Optimal. Leaf size=109 \[ \frac {2 x \sqrt {a x^2+b^2} \left (63 a^2 x^4+206 a b^2 x^2+271 b^4\right )}{693 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {4 x \left (35 a^2 b x^4+118 a b^3 x^2+211 b^5\right )}{693 \sqrt {\sqrt {a x^2+b^2}+b}} \]
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Rubi [F] time = 0.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx &=\int \left (b^4 \sqrt {b+\sqrt {b^2+a x^2}}+2 a b^2 x^2 \sqrt {b+\sqrt {b^2+a x^2}}+a^2 x^4 \sqrt {b+\sqrt {b^2+a x^2}}\right ) \, dx\\ &=a^2 \int x^4 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+\left (2 a b^2\right ) \int x^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+b^4 \int \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ &=\frac {2 a b^4 x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {2 b^5 x}{\sqrt {b+\sqrt {b^2+a x^2}}}+a^2 \int x^4 \sqrt {b+\sqrt {b^2+a x^2}} \, dx+\left (2 a b^2\right ) \int x^2 \sqrt {b+\sqrt {b^2+a x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.29, size = 144, normalized size = 1.32 \begin {gather*} \frac {2 x \left (63 a^3 x^6 \sqrt {a x^2+b^2}+196 a^3 b x^6+914 a^2 b^3 x^4+472 a^2 b^2 x^4 \sqrt {a x^2+b^2}+1848 a b^5 x^2+1386 b^6 \sqrt {a x^2+b^2}+1155 a b^4 x^2 \sqrt {a x^2+b^2}+1386 b^7\right )}{693 \left (\sqrt {a x^2+b^2}+b\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 109, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {b^2+a x^2} \left (271 b^4+206 a b^2 x^2+63 a^2 x^4\right )}{693 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 x \left (211 b^5+118 a b^3 x^2+35 a^2 b x^4\right )}{693 \sqrt {b+\sqrt {b^2+a x^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 93, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (63 \, a^{3} x^{6} + 199 \, a^{2} b^{2} x^{4} + 241 \, a b^{4} x^{2} - 151 \, b^{6} + {\left (7 \, a^{2} b x^{4} + 30 \, a b^{3} x^{2} + 151 \, b^{5}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{693 \, a x} \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 {\left (a x^{2} + b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 189, normalized size = 1.73
method | result | size |
meijerg | \(\frac {\left (b^{2}\right )^{\frac {1}{4}} a^{2} \sqrt {2}\, x^{5} \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {5}{2}\right ], \left [\frac {1}{2}, \frac {7}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{5}+\frac {2 b^{2} \left (b^{2}\right )^{\frac {1}{4}} a \sqrt {2}\, x^{3} \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{4}, \frac {3}{2}\right ], \left [\frac {1}{2}, \frac {5}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{3}-\frac {b^{4} \left (b^{2}\right )^{\frac {1}{4}} \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \sqrt {\frac {a}{b^{2}}}\, a \cosh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right )}{3 b^{2}}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \sqrt {\frac {a}{b^{2}}}\, \left (-\frac {4 x^{4} a^{2}}{3 b^{4}}-\frac {2 x^{2} a}{3 b^{2}}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {x \sqrt {a}}{b}\right )}{2}\right ) b}{\sqrt {a}\, \sqrt {\frac {x^{2} a}{b^{2}}+1}}\right )}{8 \sqrt {\pi }\, \sqrt {\frac {a}{b^{2}}}}\) | \(189\) |
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 {\left (a x^{2} + b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{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 {\left (b^2+a\,x^2\right )}^2\,\sqrt {b+\sqrt {b^2+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.33, size = 1100, normalized size = 10.09
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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