Optimal. Leaf size=406 \[ \frac {2 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} f+21 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {12 a^{5/4} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}-\frac {1}{2} a^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {\left (a+b x^4\right )^{3/2} \left (3 c-e x^2\right )}{6 x^2}+\frac {1}{4} \sqrt {a+b x^4} \left (2 a e+3 b c x^2\right )+\frac {3}{4} a \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/2} \left (7 d-f x^2\right )}{7 x}+\frac {2}{35} x \sqrt {a+b x^4} \left (5 a f+21 b d x^2\right )+\frac {12 a \sqrt {b} d x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )} \]
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Rubi [A] time = 0.34, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1833, 1252, 813, 815, 844, 217, 206, 266, 63, 208, 1272, 1177, 1198, 220, 1196} \[ \frac {2 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} f+21 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{b} \sqrt {a+b x^4}}-\frac {12 a^{5/4} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}-\frac {1}{2} a^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {\left (a+b x^4\right )^{3/2} \left (3 c-e x^2\right )}{6 x^2}+\frac {1}{4} \sqrt {a+b x^4} \left (2 a e+3 b c x^2\right )+\frac {3}{4} a \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {\left (a+b x^4\right )^{3/2} \left (7 d-f x^2\right )}{7 x}+\frac {2}{35} x \sqrt {a+b x^4} \left (5 a f+21 b d x^2\right )+\frac {12 a \sqrt {b} d x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 220
Rule 266
Rule 813
Rule 815
Rule 844
Rule 1177
Rule 1196
Rule 1198
Rule 1252
Rule 1272
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^3} \, dx &=\int \left (\frac {\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x^3}+\frac {\left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}}{x^2}\right ) \, dx\\ &=\int \frac {\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x^3} \, dx+\int \frac {\left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}}{x^2} \, dx\\ &=-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+e x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )-\frac {6}{7} \int \left (-a f-7 b d x^2\right ) \sqrt {a+b x^4} \, dx\\ &=\frac {2}{35} x \left (5 a f+21 b d x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{6 x^2}-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}-\frac {2}{35} \int \frac {-10 a^2 f-42 a b d x^2}{\sqrt {a+b x^4}} \, dx-\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-2 a e-6 b c x) \sqrt {a+b x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{4} \left (2 a e+3 b c x^2\right ) \sqrt {a+b x^4}+\frac {2}{35} x \left (5 a f+21 b d x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{6 x^2}-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}-\frac {\operatorname {Subst}\left (\int \frac {-4 a^2 b e-6 a b^2 c x}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{8 b}-\frac {1}{5} \left (12 a^{3/2} \sqrt {b} d\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{35} \left (4 a^{3/2} \left (21 \sqrt {b} d+5 \sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {12 a \sqrt {b} d x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{4} \left (2 a e+3 b c x^2\right ) \sqrt {a+b x^4}+\frac {2}{35} x \left (5 a f+21 b d x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{6 x^2}-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}-\frac {12 a^{5/4} \sqrt [4]{b} d \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 \sqrt {a+b x^4}}+\frac {2 a^{5/4} \left (21 \sqrt {b} d+5 \sqrt {a} f\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 )}{35 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {1}{4} (3 a b c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{2} \left (a^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {12 a \sqrt {b} d x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{4} \left (2 a e+3 b c x^2\right ) \sqrt {a+b x^4}+\frac {2}{35} x \left (5 a f+21 b d x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{6 x^2}-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}-\frac {12 a^{5/4} \sqrt [4]{b} d \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 \sqrt {a+b x^4}}+\frac {2 a^{5/4} \left (21 \sqrt {b} d+5 \sqrt {a} f\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 )}{35 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {1}{4} (3 a b c) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )+\frac {1}{4} \left (a^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=\frac {12 a \sqrt {b} d x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{4} \left (2 a e+3 b c x^2\right ) \sqrt {a+b x^4}+\frac {2}{35} x \left (5 a f+21 b d x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{6 x^2}-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac {3}{4} a \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {12 a^{5/4} \sqrt [4]{b} d \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 \sqrt {a+b x^4}}+\frac {2 a^{5/4} \left (21 \sqrt {b} d+5 \sqrt {a} f\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 )}{35 \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {\left (a^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{2 b}\\ &=\frac {12 a \sqrt {b} d x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{4} \left (2 a e+3 b c x^2\right ) \sqrt {a+b x^4}+\frac {2}{35} x \left (5 a f+21 b d x^2\right ) \sqrt {a+b x^4}-\frac {\left (3 c-e x^2\right ) \left (a+b x^4\right )^{3/2}}{6 x^2}-\frac {\left (7 d-f x^2\right ) \left (a+b x^4\right )^{3/2}}{7 x}+\frac {3}{4} a \sqrt {b} c \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {1}{2} a^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 a^{5/4} \sqrt [4]{b} d \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 \sqrt {a+b x^4}}+\frac {2 a^{5/4} \left (21 \sqrt {b} d+5 \sqrt {a} f\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 )}{35 \sqrt [4]{b} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 194, normalized size = 0.48 \[ \frac {x \left (e x \sqrt {\frac {b x^4}{a}+1} \left (\sqrt {a+b x^4} \left (4 a+b x^4\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )-6 a d \sqrt {a+b x^4} \, _2F_1\left (-\frac {3}{2},-\frac {1}{4};\frac {3}{4};-\frac {b x^4}{a}\right )+6 a f x^2 \sqrt {a+b x^4} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )\right )-3 a c \sqrt {a+b x^4} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b x^4}{a}\right )}{6 x^2 \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt {b x^{4} + a}}{x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 409, normalized size = 1.01 \[ \frac {\sqrt {b \,x^{4}+a}\, b f \,x^{5}}{7}+\frac {\sqrt {b \,x^{4}+a}\, b e \,x^{4}}{6}+\frac {\sqrt {b \,x^{4}+a}\, b d \,x^{3}}{5}+\frac {4 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{2} f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {3}{2}} \sqrt {b}\, d \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {12 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {3}{2}} \sqrt {b}\, d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {\sqrt {b \,x^{4}+a}\, b c \,x^{2}}{4}-\frac {a^{\frac {3}{2}} e \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2}+\frac {3 a \sqrt {b}\, c \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{4}+\frac {3 \sqrt {b \,x^{4}+a}\, a f x}{7}+\frac {2 \sqrt {b \,x^{4}+a}\, a e}{3}-\frac {\sqrt {b \,x^{4}+a}\, a d}{x}-\frac {\sqrt {b \,x^{4}+a}\, a c}{2 x^{2}} \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.48, size = 377, normalized size = 0.93 \[ - \frac {a^{\frac {3}{2}} c}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {a^{\frac {3}{2}} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2} + \frac {a^{\frac {3}{2}} f x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} b c x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} - \frac {\sqrt {a} b c x^{2}}{2 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} b f x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {a^{2} e}{2 \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {3 a \sqrt {b} c \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4} + \frac {a \sqrt {b} e x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} + b e \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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