Optimal. Leaf size=147 \[ \frac {3 a^2 c^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{32 b^{7/4}}-\frac {3 a^2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{32 b^{7/4}}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}+\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b} \]
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Rubi [A] time = 0.09, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {279, 321, 329, 331, 298, 205, 208} \[ \frac {3 a^2 c^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{32 b^{7/4}}-\frac {3 a^2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{32 b^{7/4}}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}+\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 279
Rule 298
Rule 321
Rule 329
Rule 331
Rubi steps
\begin {align*} \int (c x)^{5/2} \sqrt [4]{a+b x^2} \, dx &=\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}+\frac {1}{8} a \int \frac {(c x)^{5/2}}{\left (a+b x^2\right )^{3/4}} \, dx\\ &=\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}-\frac {\left (3 a^2 c^2\right ) \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{3/4}} \, dx}{32 b}\\ &=\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}-\frac {\left (3 a^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{16 b}\\ &=\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}-\frac {\left (3 a^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}}\right )}{16 b}\\ &=\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}-\frac {\left (3 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{c-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}}\right )}{32 b^{3/2}}+\frac {\left (3 a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{c+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}}\right )}{32 b^{3/2}}\\ &=\frac {a c (c x)^{3/2} \sqrt [4]{a+b x^2}}{16 b}+\frac {(c x)^{7/2} \sqrt [4]{a+b x^2}}{4 c}+\frac {3 a^2 c^{5/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{32 b^{7/4}}-\frac {3 a^2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a+b x^2}}\right )}{32 b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 85, normalized size = 0.58 \[ \frac {c (c x)^{3/2} \sqrt [4]{a+b x^2} \left (\left (a+b x^2\right ) \sqrt [4]{\frac {b x^2}{a}+1}-a \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{4 b \sqrt [4]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}\, dx \]
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 \[ \int {\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (c\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 9.47, size = 46, normalized size = 0.31 \[ \frac {\sqrt [4]{a} c^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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