Optimal. Leaf size=152 \[ -\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac {a^{5/2} c^2 \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A]
time = 0.08, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {285, 327, 335,
243, 342, 281, 237} \begin {gather*} -\frac {a^{5/2} c^2 (c x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 285
Rule 327
Rule 335
Rule 342
Rubi steps
\begin {align*} \int (c x)^{7/2} \sqrt [4]{a+b x^2} \, dx &=\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac {1}{10} a \int \frac {(c x)^{7/2}}{\left (a+b x^2\right )^{3/4}} \, dx\\ &=\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac {\left (a^2 c^2\right ) \int \frac {(c x)^{3/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{12 b}\\ &=-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac {\left (a^3 c^4\right ) \int \frac {1}{\sqrt {c x} \left (a+b x^2\right )^{3/4}} \, dx}{24 b^2}\\ &=-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac {\left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{12 b^2}\\ &=-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}+\frac {\left (a^3 c^3 \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {c x}\right )}{12 b^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac {\left (a^3 c^3 \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{12 b^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac {\left (a^3 c^3 \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{c x}\right )}{24 b^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {a^2 c^3 \sqrt {c x} \sqrt [4]{a+b x^2}}{12 b^2}+\frac {a c (c x)^{5/2} \sqrt [4]{a+b x^2}}{30 b}+\frac {(c x)^{9/2} \sqrt [4]{a+b x^2}}{5 c}-\frac {a^{5/2} c^2 \left (1+\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 102, normalized size = 0.67 \begin {gather*} \frac {c^3 \sqrt {c x} \sqrt [4]{a+b x^2} \left (\sqrt [4]{1+\frac {b x^2}{a}} \left (-5 a^2+a b x^2+6 b^2 x^4\right )+5 a^2 \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{30 b^2 \sqrt [4]{1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (c x \right )^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [C] Result contains complex when optimal does not.
time = 16.98, size = 46, normalized size = 0.30 \begin {gather*} \frac {\sqrt [4]{a} c^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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