Optimal. Leaf size=153 \[ -\frac {9 a x \left (a-b x^2\right ) \sqrt {a+b x^2}}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt {a^2-b^2 x^4}}+\frac {19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{8 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1152, 416, 388, 217, 203} \begin {gather*} -\frac {9 a x \left (a-b x^2\right ) \sqrt {a+b x^2}}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt {a^2-b^2 x^4}}+\frac {19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-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 203
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 ) \left (a+b x^2\right )^{3/2}}{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 \left (a-b x^2\right ) \sqrt {a+b x^2}}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{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 \left (a-b x^2\right ) \sqrt {a+b x^2}}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{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 \left (a-b x^2\right ) \sqrt {a+b x^2}}{8 \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \left (a+b x^2\right )^{3/2}}{4 \sqrt {a^2-b^2 x^4}}+\frac {19 a^2 \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-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 [C] time = 0.17, size = 98, normalized size = 0.64 \begin {gather*} -\frac {\left (11 a x+2 b x^3\right ) \sqrt {a^2-b^2 x^4}}{8 \sqrt {a+b x^2}}+\frac {19 i a^2 \log \left (\frac {2 \sqrt {a^2-b^2 x^4}}{\sqrt {a+b x^2}}-2 i \sqrt {b} x\right )}{8 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 3.03, 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 = 1.13, size = 251, normalized size = 1.64 \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 = 0.07, size = 132, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (2 \sqrt {-b \,x^{2}+a}\, b^{\frac {3}{2}} x^{3}-32 a^{2} \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right )+13 a^{2} \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right )+11 \sqrt {-b \,x^{2}+a}\, a \sqrt {b}\, x \right )}{8 \sqrt {b \,x^{2}+a}\, \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 (b\,x^2+a\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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