Optimal. Leaf size=140 \[ -\frac {\tanh ^{-1}\left (\frac {\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^3-b x}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3-b x}}{-2 \sqrt {a} \sqrt {b} x+a x^2-b}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}} \]
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Rubi [A] time = 0.22, antiderivative size = 176, normalized size of antiderivative = 1.26, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 466, 405} \begin {gather*} -\frac {\sqrt {x} \sqrt {a x^2-b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3-b x}}-\frac {\sqrt {x} \sqrt {a x^2-b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {a} x+\sqrt {b}\right )}{\sqrt [4]{b} \sqrt {a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 405
Rule 466
Rule 2056
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {-b x+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {\sqrt {-b+a x^2}}{\sqrt {x} \left (b+a x^2\right )} \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b+a x^4}}{b+a x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=-\frac {\sqrt {x} \sqrt {-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {-b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}-\frac {\sqrt {x} \sqrt {-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right )}{\sqrt [4]{b} \sqrt {-b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 62, normalized size = 0.44 \begin {gather*} \frac {2 \sqrt {a x^3-b x} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a x^2}{b},-\frac {a x^2}{b}\right )}{b \sqrt {1-\frac {a x^2}{b}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.42, size = 150, normalized size = 1.07 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}-\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 344, normalized size = 2.46 \begin {gather*} \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \arctan \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {a x^{3} - b x} a b \left (-\frac {1}{a b}\right )^{\frac {3}{4}}}{a x^{2} - b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} + 8 \, \sqrt {a x^{3} - b x} {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{2} b x^{2} - a b^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {3}{4}}\right )} - 4 \, {\left (a^{2} b x^{3} - a b^{2} x\right )} \sqrt {-\frac {1}{a b}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} - 8 \, \sqrt {a x^{3} - b x} {\left (\left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a b}\right )^{\frac {1}{4}} + \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (a^{2} b x^{2} - a b^{2}\right )} \left (-\frac {1}{a b}\right )^{\frac {3}{4}}\right )} - 4 \, {\left (a^{2} b x^{3} - a b^{2} x\right )} \sqrt {-\frac {1}{a b}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) \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}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 380, normalized size = 2.71
method | result | size |
elliptic | \(\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\) | \(380\) |
default | \(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-2 b \left (\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-a b}\, a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\right )\) | \(400\) |
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}{\sqrt {a x^{3} - b x} {\left (a x^{2} + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \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 {x \left (a x^{2} - b\right )} \left (a x^{2} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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