Optimal. Leaf size=369 \[ \frac {a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {b} c-5 \sqrt {a} e\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{5/4} \sqrt {a+b x^4}}-\frac {2 a^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}-\frac {a^2 f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {1}{35} x^3 \sqrt {a+b x^4} \left (7 c+5 e x^2\right )+\frac {2 a c x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {\left (a+b x^4\right )^{3/2} \left (4 d+3 f x^2\right )}{24 b}+\frac {2 a e x \sqrt {a+b x^4}}{21 b}-\frac {a f x^2 \sqrt {a+b x^4}}{16 b} \]
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Rubi [A] time = 0.30, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1833, 1274, 1280, 1198, 220, 1196, 1252, 780, 195, 217, 206} \[ \frac {a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {b} c-5 \sqrt {a} e\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{5/4} \sqrt {a+b x^4}}-\frac {2 a^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 b^{3/4} \sqrt {a+b x^4}}-\frac {a^2 f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}+\frac {1}{35} x^3 \sqrt {a+b x^4} \left (7 c+5 e x^2\right )+\frac {2 a c x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {\left (a+b x^4\right )^{3/2} \left (4 d+3 f x^2\right )}{24 b}+\frac {2 a e x \sqrt {a+b x^4}}{21 b}-\frac {a f x^2 \sqrt {a+b x^4}}{16 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 220
Rule 780
Rule 1196
Rule 1198
Rule 1252
Rule 1274
Rule 1280
Rule 1833
Rubi steps
\begin {align*} \int x^2 \left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4} \, dx &=\int \left (x^2 \left (c+e x^2\right ) \sqrt {a+b x^4}+x^3 \left (d+f x^2\right ) \sqrt {a+b x^4}\right ) \, dx\\ &=\int x^2 \left (c+e x^2\right ) \sqrt {a+b x^4} \, dx+\int x^3 \left (d+f x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {1}{35} x^3 \left (7 c+5 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{2} \operatorname {Subst}\left (\int x (d+f x) \sqrt {a+b x^2} \, dx,x,x^2\right )+\frac {1}{35} (2 a) \int \frac {x^2 \left (7 c+5 e x^2\right )}{\sqrt {a+b x^4}} \, dx\\ &=\frac {2 a e x \sqrt {a+b x^4}}{21 b}+\frac {1}{35} x^3 \left (7 c+5 e x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {(2 a) \int \frac {5 a e-21 b c x^2}{\sqrt {a+b x^4}} \, dx}{105 b}-\frac {(a f) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )}{8 b}\\ &=\frac {2 a e x \sqrt {a+b x^4}}{21 b}-\frac {a f x^2 \sqrt {a+b x^4}}{16 b}+\frac {1}{35} x^3 \left (7 c+5 e x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {\left (2 a^{3/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{5 \sqrt {b}}+\frac {\left (2 a^{3/2} \left (21 \sqrt {b} c-5 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{105 b}-\frac {\left (a^2 f\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 b}\\ &=\frac {2 a e x \sqrt {a+b x^4}}{21 b}-\frac {a f x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a c x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{35} x^3 \left (7 c+5 e x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {2 a^{5/4} c \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^{5/4} \left (21 \sqrt {b} c-5 \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 )}{105 b^{5/4} \sqrt {a+b x^4}}-\frac {\left (a^2 f\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{16 b}\\ &=\frac {2 a e x \sqrt {a+b x^4}}{21 b}-\frac {a f x^2 \sqrt {a+b x^4}}{16 b}+\frac {2 a c x \sqrt {a+b x^4}}{5 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{35} x^3 \left (7 c+5 e x^2\right ) \sqrt {a+b x^4}+\frac {\left (4 d+3 f x^2\right ) \left (a+b x^4\right )^{3/2}}{24 b}-\frac {a^2 f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 b^{3/2}}-\frac {2 a^{5/4} c \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^{5/4} \left (21 \sqrt {b} c-5 \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 )}{105 b^{5/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.78, size = 182, normalized size = 0.49 \[ \frac {1}{336} \sqrt {a+b x^4} \left (-\frac {21 a^{3/2} f \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{b^{3/2} \sqrt {\frac {b x^4}{a}+1}}+\frac {112 c x^3 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+\frac {56 d \left (a+b x^4\right )}{b}-\frac {48 a e x \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{b \sqrt {\frac {b x^4}{a}+1}}+\frac {48 e x \left (a+b x^4\right )}{b}+\frac {21 f x^2 \left (a+2 b x^4\right )}{b}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f x^{5} + e x^{4} + d x^{3} + c x^{2}\right )} \sqrt {b x^{4} + a}, x\right ) \]
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 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 361, normalized size = 0.98 \[ \frac {\sqrt {b \,x^{4}+a}\, e \,x^{5}}{7}+\frac {\sqrt {b \,x^{4}+a}\, c \,x^{3}}{5}-\frac {2 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{2} e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{21 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b}-\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {3}{2}} c \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {2 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {3}{2}} c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}-\frac {\sqrt {b \,x^{4}+a}\, a f \,x^{2}}{16 b}-\frac {a^{2} f \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{16 b^{\frac {3}{2}}}+\frac {2 \sqrt {b \,x^{4}+a}\, a e x}{21 b}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} f \,x^{2}}{8 b}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}} d}{6 b} \]
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 \[ \int \sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{2}\,{d x} \]
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 \[ \int x^2\,\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.09, size = 212, normalized size = 0.57 \[ \frac {a^{\frac {3}{2}} f x^{2}}{16 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} c x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} f x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a^{2} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 b^{\frac {3}{2}}} + d \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + \frac {b f x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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