Optimal. Leaf size=403 \[ \frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} f+15 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}+\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \sqrt {a+b x^4} \left (8 c+3 e x^2\right )+\frac {1}{24} \left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )+\frac {2}{105} a x \sqrt {a+b x^4} \left (15 d+7 f x^2\right )+\frac {1}{63} x \left (a+b x^4\right )^{3/2} \left (9 d+7 f x^2\right ) \]
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Rubi [A] time = 0.35, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {1833, 1252, 815, 844, 217, 206, 266, 63, 208, 1177, 1198, 220, 1196} \[ \frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} f+15 \sqrt {b} d\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}-\frac {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}+\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \sqrt {a+b x^4} \left (8 c+3 e x^2\right )+\frac {1}{24} \left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )+\frac {2}{105} a x \sqrt {a+b x^4} \left (15 d+7 f x^2\right )+\frac {1}{63} x \left (a+b x^4\right )^{3/2} \left (9 d+7 f x^2\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 208
Rule 217
Rule 220
Rule 266
Rule 815
Rule 844
Rule 1177
Rule 1196
Rule 1198
Rule 1252
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} \, dx &=\int \left (\frac {\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x}+\left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}\right ) \, dx\\ &=\int \frac {\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x} \, dx+\int \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx\\ &=\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{21} \int \left (18 a d+14 a f x^2\right ) \sqrt {a+b x^4} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {(c+e x) \left (a+b x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{315} \int \frac {180 a^2 d+84 a^2 f x^2}{\sqrt {a+b x^4}} \, dx+\frac {\operatorname {Subst}\left (\int \frac {(4 a b c+3 a b e x) \sqrt {a+b x^2}}{x} \, dx,x,x^2\right )}{8 b}\\ &=\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\operatorname {Subst}\left (\int \frac {8 a^2 b^2 c+3 a^2 b^2 e x}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 b^2}-\frac {\left (4 a^{5/2} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {b}}+\frac {1}{105} \left (4 a^2 \left (15 d+\frac {7 \sqrt {a} f}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac {4 a^{9/4} f \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^{7/4} \left (15 \sqrt {b} d+7 \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 )}{105 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{2} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (3 a^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac {4 a^{9/4} f \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^{7/4} \left (15 \sqrt {b} d+7 \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 )}{105 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{4} \left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )+\frac {1}{16} \left (3 a^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {4 a^{9/4} f \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^{7/4} \left (15 \sqrt {b} d+7 \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 )}{105 b^{3/4} \sqrt {a+b x^4}}+\frac {\left (a^2 c\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 {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {4 a^{9/4} f \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^{7/4} \left (15 \sqrt {b} d+7 \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 )}{105 b^{3/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 224, normalized size = 0.56 \[ \frac {1}{6} c \left (\sqrt {a+b x^4} \left (4 a+b x^4\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+\frac {1}{16} e \sqrt {a+b x^4} \left (\frac {3 a^{3/2} \sinh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {\frac {b x^4}{a}+1}}+5 a x^2+2 b x^6\right )+\frac {a d x \sqrt {a+b x^4} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt {\frac {b x^4}{a}+1}}+\frac {a f x^3 \sqrt {a+b x^4} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {b x^4}{a}\right )}{3 \sqrt {\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt {b x^{4} + a}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 411, normalized size = 1.02 \[ \frac {\sqrt {b \,x^{4}+a}\, b f \,x^{7}}{9}+\frac {\sqrt {b \,x^{4}+a}\, b e \,x^{6}}{8}+\frac {\sqrt {b \,x^{4}+a}\, b d \,x^{5}}{7}+\frac {\sqrt {b \,x^{4}+a}\, b c \,x^{4}}{6}+\frac {11 \sqrt {b \,x^{4}+a}\, a f \,x^{3}}{45}-\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} f \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{\frac {5}{2}} f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {4 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, a^{2} d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {5 \sqrt {b \,x^{4}+a}\, a e \,x^{2}}{16}+\frac {3 a^{2} e \ln \left (\sqrt {b}\, x^{2}+\sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}-\frac {a^{\frac {3}{2}} c \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2}+\frac {3 \sqrt {b \,x^{4}+a}\, a d x}{7}+\frac {2 \sqrt {b \,x^{4}+a}\, a c}{3} \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 31.32, size = 405, normalized size = 1.00 \[ - \frac {a^{\frac {3}{2}} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2} + \frac {a^{\frac {3}{2}} d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} e x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} + \frac {a^{\frac {3}{2}} e x^{2}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} f x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} b d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} b e x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b f x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {a^{2} c}{2 \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {3 a^{2} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {a \sqrt {b} c x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} + b c \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} e x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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