Optimal. Leaf size=148 \[ \frac {4 a^2 x \sqrt {a+c x^4}}{77 c}+\frac {6}{77} a x^5 \sqrt {a+c x^4}+\frac {1}{11} x^5 \left (a+c x^4\right )^{3/2}-\frac {2 a^{11/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{77 c^{5/4} \sqrt {a+c x^4}} \]
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Rubi [A]
time = 0.04, 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 = {285, 327, 226}
\begin {gather*} -\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 \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 285
Rule 327
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.61, size = 67, normalized size = 0.45 \begin {gather*} \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 {1+\frac {c x^4}{a}}}\right )}{11 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 126, normalized size = 0.85
method | result | size |
risch | \(\frac {x \left (7 c^{2} x^{8}+13 a c \,x^{4}+4 a^{2}\right ) \sqrt {x^{4} c +a}}{77 c}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(114\) |
default | \(\frac {c \,x^{9} \sqrt {x^{4} c +a}}{11}+\frac {13 a \,x^{5} \sqrt {x^{4} c +a}}{77}+\frac {4 a^{2} x \sqrt {x^{4} c +a}}{77 c}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(126\) |
elliptic | \(\frac {c \,x^{9} \sqrt {x^{4} c +a}}{11}+\frac {13 a \,x^{5} \sqrt {x^{4} c +a}}{77}+\frac {4 a^{2} x \sqrt {x^{4} c +a}}{77 c}-\frac {4 a^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {x^{4} c +a}}\) | \(126\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 70, normalized size = 0.47 \begin {gather*} -\frac {4 \, a^{2} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (7 \, c^{2} x^{9} + 13 \, a c x^{5} + 4 \, a^{2} x\right )} \sqrt {c x^{4} + a}}{77 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.50, size = 39, normalized size = 0.26 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^4\,{\left (c\,x^4+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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