Optimal. Leaf size=326 \[ \frac {1}{2} d e x^2 \sqrt {a+c x^4}+\frac {2 a e^2 x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{15} x \left (5 d^2+3 e^2 x^2\right ) \sqrt {a+c x^4}+\frac {a d e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}}-\frac {2 a^{5/4} e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {a^{3/4} \left (5 \sqrt {c} d^2+3 \sqrt {a} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 c^{3/4} \sqrt {a+c x^4}} \]
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Rubi [A]
time = 0.12, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1899, 281,
201, 223, 212, 1191, 1212, 226, 1210} \begin {gather*} \frac {a^{3/4} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (3 \sqrt {a} e^2+5 \sqrt {c} d^2\right ) F\left (2 \text {ArcTan}\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^{5/4} e^2 \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 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {1}{15} x \sqrt {a+c x^4} \left (5 d^2+3 e^2 x^2\right )+\frac {1}{2} d e x^2 \sqrt {a+c x^4}+\frac {a d e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}}+\frac {2 a e^2 x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 226
Rule 281
Rule 1191
Rule 1210
Rule 1212
Rule 1899
Rubi steps
\begin {align*} \int (d+e x)^2 \sqrt {a+c x^4} \, dx &=\int \left (2 d e x \sqrt {a+c x^4}+\left (d^2+e^2 x^2\right ) \sqrt {a+c x^4}\right ) \, dx\\ &=(2 d e) \int x \sqrt {a+c x^4} \, dx+\int \left (d^2+e^2 x^2\right ) \sqrt {a+c x^4} \, dx\\ &=\frac {1}{15} x \left (5 d^2+3 e^2 x^2\right ) \sqrt {a+c x^4}+\frac {1}{15} \int \frac {10 a d^2+6 a e^2 x^2}{\sqrt {a+c x^4}} \, dx+(d e) \text {Subst}\left (\int \sqrt {a+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} d e x^2 \sqrt {a+c x^4}+\frac {1}{15} x \left (5 d^2+3 e^2 x^2\right ) \sqrt {a+c x^4}+\frac {1}{2} (a d e) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )-\frac {\left (2 a^{3/2} e^2\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{5 \sqrt {c}}+\frac {1}{15} \left (2 a \left (5 d^2+\frac {3 \sqrt {a} e^2}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx\\ &=\frac {1}{2} d e x^2 \sqrt {a+c x^4}+\frac {2 a e^2 x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{15} x \left (5 d^2+3 e^2 x^2\right ) \sqrt {a+c x^4}-\frac {2 a^{5/4} e^2 \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 )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {a^{3/4} \left (5 \sqrt {c} d^2+3 \sqrt {a} e^2\right ) \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 {1}{2} (a d e) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=\frac {1}{2} d e x^2 \sqrt {a+c x^4}+\frac {2 a e^2 x \sqrt {a+c x^4}}{5 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {1}{15} x \left (5 d^2+3 e^2 x^2\right ) \sqrt {a+c x^4}+\frac {a d e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}}-\frac {2 a^{5/4} e^2 \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 )}{5 c^{3/4} \sqrt {a+c x^4}}+\frac {a^{3/4} \left (5 \sqrt {c} d^2+3 \sqrt {a} e^2\right ) \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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 9.01, size = 146, normalized size = 0.45 \begin {gather*} \frac {\sqrt {a+c x^4} \left (6 \sqrt {c} d^2 x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )+e \left (3 d \left (\sqrt {c} x^2 \sqrt {1+\frac {c x^4}{a}}+\sqrt {a} \sinh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )\right )+2 \sqrt {c} e x^3 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {c x^4}{a}\right )\right )\right )}{6 \sqrt {c} \sqrt {1+\frac {c x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.24, size = 248, normalized size = 0.76
method | result | size |
default | \(e^{2} \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5}+\frac {2 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+2 d e \left (\frac {x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {a \ln \left (x^{2} \sqrt {c}+\sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}\right )+d^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\) | \(248\) |
elliptic | \(\frac {e^{2} x^{3} \sqrt {c \,x^{4}+a}}{5}+\frac {d e \,x^{2} \sqrt {c \,x^{4}+a}}{2}+\frac {d^{2} x \sqrt {c \,x^{4}+a}}{3}+\frac {2 a \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {a d e \ln \left (2 x^{2} \sqrt {c}+2 \sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {2 i a^{\frac {3}{2}} e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(252\) |
risch | \(\frac {x \left (6 e^{2} x^{2}+15 d e x +10 d^{2}\right ) \sqrt {c \,x^{4}+a}}{30}+\frac {2 i a^{\frac {3}{2}} e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {2 i a^{\frac {3}{2}} e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\frac {a d e \ln \left (x^{2} \sqrt {c}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {2 a \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(293\) |
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 [C] Result contains complex when optimal does not.
time = 2.02, size = 138, normalized size = 0.42 \begin {gather*} \frac {\sqrt {a} d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{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 )} + \frac {\sqrt {a} d e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{2} + \frac {\sqrt {a} e^{2} 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^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a d e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{2 \sqrt {c}} \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 = -0.00 \begin {gather*} \int \sqrt {c\,x^4+a}\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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