Optimal. Leaf size=111 \[ \frac {8 \sqrt {a^2 x^2-b} \left (a^2 x^3-4 b x\right )}{7 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}}+\frac {4 \left (18 a^4 x^4-81 a^2 b x^2+32 b^2\right )}{63 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}} \]
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Rubi [A] time = 0.35, antiderivative size = 93, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2120, 270} \begin {gather*} -\frac {b^2}{9 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/4}}-\frac {2 b}{a^3 \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}+\frac {\left (\sqrt {a^2 x^2-b}+a x\right )^{7/4}}{7 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 2120
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b+x^2\right )^2}{x^{13/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{4 a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^2}{x^{13/4}}+\frac {2 b}{x^{5/4}}+x^{3/4}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{4 a^3}\\ &=-\frac {b^2}{9 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}-\frac {2 b}{a^3 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{7/4}}{7 a^3}\\ \end {align*}
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Mathematica [B] time = 2.52, size = 629, normalized size = 5.67 \begin {gather*} \frac {4 \sqrt {a^2 x^2-b} \left (294912 a^{19} x^{19}-2506752 a^{17} b x^{17}+7745536 a^{15} b^2 x^{15}-12199936 a^{13} b^3 x^{13}+10988800 a^{11} b^4 x^{11}-5854336 a^9 b^5 x^9+1812000 a^7 b^6 x^7-302768 a^5 b^7 x^5+23064 a^3 b^8 x^3-32 b^9 \sqrt {a^2 x^2-b}+4225 a^2 b^8 x^2 \sqrt {a^2 x^2-b}+294912 a^{18} x^{18} \sqrt {a^2 x^2-b}-2359296 a^{16} b x^{16} \sqrt {a^2 x^2-b}+6602752 a^{14} b^2 x^{14} \sqrt {a^2 x^2-b}-9175040 a^{12} b^3 x^{12} \sqrt {a^2 x^2-b}+7090688 a^{10} b^4 x^{10} \sqrt {a^2 x^2-b}-3127296 a^8 b^5 x^8 \sqrt {a^2 x^2-b}+760704 a^6 b^6 x^6 \sqrt {a^2 x^2-b}-91648 a^4 b^7 x^4 \sqrt {a^2 x^2-b}-520 a b^9 x\right )}{63 a^3 \left (\sqrt {a^2 x^2-b}+a x\right )^{21/4} \left (512 a^{11} x^{11}-1536 a^9 b x^9+1696 a^7 b^2 x^7-832 a^5 b^3 x^5+170 a^3 b^4 x^3-b^5 \sqrt {a^2 x^2-b}+50 a^2 b^4 x^2 \sqrt {a^2 x^2-b}+512 a^{10} x^{10} \sqrt {a^2 x^2-b}-1280 a^8 b x^8 \sqrt {a^2 x^2-b}+1120 a^6 b^2 x^6 \sqrt {a^2 x^2-b}-400 a^4 b^3 x^4 \sqrt {a^2 x^2-b}-10 a b^5 x\right ) \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 111, normalized size = 1.00 \begin {gather*} \frac {8 \sqrt {-b+a^2 x^2} \left (-4 b x+a^2 x^3\right )}{7 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}}+\frac {4 \left (32 b^2-81 a^2 b x^2+18 a^4 x^4\right )}{63 a^3 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 68, normalized size = 0.61 \begin {gather*} -\frac {4 \, {\left (7 \, a^{3} x^{3} + 24 \, a b x - {\left (7 \, a^{2} x^{2} + 32 \, b\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{63 \, a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}\, dx\]
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 {x^{2}}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}}\,{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 {x^2}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,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 {x^{2}}{\sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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