Optimal. Leaf size=255 \[ \frac {2 a^{9/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}-\frac {4 a^{9/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}+\frac {4 a^2 x \sqrt {a+c x^4}}{15 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac {2}{15} a x^3 \sqrt {a+c x^4} \]
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Rubi [A] time = 0.09, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 305, 220, 1196} \[ \frac {2 a^{9/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}-\frac {4 a^{9/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}+\frac {4 a^2 x \sqrt {a+c x^4}}{15 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac {2}{15} a x^3 \sqrt {a+c x^4} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 305
Rule 1196
Rubi steps
\begin {align*} \int x^2 \left (a+c x^4\right )^{3/2} \, dx &=\frac {1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac {1}{3} (2 a) \int x^2 \sqrt {a+c x^4} \, dx\\ &=\frac {2}{15} a x^3 \sqrt {a+c x^4}+\frac {1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac {1}{15} \left (4 a^2\right ) \int \frac {x^2}{\sqrt {a+c x^4}} \, dx\\ &=\frac {2}{15} a x^3 \sqrt {a+c x^4}+\frac {1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac {\left (4 a^{5/2}\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{15 \sqrt {c}}-\frac {\left (4 a^{5/2}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{15 \sqrt {c}}\\ &=\frac {2}{15} a x^3 \sqrt {a+c x^4}+\frac {4 a^2 x \sqrt {a+c x^4}}{15 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{9} x^3 \left (a+c x^4\right )^{3/2}-\frac {4 a^{9/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}}+\frac {2 a^{9/4} \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 )}{15 c^{3/4} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.20 \[ \frac {a x^3 \sqrt {a+c x^4} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^4}{a}\right )}{3 \sqrt {\frac {c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{6} + a x^{2}\right )} \sqrt {c x^{4} + a}, x\right ) \]
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 \[ \int {\left (c x^{4} + a\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 128, normalized size = 0.50 \[ \frac {\sqrt {c \,x^{4}+a}\, c \,x^{7}}{9}+\frac {11 \sqrt {c \,x^{4}+a}\, a \,x^{3}}{45}+\frac {4 i \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {5}{2}}}{15 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}} \]
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 \[ \int {\left (c x^{4} + a\right )}^{\frac {3}{2}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (c\,x^4+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.87, size = 39, normalized size = 0.15 \[ \frac {a^{\frac {3}{2}} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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