Optimal. Leaf size=83 \[ \frac {2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac {64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {279, 270}
\begin {gather*} \frac {64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}-\frac {16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac {2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 279
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{9/2} \left (a+b x^2\right )^{5/4}} \, dx &=\frac {2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}+\frac {8 \int \frac {1}{(c x)^{9/2} \sqrt [4]{a+b x^2}} \, dx}{a}\\ &=\frac {2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}-\frac {32 \int \frac {\left (a+b x^2\right )^{3/4}}{(c x)^{9/2}} \, dx}{3 a^2}\\ &=\frac {2}{a c (c x)^{7/2} \sqrt [4]{a+b x^2}}-\frac {16 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{7/2}}+\frac {64 \left (a+b x^2\right )^{7/4}}{21 a^3 c (c x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 47, normalized size = 0.57 \begin {gather*} -\frac {2 x \left (3 a^2-8 a b x^2-32 b^2 x^4\right )}{21 a^3 (c x)^{9/2} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 42, normalized size = 0.51
method | result | size |
gosper | \(-\frac {2 x \left (-32 b^{2} x^{4}-8 a b \,x^{2}+3 a^{2}\right )}{21 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} \left (c x \right )^{\frac {9}{2}}}\) | \(42\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (-11 b \,x^{2}+3 a \right )}{21 a^{3} x^{3} c^{4} \sqrt {c x}}+\frac {2 b^{2} x}{a^{3} c^{4} \sqrt {c x}\, \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\) | \(63\) |
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 = 1.04, size = 61, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (32 \, b^{2} x^{4} + 8 \, a b x^{2} - 3 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{21 \, {\left (a^{3} b c^{5} x^{6} + a^{4} c^{5} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs.
\(2 (71) = 142\).
time = 80.89, size = 384, normalized size = 4.63 \begin {gather*} - \frac {3 a^{3} b^{\frac {19}{4}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{32 a^{5} b^{4} c^{\frac {9}{2}} x^{2} \Gamma \left (\frac {5}{4}\right ) + 64 a^{4} b^{5} c^{\frac {9}{2}} x^{4} \Gamma \left (\frac {5}{4}\right ) + 32 a^{3} b^{6} c^{\frac {9}{2}} x^{6} \Gamma \left (\frac {5}{4}\right )} + \frac {5 a^{2} b^{\frac {23}{4}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{32 a^{5} b^{4} c^{\frac {9}{2}} x^{2} \Gamma \left (\frac {5}{4}\right ) + 64 a^{4} b^{5} c^{\frac {9}{2}} x^{4} \Gamma \left (\frac {5}{4}\right ) + 32 a^{3} b^{6} c^{\frac {9}{2}} x^{6} \Gamma \left (\frac {5}{4}\right )} + \frac {40 a b^{\frac {27}{4}} x^{4} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{32 a^{5} b^{4} c^{\frac {9}{2}} x^{2} \Gamma \left (\frac {5}{4}\right ) + 64 a^{4} b^{5} c^{\frac {9}{2}} x^{4} \Gamma \left (\frac {5}{4}\right ) + 32 a^{3} b^{6} c^{\frac {9}{2}} x^{6} \Gamma \left (\frac {5}{4}\right )} + \frac {32 b^{\frac {31}{4}} x^{6} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{32 a^{5} b^{4} c^{\frac {9}{2}} x^{2} \Gamma \left (\frac {5}{4}\right ) + 64 a^{4} b^{5} c^{\frac {9}{2}} x^{4} \Gamma \left (\frac {5}{4}\right ) + 32 a^{3} b^{6} c^{\frac {9}{2}} x^{6} \Gamma \left (\frac {5}{4}\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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.15, size = 70, normalized size = 0.84 \begin {gather*} \frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {16\,x^2}{21\,a^2\,c^4}-\frac {2}{7\,a\,b\,c^4}+\frac {64\,b\,x^4}{21\,a^3\,c^4}\right )}{x^5\,\sqrt {c\,x}+\frac {a\,x^3\,\sqrt {c\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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