Optimal. Leaf size=60 \[ -\frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b x+a^2 x^3}}{b+a^2 x^2}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 1.40, antiderivative size = 290, normalized size of antiderivative = 4.83, number of steps
used = 13, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2081, 6860,
335, 226, 947, 174, 551} \begin {gather*} -\frac {2 \sqrt [4]{-b} \sqrt {x} \sqrt {\frac {a^2 x^2}{b}+1} \Pi \left (\frac {\sqrt {-b}}{\left (\sqrt {b-1}-\sqrt {b}\right ) \sqrt {b}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{-b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3+b x}}-\frac {2 \sqrt [4]{-b} \sqrt {x} \sqrt {\frac {a^2 x^2}{b}+1} \Pi \left (-\frac {\sqrt {-b}}{\left (\sqrt {b-1}+\sqrt {b}\right ) \sqrt {b}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{-b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {a^2 x^3+b x}}+\frac {\sqrt {x} \left (a x+\sqrt {b}\right ) \sqrt {\frac {a^2 x^2+b}{\left (a x+\sqrt {b}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {a^2 x^3+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 174
Rule 226
Rule 335
Rule 551
Rule 947
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {-b+a^2 x^2}{\left (b+2 a b x+a^2 x^2\right ) \sqrt {b x+a^2 x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {-b+a^2 x^2}{\sqrt {x} \sqrt {b+a^2 x^2} \left (b+2 a b x+a^2 x^2\right )} \, dx}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {b+a^2 x^2}}-\frac {2 (b+a b x)}{\sqrt {x} \sqrt {b+a^2 x^2} \left (b+2 a b x+a^2 x^2\right )}\right ) \, dx}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+a^2 x^2}} \, dx}{\sqrt {b x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {b+a b x}{\sqrt {x} \sqrt {b+a^2 x^2} \left (b+2 a b x+a^2 x^2\right )} \, dx}{\sqrt {b x+a^2 x^3}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \left (\frac {-a \sqrt {-1+b} \sqrt {b}+a b}{\sqrt {x} \left (-2 a \sqrt {-1+b} \sqrt {b}+2 a b+2 a^2 x\right ) \sqrt {b+a^2 x^2}}+\frac {a \sqrt {-1+b} \sqrt {b}+a b}{\sqrt {x} \left (2 a \sqrt {-1+b} \sqrt {b}+2 a b+2 a^2 x\right ) \sqrt {b+a^2 x^2}}\right ) \, dx}{\sqrt {b x+a^2 x^3}}+\frac {\left (2 \sqrt {x} \sqrt {b+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (2 a \left (\sqrt {-1+b}-\sqrt {b}\right ) \sqrt {b} \sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {1}{\sqrt {x} \left (-2 a \sqrt {-1+b} \sqrt {b}+2 a b+2 a^2 x\right ) \sqrt {b+a^2 x^2}} \, dx}{\sqrt {b x+a^2 x^3}}-\frac {\left (2 a \left (\sqrt {-1+b}+\sqrt {b}\right ) \sqrt {b} \sqrt {x} \sqrt {b+a^2 x^2}\right ) \int \frac {1}{\sqrt {x} \left (2 a \sqrt {-1+b} \sqrt {b}+2 a b+2 a^2 x\right ) \sqrt {b+a^2 x^2}} \, dx}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}+\frac {\left (2 a \left (\sqrt {-1+b}-\sqrt {b}\right ) \sqrt {b} \sqrt {x} \sqrt {1+\frac {a^2 x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (-2 a \sqrt {-1+b} \sqrt {b}+2 a b+2 a^2 x\right ) \sqrt {1-\frac {a x}{\sqrt {-b}}} \sqrt {1+\frac {a x}{\sqrt {-b}}}} \, dx}{\sqrt {b x+a^2 x^3}}-\frac {\left (2 a \left (\sqrt {-1+b}+\sqrt {b}\right ) \sqrt {b} \sqrt {x} \sqrt {1+\frac {a^2 x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (2 a \sqrt {-1+b} \sqrt {b}+2 a b+2 a^2 x\right ) \sqrt {1-\frac {a x}{\sqrt {-b}}} \sqrt {1+\frac {a x}{\sqrt {-b}}}} \, dx}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {\left (4 a \left (\sqrt {-1+b}-\sqrt {b}\right ) \sqrt {b} \sqrt {x} \sqrt {1+\frac {a^2 x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (2 a \left (\sqrt {-1+b}-\sqrt {b}\right ) \sqrt {b}-2 a^2 x^2\right ) \sqrt {1-\frac {a x^2}{\sqrt {-b}}} \sqrt {1+\frac {a x^2}{\sqrt {-b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}}+\frac {\left (4 a \left (\sqrt {-1+b}+\sqrt {b}\right ) \sqrt {b} \sqrt {x} \sqrt {1+\frac {a^2 x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-2 a \left (\sqrt {-1+b}+\sqrt {b}\right ) \sqrt {b}-2 a^2 x^2\right ) \sqrt {1-\frac {a x^2}{\sqrt {-b}}} \sqrt {1+\frac {a x^2}{\sqrt {-b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a^2 x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+a x\right ) \sqrt {\frac {b+a^2 x^2}{\left (\sqrt {b}+a x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt {a} \sqrt [4]{b} \sqrt {b x+a^2 x^3}}-\frac {2 \sqrt [4]{-b} \sqrt {x} \sqrt {1+\frac {a^2 x^2}{b}} \Pi \left (\frac {\sqrt {-b}}{\left (\sqrt {-1+b}-\sqrt {b}\right ) \sqrt {b}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{-b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {b x+a^2 x^3}}-\frac {2 \sqrt [4]{-b} \sqrt {x} \sqrt {1+\frac {a^2 x^2}{b}} \Pi \left (-\frac {\sqrt {-b}}{\left (\sqrt {-1+b}+\sqrt {b}\right ) \sqrt {b}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt [4]{-b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {b x+a^2 x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.85, size = 85, normalized size = 1.42 \begin {gather*} -\frac {\sqrt {2} \sqrt {x} \sqrt {b+a^2 x^2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b+a^2 x^2}}\right )}{\sqrt {a} \sqrt {b} \sqrt {x \left (b+a^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.00, size = 1096, normalized size = 18.27 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 219, normalized size = 3.65 \begin {gather*} \left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} - 12 \, a b^{2} x + 2 \, {\left (2 \, a^{2} b^{2} + a^{2} b\right )} x^{2} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{2}\right )} \sqrt {a^{2} x^{3} + b x} \sqrt {-\frac {1}{a b}} + b^{2}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 4 \, a b^{2} x + 2 \, {\left (2 \, a^{2} b^{2} + a^{2} b\right )} x^{2} + b^{2}}\right ), \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a b}} \arctan \left (\frac {\sqrt {2} {\left (a^{2} x^{2} - 2 \, a b x + b\right )} \sqrt {\frac {1}{a b}}}{4 \, \sqrt {a^{2} x^{3} + b x}}\right )\right ] \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^{2} x^{2} - b}{\sqrt {x \left (a^{2} x^{2} + b\right )} \left (a^{2} x^{2} + 2 a b x + b\right )}\, 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 [B]
time = 3.28, size = 78, normalized size = 1.30 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,b}{4}+\frac {\sqrt {2}\,a^2\,x^2}{4}-\frac {\sqrt {2}\,a\,b\,x}{2}+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3+b\,x}\,1{}\mathrm {i}}{a^2\,x^2+2\,b\,a\,x+b}\right )\,1{}\mathrm {i}}{2\,\sqrt {a}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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