Optimal. Leaf size=148 \[ -\frac {2 a^{11/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 c^{5/4} \sqrt {a+c x^4}}+\frac {4 a^2 x \sqrt {a+c x^4}}{77 c}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac {6}{77} a x^5 \sqrt {a+c x^4} \]
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Rubi [A] time = 0.05, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {279, 321, 220} \[ -\frac {2 a^{11/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 c^{5/4} \sqrt {a+c x^4}}+\frac {4 a^2 x \sqrt {a+c x^4}}{77 c}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac {6}{77} a x^5 \sqrt {a+c x^4} \]
Antiderivative was successfully verified.
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Rule 220
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^4 \left (a+c x^4\right )^{3/2} \, dx &=\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac {1}{11} (6 a) \int x^4 \sqrt {a+c x^4} \, dx\\ &=\frac {6}{77} a x^5 \sqrt {a+c x^4}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac {1}{77} \left (12 a^2\right ) \int \frac {x^4}{\sqrt {a+c x^4}} \, dx\\ &=\frac {4 a^2 x \sqrt {a+c x^4}}{77 c}+\frac {6}{77} a x^5 \sqrt {a+c x^4}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}-\frac {\left (4 a^3\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{77 c}\\ &=\frac {4 a^2 x \sqrt {a+c x^4}}{77 c}+\frac {6}{77} a x^5 \sqrt {a+c x^4}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}-\frac {2 a^{11/4} \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 )}{77 c^{5/4} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 67, normalized size = 0.45 \[ \frac {x \sqrt {a+c x^4} \left (\left (a+c x^4\right )^2-\frac {a^2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )}{\sqrt {\frac {c x^4}{a}+1}}\right )}{11 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{8} + a x^{4}\right )} \sqrt {c x^{4} + a}, x\right ) \]
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 \[ \int {\left (c x^{4} + a\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 126, normalized size = 0.85 \[ \frac {\sqrt {c \,x^{4}+a}\, c \,x^{9}}{11}+\frac {13 \sqrt {c \,x^{4}+a}\, a \,x^{5}}{77}-\frac {4 \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, a^{3} \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{77 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, c}+\frac {4 \sqrt {c \,x^{4}+a}\, a^{2} x}{77 c} \]
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 \[ \int {\left (c x^{4} + a\right )}^{\frac {3}{2}} x^{4}\,{d x} \]
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 \[ \int x^4\,{\left (c\,x^4+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.77, size = 39, normalized size = 0.26 \[ \frac {a^{\frac {3}{2}} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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