Optimal. Leaf size=62 \[ \frac {2 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{3 a^2}-\frac {2 b}{5 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \]
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Rubi [A] time = 0.25, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2120, 14} \begin {gather*} \frac {2 \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{3 a^2}-\frac {2 b}{5 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2120
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b+x^2}{x^{9/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b}{x^{9/4}}+\frac {1}{\sqrt [4]{x}}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{2 a^2}\\ &=-\frac {2 b}{5 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{3 a^2}\\ \end {align*}
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Mathematica [B] time = 2.44, size = 510, normalized size = 8.23 \begin {gather*} \frac {4 \sqrt {a^2 x^2-b} \left (20480 a^{15} x^{15}-84992 a^{13} b x^{13}+142592 a^{11} b^2 x^{11}-123392 a^9 b^3 x^9+58080 a^7 b^4 x^7-14308 a^5 b^5 x^5+1593 a^3 b^6 x^3-4 b^7 \sqrt {a^2 x^2-b}+353 a^2 b^6 x^2 \sqrt {a^2 x^2-b}+20480 a^{14} x^{14} \sqrt {a^2 x^2-b}-74752 a^{12} b x^{12} \sqrt {a^2 x^2-b}+107776 a^{10} b^2 x^{10} \sqrt {a^2 x^2-b}-77568 a^8 b^3 x^8 \sqrt {a^2 x^2-b}+28896 a^6 b^4 x^6 \sqrt {a^2 x^2-b}-5180 a^4 b^5 x^4 \sqrt {a^2 x^2-b}-53 a b^7 x\right )}{15 a^2 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4} \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right ) \left (a b^4 x \left (9 \sqrt {a^2 x^2-b}+41 a x\right )+256 a^9 x^9 \left (\sqrt {a^2 x^2-b}+a x\right )-64 a^7 b x^7 \left (9 \sqrt {a^2 x^2-b}+11 a x\right )+16 a^5 b^2 x^5 \left (27 \sqrt {a^2 x^2-b}+43 a x\right )-40 a^3 b^3 x^3 \left (3 \sqrt {a^2 x^2-b}+7 a x\right )-b^5\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 62, normalized size = 1.00 \begin {gather*} -\frac {2 b}{5 a^2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {2 \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 56, normalized size = 0.90 \begin {gather*} -\frac {4 \, {\left (3 \, a^{2} x^{2} - 3 \, \sqrt {a^{2} x^{2} - b} a x - 4 \, b\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{15 \, a^{2} 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 \[\int \frac {x}{\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}\, dx\]
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}{\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 = -0.02 \begin {gather*} \int \frac {x}{{\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}{\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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