Optimal. Leaf size=307 \[ \frac {a \sqrt {c} \log \left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {c}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \sqrt {c} \log \left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {c}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \sqrt {c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt {2} b^{3/4}}+\frac {a \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt {2} b^{3/4}}+\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c} \]
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Rubi [A] time = 0.27, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {279, 329, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {a \sqrt {c} \log \left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {c}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \sqrt {c} \log \left (\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {c}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \sqrt {c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt {2} b^{3/4}}+\frac {a \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{4 \sqrt {2} b^{3/4}}+\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 279
Rule 297
Rule 329
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \sqrt {c x} \sqrt [4]{a-b x^2} \, dx &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac {1}{4} a \int \frac {\sqrt {c x}}{\left (a-b x^2\right )^{3/4}} \, dx\\ &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac {a \operatorname {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{2 c}\\ &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac {a \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 )}{2 c}\\ &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}-\frac {a \operatorname {Subst}\left (\int \frac {c-\sqrt {b} x^2}{1+\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{4 \sqrt {b} c}+\frac {a \operatorname {Subst}\left (\int \frac {c+\sqrt {b} x^2}{1+\frac {b x^4}{c^2}} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{4 \sqrt {b} c}\\ &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac {\left (a \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt [4]{b}}+2 x}{-\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt {2} b^{3/4}}+\frac {\left (a \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c}}{\sqrt [4]{b}}-2 x}{-\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt {2} b^{3/4}}+\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 b}+\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 b}\\ &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}+\frac {a \sqrt {c} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \sqrt {c} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt {2} b^{3/4}}+\frac {\left (a \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt {2} b^{3/4}}-\frac {\left (a \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt {2} b^{3/4}}\\ &=\frac {(c x)^{3/2} \sqrt [4]{a-b x^2}}{2 c}-\frac {a \sqrt {c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt {2} b^{3/4}}+\frac {a \sqrt {c} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{4 \sqrt {2} b^{3/4}}+\frac {a \sqrt {c} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt {2} b^{3/4}}-\frac {a \sqrt {c} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{8 \sqrt {2} b^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.19 \[ \frac {2 x \sqrt {c x} \sqrt [4]{a-b x^2} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {b x^2}{a}\right )}{3 \sqrt [4]{1-\frac {b x^2}{a}}} \]
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 \[ \int {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}\,{d x} \]
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 \[ \int \sqrt {c x}\, \left (-b \,x^{2}+a \right )^{\frac {1}{4}}\, dx \]
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 \[ \int {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {c\,x}\,{\left (a-b\,x^2\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.78, size = 48, normalized size = 0.16 \[ \frac {\sqrt [4]{a} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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