Optimal. Leaf size=122 \[ \frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {195, 220} \[ \frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 220
Rubi steps
\begin {align*} \int \left (a+c x^4\right )^{3/2} \, dx &=\frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {1}{7} (6 a) \int \sqrt {a+c x^4} \, dx\\ &=\frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {1}{7} \left (4 a^2\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx\\ &=\frac {2}{7} a x \sqrt {a+c x^4}+\frac {1}{7} x \left (a+c x^4\right )^{3/2}+\frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{7 \sqrt [4]{c} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 47, normalized size = 0.39 \[ \frac {a x \sqrt {a+c x^4} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )}{\sqrt {\frac {c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{4} + a\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (c x^{4} + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 103, normalized size = 0.84 \[ \frac {\sqrt {c \,x^{4}+a}\, c \,x^{5}}{7}+\frac {4 \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, a^{2} \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{7 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 \sqrt {c \,x^{4}+a}\, a x}{7} \]
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 \[ \int {\left (c x^{4} + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 37, normalized size = 0.30 \[ \frac {x\,{\left (c\,x^4+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ -\frac {c\,x^4}{a}\right )}{{\left (\frac {c\,x^4}{a}+1\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.25, size = 37, normalized size = 0.30 \[ \frac {a^{\frac {3}{2}} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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