Optimal. Leaf size=88 \[ -\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac {16 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac {64 \left (a-b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {279, 270}
\begin {gather*} -\frac {64 \left (a-b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}+\frac {16 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 279
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{13/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}-\frac {8 \int \frac {\left (a-b x^2\right )^{3/4}}{(c x)^{13/2}} \, dx}{3 a}\\ &=-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac {16 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}+\frac {32 \int \frac {\left (a-b x^2\right )^{7/4}}{(c x)^{13/2}} \, dx}{21 a^2}\\ &=-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{11/2}}+\frac {16 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{11/2}}-\frac {64 \left (a-b x^2\right )^{11/4}}{231 a^3 c (c x)^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 48, normalized size = 0.55 \begin {gather*} -\frac {2 x \left (a-b x^2\right )^{3/4} \left (21 a^2+24 a b x^2+32 b^2 x^4\right )}{231 a^3 (c x)^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 43, normalized size = 0.49
method | result | size |
gosper | \(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{4}+24 a b \,x^{2}+21 a^{2}\right )}{231 a^{3} \left (c x \right )^{\frac {13}{2}}}\) | \(43\) |
risch | \(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (32 b^{2} x^{4}+24 a b \,x^{2}+21 a^{2}\right )}{231 c^{6} \sqrt {c x}\, a^{3} x^{5}}\) | \(48\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.32, size = 47, normalized size = 0.53 \begin {gather*} -\frac {2 \, {\left (32 \, b^{2} x^{4} + 24 \, a b x^{2} + 21 \, a^{2}\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{231 \, a^{3} c^{7} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.25, size = 55, normalized size = 0.62 \begin {gather*} -\frac {{\left (a-b\,x^2\right )}^{3/4}\,\left (\frac {2}{11\,a\,c^6}+\frac {16\,b\,x^2}{77\,a^2\,c^6}+\frac {64\,b^2\,x^4}{231\,a^3\,c^6}\right )}{x^5\,\sqrt {c\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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