Optimal. Leaf size=331 \[ \frac {a^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (3 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {2 a^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{15} x \sqrt {a+b x^4} \left (5 c+3 e x^2\right )+\frac {1}{4} d x^2 \sqrt {a+b x^4}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 \sqrt {b}}+\frac {2 a e x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {f \left (a+b x^4\right )^{3/2}}{6 b} \]
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Rubi [A] time = 0.19, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1885, 1177, 1198, 220, 1196, 1248, 641, 195, 217, 206} \[ \frac {a^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (3 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {2 a^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{15} x \sqrt {a+b x^4} \left (5 c+3 e x^2\right )+\frac {1}{4} d x^2 \sqrt {a+b x^4}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 \sqrt {b}}+\frac {2 a e x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {f \left (a+b x^4\right )^{3/2}}{6 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 220
Rule 641
Rule 1177
Rule 1196
Rule 1198
Rule 1248
Rule 1885
Rubi steps
\begin {align*} \int \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx &=\int \left (\left (c+e x^2\right ) \sqrt {a+b x^4}+x \left (d+f x^2\right ) \sqrt {a+b x^4}\right ) \, dx\\ &=\int \left (c+e x^2\right ) \sqrt {a+b x^4} \, dx+\int x \left (d+f x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {1}{15} x \left (5 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{15} \int \frac {10 a c+6 a e x^2}{\sqrt {a+b x^4}} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int (d+f x) \sqrt {a+b x^2} \, dx,x,x^2\right )\\ &=\frac {1}{15} x \left (5 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {f \left (a+b x^4\right )^{3/2}}{6 b}+\frac {1}{2} d \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )-\frac {\left (2 a^{3/2} e\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{5 \sqrt {b}}+\frac {1}{15} \left (2 a \left (5 c+\frac {3 \sqrt {a} e}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {1}{4} d x^2 \sqrt {a+b x^4}+\frac {2 a e x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{15} x \left (5 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {f \left (a+b x^4\right )^{3/2}}{6 b}-\frac {2 a^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {a^{3/4} \left (5 \sqrt {b} c+3 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{4} (a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{4} d x^2 \sqrt {a+b x^4}+\frac {2 a e x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{15} x \left (5 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {f \left (a+b x^4\right )^{3/2}}{6 b}-\frac {2 a^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {a^{3/4} \left (5 \sqrt {b} c+3 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{4} (a d) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=\frac {1}{4} d x^2 \sqrt {a+b x^4}+\frac {2 a e x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{15} x \left (5 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {f \left (a+b x^4\right )^{3/2}}{6 b}+\frac {a d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{4 \sqrt {b}}-\frac {2 a^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}+\frac {a^{3/4} \left (5 \sqrt {b} c+3 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 171, normalized size = 0.52 \[ \frac {\sqrt {a+b x^4} \left (12 b c x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )+3 \sqrt {a} \sqrt {b} d \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )+3 b d x^2 \sqrt {\frac {b x^4}{a}+1}+4 b e x^3 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )+2 b f x^4 \sqrt {\frac {b x^4}{a}+1}+2 a f \sqrt {\frac {b x^4}{a}+1}\right )}{12 b \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}, x\right ) \]
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 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 313, normalized size = 0.95 \[ \frac {\sqrt {b \,x^{4}+a}\, e \,x^{3}}{5}-\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {3}{2}} e \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {3}{2}} e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {\sqrt {b \,x^{4}+a}\, d \,x^{2}}{4}+\frac {a d \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{4 \sqrt {b}}+\frac {\sqrt {b \,x^{4}+a}\, c x}{3}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} f}{6 b} \]
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 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.97, size = 156, normalized size = 0.47 \[ \frac {\sqrt {a} c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} d x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} + \frac {\sqrt {a} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4 \sqrt {b}} + f \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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