Optimal. Leaf size=152 \[ -\frac {9 a x \sqrt {a-b x^2} \left (a+b x^2\right )}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt {a^2-b^2 x^4}}+\frac {19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1152, 416, 388, 217, 206} \begin {gather*} -\frac {9 a x \sqrt {a-b x^2} \left (a+b x^2\right )}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt {a^2-b^2 x^4}}+\frac {19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} \sqrt {a^2-b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 416
Rule 1152
Rubi steps
\begin {align*} \int \frac {\left (a-b x^2\right )^{5/2}}{\sqrt {a^2-b^2 x^4}} \, dx &=\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {\left (a-b x^2\right )^2}{\sqrt {a+b x^2}} \, dx}{\sqrt {a^2-b^2 x^4}}\\ &=-\frac {x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt {a^2-b^2 x^4}}+\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {5 a^2 b-9 a b^2 x^2}{\sqrt {a+b x^2}} \, dx}{4 b \sqrt {a^2-b^2 x^4}}\\ &=-\frac {9 a x \sqrt {a-b x^2} \left (a+b x^2\right )}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt {a^2-b^2 x^4}}+\frac {\left (19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 \sqrt {a^2-b^2 x^4}}\\ &=-\frac {9 a x \sqrt {a-b x^2} \left (a+b x^2\right )}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt {a^2-b^2 x^4}}+\frac {\left (19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {a^2-b^2 x^4}}\\ &=-\frac {9 a x \sqrt {a-b x^2} \left (a+b x^2\right )}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )}{4 \sqrt {a^2-b^2 x^4}}+\frac {19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} \sqrt {a^2-b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 123, normalized size = 0.81 \begin {gather*} \frac {1}{8} \left (\frac {x \left (2 b x^2-11 a\right ) \sqrt {a^2-b^2 x^4}}{\sqrt {a-b x^2}}+\frac {19 a^2 \log \left (\sqrt {b} \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}+a b x-b^2 x^3\right )}{\sqrt {b}}-\frac {19 a^2 \log \left (b x^2-a\right )}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a-b x^2\right )^{5/2}}{\sqrt {a^2-b^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.32, size = 265, normalized size = 1.74 \begin {gather*} \left [\frac {19 \, {\left (a^{2} b x^{2} - a^{3}\right )} \sqrt {b} \log \left (\frac {2 \, b^{2} x^{4} - a b x^{2} - 2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {b} x - a^{2}}{b x^{2} - a}\right ) - 2 \, \sqrt {-b^{2} x^{4} + a^{2}} {\left (2 \, b^{2} x^{3} - 11 \, a b x\right )} \sqrt {-b x^{2} + a}}{16 \, {\left (b^{2} x^{2} - a b\right )}}, \frac {19 \, {\left (a^{2} b x^{2} - a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {-b}}{b^{2} x^{3} - a b x}\right ) - \sqrt {-b^{2} x^{4} + a^{2}} {\left (2 \, b^{2} x^{3} - 11 \, a b x\right )} \sqrt {-b x^{2} + a}}{8 \, {\left (b^{2} x^{2} - a b\right )}}\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 {{\left (-b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {-b^{2} x^{4} + a^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 105, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \left (2 \sqrt {b \,x^{2}+a}\, b^{\frac {3}{2}} x^{3}+19 a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-11 \sqrt {b \,x^{2}+a}\, a \sqrt {b}\, x \right )}{8 \left (b \,x^{2}-a \right ) \sqrt {b \,x^{2}+a}\, \sqrt {b}} \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 {{\left (-b x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {-b^{2} x^{4} + a^{2}}}\,{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 {{\left (a-b\,x^2\right )}^{5/2}}{\sqrt {a^2-b^2\,x^4}} \,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 {\left (a - b x^{2}\right )^{\frac {5}{2}}}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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