Optimal. Leaf size=158 \[ \frac {a^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}} \]
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Rubi [A] time = 0.09, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1885, 195, 220, 275, 217, 206} \[ \frac {a^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 220
Rule 275
Rule 1885
Rubi steps
\begin {align*} \int (d+e x) \sqrt {a+c x^4} \, dx &=\int \left (d \sqrt {a+c x^4}+e x \sqrt {a+c x^4}\right ) \, dx\\ &=d \int \sqrt {a+c x^4} \, dx+e \int x \sqrt {a+c x^4} \, dx\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{3} (2 a d) \int \frac {1}{\sqrt {a+c x^4}} \, dx+\frac {1}{2} e \operatorname {Subst}\left (\int \sqrt {a+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a^{3/4} d \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 )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{4} (a e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a^{3/4} d \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 )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {1}{4} (a e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=\frac {1}{3} d x \sqrt {a+c x^4}+\frac {1}{4} e x^2 \sqrt {a+c x^4}+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 \sqrt {c}}+\frac {a^{3/4} d \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 )}{3 \sqrt [4]{c} \sqrt {a+c x^4}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 109, normalized size = 0.69 \[ \frac {\sqrt {a+c x^4} \left (4 \sqrt {c} d x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^4}{a}\right )+\sqrt {a} e \sinh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )+\sqrt {c} e x^2 \sqrt {\frac {c x^4}{a}+1}\right )}{4 \sqrt {c} \sqrt {\frac {c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{4} + a} {\left (e x + d\right )}, x\right ) \]
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 \[ \int \sqrt {c x^{4} + a} {\left (e x + d\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 127, normalized size = 0.80 \[ \frac {2 \sqrt {-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}+1}\, a d \EllipticF \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, x , i\right )}{3 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\sqrt {c \,x^{4}+a}\, e \,x^{2}}{4}+\frac {a e \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{4 \sqrt {c}}+\frac {\sqrt {c \,x^{4}+a}\, d x}{3} \]
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 \[ \int \sqrt {c x^{4} + a} {\left (e x + d\right )}\,{d x} \]
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 \[ \int \sqrt {c\,x^4+a}\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.45, size = 88, normalized size = 0.56 \[ \frac {\sqrt {a} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} e x^{2} \sqrt {1 + \frac {c x^{4}}{a}}}{4} + \frac {a e \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{4 \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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