Optimal. Leaf size=110 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3+b x^2}}{\sqrt {a x^3+b x^2}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3+b x^2}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3+b x^2}}\right ) \]
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Rubi [C] time = 1.39, antiderivative size = 573, normalized size of antiderivative = 5.21, number of steps used = 13, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2056, 6728, 107, 106, 490, 1218} \begin {gather*} \frac {\sqrt {2} \sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 106
Rule 107
Rule 490
Rule 1218
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {2 b+a x}{\left (b+a x+x^2\right ) \sqrt [4]{b x^2+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {2 b+a x}{\sqrt {x} \sqrt [4]{b+a x} \left (b+a x+x^2\right )} \, dx}{\sqrt [4]{b x^2+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x}\right ) \int \left (\frac {a-\sqrt {a^2-4 b}}{\sqrt {x} \left (a-\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}}+\frac {a+\sqrt {a^2-4 b}}{\sqrt {x} \left (a+\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}}\right ) \, dx}{\sqrt [4]{b x^2+a x^3}}\\ &=\frac {\left (\left (a-\sqrt {a^2-4 b}\right ) \sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a-\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2-4 b}\right ) \sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {x} \left (a+\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}}\\ &=\frac {\left (\left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {-\frac {a x}{b}} \left (a-\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \int \frac {1}{\sqrt {-\frac {a x}{b}} \left (a+\sqrt {a^2-4 b}+2 x\right ) \sqrt [4]{b+a x}} \, dx}{\sqrt [4]{b x^2+a x^3}}\\ &=-\frac {\left (4 \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a \left (a-\sqrt {a^2-4 b}\right )+2 b-2 x^4\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}-\frac {\left (4 \left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-a \left (a+\sqrt {a^2-4 b}\right )+2 b-2 x^4\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}\\ &=-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}-\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}+\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}-\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}+\frac {\left (\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}+\sqrt {2} x^2\right ) \sqrt {1-\frac {x^4}{b}}} \, dx,x,\sqrt [4]{b+a x}\right )}{\sqrt [4]{b x^2+a x^3}}\\ &=\frac {\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \Pi \left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}-\frac {\sqrt {2} \left (a+\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \Pi \left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a^2-a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}+\frac {\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \Pi \left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}-\frac {\sqrt {2} \left (a-\sqrt {a^2-4 b}\right ) \sqrt [4]{b} \sqrt {-\frac {a x}{b}} \sqrt [4]{b+a x} \Pi \left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a^2+a \sqrt {a^2-4 b}+2 b} \sqrt [4]{b x^2+a x^3}}\\ \end {align*}
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Mathematica [C] time = 7.21, size = 585, normalized size = 5.32 \begin {gather*} \frac {i \sqrt {2} a (a x+b)^{3/4} \sqrt {1-\frac {b}{a x+b}} \left (\left (a^2+\sqrt {a^4-4 a^2 b}-4 b\right ) \sqrt {\frac {-a^2+\sqrt {a^4-4 a^2 b}+2 b}{b^2}} \Pi \left (-\frac {\sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2-2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )+\left (-a^2-\sqrt {a^4-4 a^2 b}+4 b\right ) \sqrt {\frac {-a^2+\sqrt {a^4-4 a^2 b}+2 b}{b^2}} \Pi \left (\frac {\sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2-2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )+\sqrt {-\frac {a^2+\sqrt {a^4-4 a^2 b}-2 b}{b^2}} \left (-a^2+\sqrt {a^4-4 a^2 b}+4 b\right ) \left (\Pi \left (-\frac {\sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2+2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2+2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )\right )\right )}{\sqrt {-\sqrt {b}} b \sqrt {a^4-4 a^2 b} \sqrt {-\frac {a^2+\sqrt {a^4-4 a^2 b}-2 b}{b^2}} \sqrt {\frac {-a^2+\sqrt {a^4-4 a^2 b}+2 b}{b^2}} \sqrt [4]{x^2 (a x+b)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.65, size = 110, normalized size = 1.00 \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3+b x^2}}{\sqrt {a x^3+b x^2}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3+b x^2}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3+b x^2}}\right ) \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 {a x + 2 \, b}{{\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x +2 b}{\left (a x +x^{2}+b \right ) \left (a \,x^{3}+b \,x^{2}\right )^{\frac {1}{4}}}\, 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 {a x + 2 \, b}{{\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} + 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 = -0.01 \begin {gather*} \int \frac {2\,b+a\,x}{{\left (a\,x^3+b\,x^2\right )}^{1/4}\,\left (x^2+a\,x+b\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 {a x + 2 b}{\sqrt [4]{x^{2} \left (a x + b\right )} \left (a x + b + x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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