Optimal. Leaf size=427 \[ \frac {2 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (77 \sqrt {b} c-15 \sqrt {a} e\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 b^{5/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {a^3 f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}+\frac {4 a^2 c x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {2 a x^3 \sqrt {a+b x^4} \left (77 c+45 e x^2\right )}{1155}+\frac {1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 c+9 e x^2\right )+\frac {\left (a+b x^4\right )^{5/2} \left (6 d+5 f x^2\right )}{60 b}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b} \]
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Rubi [A] time = 0.35, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \frac {2 a^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (77 \sqrt {b} c-15 \sqrt {a} e\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 b^{5/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {a^3 f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}+\frac {4 a^2 c x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {2 a x^3 \sqrt {a+b x^4} \left (77 c+45 e x^2\right )}{1155}+\frac {1}{99} x^3 \left (a+b x^4\right )^{3/2} \left (11 c+9 e x^2\right )+\frac {\left (a+b x^4\right )^{5/2} \left (6 d+5 f x^2\right )}{60 b}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b} \]
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 ) \left (a+b x^4\right )^{3/2} \, dx &=\int \left (x^2 \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}+x^3 \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}\right ) \, dx\\ &=\int x^2 \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2} \, dx+\int x^3 \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx\\ &=\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} \operatorname {Subst}\left (\int x (d+f x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )+\frac {1}{33} (2 a) \int x^2 \left (11 c+9 e x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {2 a x^3 \left (77 c+45 e x^2\right ) \sqrt {a+b x^4}}{1155}+\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 d+5 f x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}+\frac {\left (4 a^2\right ) \int \frac {x^2 \left (77 c+45 e x^2\right )}{\sqrt {a+b x^4}} \, dx}{1155}-\frac {(a f) \operatorname {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right )}{12 b}\\ &=\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}+\frac {2 a x^3 \left (77 c+45 e x^2\right ) \sqrt {a+b x^4}}{1155}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 d+5 f x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}-\frac {\left (4 a^2\right ) \int \frac {45 a e-231 b c x^2}{\sqrt {a+b x^4}} \, dx}{3465 b}-\frac {\left (a^2 f\right ) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )}{16 b}\\ &=\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {2 a x^3 \left (77 c+45 e x^2\right ) \sqrt {a+b x^4}}{1155}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 d+5 f x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}-\frac {\left (4 a^{5/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {b}}+\frac {\left (4 a^{5/2} \left (77 \sqrt {b} c-15 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{1155 b}-\frac {\left (a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )}{32 b}\\ &=\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {4 a^2 c x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a x^3 \left (77 c+45 e x^2\right ) \sqrt {a+b x^4}}{1155}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 d+5 f x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}-\frac {4 a^{9/4} 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 )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} \left (77 \sqrt {b} c-15 \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 )}{1155 b^{5/4} \sqrt {a+b x^4}}-\frac {\left (a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )}{32 b}\\ &=\frac {4 a^2 e x \sqrt {a+b x^4}}{77 b}-\frac {a^2 f x^2 \sqrt {a+b x^4}}{32 b}+\frac {4 a^2 c x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2 a x^3 \left (77 c+45 e x^2\right ) \sqrt {a+b x^4}}{1155}-\frac {a f x^2 \left (a+b x^4\right )^{3/2}}{48 b}+\frac {1}{99} x^3 \left (11 c+9 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\left (6 d+5 f x^2\right ) \left (a+b x^4\right )^{5/2}}{60 b}-\frac {a^3 f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{32 b^{3/2}}-\frac {4 a^{9/4} 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 )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{9/4} \left (77 \sqrt {b} c-15 \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 )}{1155 b^{5/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.94, size = 205, normalized size = 0.48 \[ \frac {\sqrt {a+b x^4} \left (-\frac {480 a^2 e x \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{b \sqrt {\frac {b x^4}{a}+1}}+\frac {55 f \left (\sqrt {b} x^2 \left (3 a^2+14 a b x^4+8 b^2 x^8\right )-\frac {3 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {\frac {b x^4}{a}+1}}\right )}{b^{3/2}}+\frac {1760 a c x^3 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+\frac {528 d \left (a+b x^4\right )^2}{b}+\frac {480 e x \left (a+b x^4\right )^2}{b}\right )}{5280} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b f x^{9} + b e x^{8} + b d x^{7} + b c x^{6} + a f x^{5} + a e x^{4} + a d x^{3} + a c x^{2}\right )} \sqrt {b x^{4} + a}, x\right ) \]
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 \[ \int {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 413, normalized size = 0.97 \[ \frac {\sqrt {b \,x^{4}+a}\, b f \,x^{10}}{12}+\frac {\sqrt {b \,x^{4}+a}\, b e \,x^{9}}{11}+\frac {\sqrt {b \,x^{4}+a}\, b c \,x^{7}}{9}+\frac {7 \sqrt {b \,x^{4}+a}\, a f \,x^{6}}{48}+\frac {13 \sqrt {b \,x^{4}+a}\, a e \,x^{5}}{77}+\frac {11 \sqrt {b \,x^{4}+a}\, a c \,x^{3}}{45}-\frac {4 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{3} e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{77 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b}-\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} c \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {\sqrt {b \,x^{4}+a}\, a^{2} f \,x^{2}}{32 b}-\frac {a^{3} f \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{32 b^{\frac {3}{2}}}+\frac {4 \sqrt {b \,x^{4}+a}\, a^{2} e x}{77 b}+\frac {\left (b \,x^{4}+a \right )^{\frac {5}{2}} d}{10 b} \]
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 \[ \int {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.74, size = 398, normalized size = 0.93 \[ \frac {a^{\frac {5}{2}} f x^{2}}{32 b \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} c x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {17 a^{\frac {3}{2}} f x^{6}}{96 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b c x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {a} b e x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} + \frac {11 \sqrt {a} b f x^{10}}{48 \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a^{3} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{32 b^{\frac {3}{2}}} + a d \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + b d \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} f x^{14}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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