Optimal. Leaf size=213 \[ -\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {3 a^{3/4} e \left (5 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {a^{3/4} \left (\frac {5 \sqrt {c} d \left (c d^2+a e^2\right )}{\sqrt {a}}-3 e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}} \]
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Rubi [A]
time = 0.17, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1221, 1902,
1215, 230, 227, 1214, 1213, 435} \begin {gather*} \frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (\frac {5 \sqrt {c} d \left (a e^2+c d^2\right )}{\sqrt {a}}-3 e \left (a e^2+5 c d^2\right )\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+5 c d^2\right ) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 227
Rule 230
Rule 435
Rule 1213
Rule 1214
Rule 1215
Rule 1221
Rule 1902
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx &=-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}-\frac {\int \frac {-5 c d^3-3 e \left (5 c d^2+a e^2\right ) x^2-15 c d e^2 x^4}{\sqrt {a-c x^4}} \, dx}{5 c}\\ &=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\int \frac {15 c d \left (c d^2+a e^2\right )+9 c e \left (5 c d^2+a e^2\right ) x^2}{\sqrt {a-c x^4}} \, dx}{15 c^2}\\ &=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\left (3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a-c x^4}} \, dx}{5 c^{3/2}}+\frac {\left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {a-c x^4}} \, dx}{5 c^{3/2}}\\ &=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\left (3 \sqrt {a} e \left (5 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{5 c^{3/2} \sqrt {a-c x^4}}+\frac {\left (\left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{5 c^{3/2} \sqrt {a-c x^4}}\\ &=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\left (3 \sqrt {a} e \left (5 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}} \, dx}{5 c^{3/2} \sqrt {a-c x^4}}\\ &=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {3 a^{3/4} e \left (5 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \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 = 10.11, size = 141, normalized size = 0.66 \begin {gather*} \frac {5 d \left (c d^2+a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {c x^4}{a}\right )+e x \left (e \left (5 d+e x^2\right ) \left (-a+c x^4\right )+\left (5 c d^2+a e^2\right ) x^2 \sqrt {1-\frac {c x^4}{a}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {c x^4}{a}\right )\right )}{5 c \sqrt {a-c x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 359 vs. \(2 (175 ) = 350\).
time = 0.13, size = 360, normalized size = 1.69
method | result | size |
elliptic | \(-\frac {e^{3} x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {d \,e^{2} x \sqrt {-c \,x^{4}+a}}{c}+\frac {\left (d^{3}+\frac {a d \,e^{2}}{c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (3 d^{2} e +\frac {3 a \,e^{3}}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(222\) |
risch | \(-\frac {e^{2} x \left (e \,x^{2}+5 d \right ) \sqrt {-c \,x^{4}+a}}{5 c}+\frac {-\frac {\left (3 a \,e^{3}+15 c \,d^{2} e \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {5 d \,e^{2} a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 c \,d^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{5 c}\) | \(274\) |
default | \(e^{3} \left (-\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+3 d \,e^{2} \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-\frac {3 d^{2} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {d^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(360\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.21, size = 180, normalized size = 0.85 \begin {gather*} \frac {d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {3 d^{2} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \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 = -0.00 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^3}{\sqrt {a-c\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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