gather*} \frac {4 x \sqrt [4]{\frac {a^2 x^2}{b^2}+1} F_1\left (\frac {3}{8};1,\frac {1}{4};\frac {11}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{a^2 x^3+b^2 x}}-\frac {4 a x^2 \sqrt [4]{\frac {a^2 x^2}{b^2}+1} F_1\left (\frac {7}{8};1,\frac {1}{4};\frac {15}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{a^2 x^3+b^2 x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 466
Rule 510
Rule 511
Rule 959
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {\left (a \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {x^{3/4}}{\left (b^2-a^2 x^2\right ) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{x} \left (b^2-a^2 x^2\right ) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b^2-a^2 x^8\right ) \sqrt [4]{b^2+a^2 x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b^2-a^2 x^8\right ) \sqrt [4]{b^2+a^2 x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{1+\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b^2-a^2 x^8\right ) \sqrt [4]{1+\frac {a^2 x^8}{b^2}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt [4]{x} \sqrt [4]{1+\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b^2-a^2 x^8\right ) \sqrt [4]{1+\frac {a^2 x^8}{b^2}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}\\ &=\frac {4 x \sqrt [4]{1+\frac {a^2 x^2}{b^2}} F_1\left (\frac {3}{8};1,\frac {1}{4};\frac {11}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{b^2 x+a^2 x^3}}-\frac {4 a x^2 \sqrt [4]{1+\frac {a^2 x^2}{b^2}} F_1\left (\frac {7}{8};1,\frac {1}{4};\frac {15}{8};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{b^2 x+a^2 x^3}}\\ \end {align*}
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Mathematica [F] time = 0.83, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.24, size = 345, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}{2^{3/4} b-2^{3/4} a x+2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}{-2^{3/4} b+2^{3/4} a x+2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^2}{\sqrt [4]{2}}+2^{3/4} a b x-\frac {a^2 x^2}{\sqrt [4]{2}}-2 \sqrt [4]{2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{-2 \sqrt [4]{a} b^{5/4} \sqrt [4]{b^2 x+a^2 x^3}+2 a^{5/4} \sqrt [4]{b} x \sqrt [4]{b^2 x+a^2 x^3}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \int \frac {1}{{\left (a^{2} x^{3} + b^{2} x\right )}^{\frac {1}{4}} {\left (a x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x +b \right ) \left (a^{2} x^{3}+b^{2} x \right )^{\frac {1}{4}}}\, dx\]
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 {1}{{\left (a^{2} x^{3} + b^{2} x\right )}^{\frac {1}{4}} {\left (a x + b\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 = -0.00 \begin {gather*} \int \frac {1}{{\left (a^2\,x^3+b^2\,x\right )}^{1/4}\,\left (b+a\,x\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 {1}{\sqrt [4]{x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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