Optimal. Leaf size=387 \[ -\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {b} c-9 \sqrt {a} e\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 a^{9/4} \sqrt {a+b x^4}}-\frac {3 \sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}-\frac {f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} e x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {f \sqrt {a+b x^4}}{2 a^2} \]
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Rubi [A] time = 0.61, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {1829, 1833, 1835, 1585, 1584, 1198, 220, 1196, 21, 266, 50, 63, 208} \[ -\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {b} c-9 \sqrt {a} e\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{12 a^{9/4} \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} e x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{b} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} \sqrt {a+b x^4}}+\frac {f \sqrt {a+b x^4}}{2 a^2}-\frac {f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 50
Rule 63
Rule 208
Rule 220
Rule 266
Rule 1196
Rule 1198
Rule 1584
Rule 1585
Rule 1829
Rule 1833
Rule 1835
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3}{x^4 \left (a+b x^4\right )^{3/2}} \, dx &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int \frac {-2 b c-2 b d x-2 b e x^2-2 b f x^3+\frac {b^2 c x^4}{a}-\frac {b^2 e x^6}{a}-\frac {2 b^2 f x^7}{a}}{x^4 \sqrt {a+b x^4}} \, dx}{2 a b}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int \left (\frac {-2 b c-2 b e x^2+\frac {b^2 c x^4}{a}-\frac {b^2 e x^6}{a}}{x^4 \sqrt {a+b x^4}}+\frac {-2 b d-2 b f x^2-\frac {2 b^2 f x^6}{a}}{x^3 \sqrt {a+b x^4}}\right ) \, dx}{2 a b}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {\int \frac {-2 b c-2 b e x^2+\frac {b^2 c x^4}{a}-\frac {b^2 e x^6}{a}}{x^4 \sqrt {a+b x^4}} \, dx}{2 a b}-\frac {\int \frac {-2 b d-2 b f x^2-\frac {2 b^2 f x^6}{a}}{x^3 \sqrt {a+b x^4}} \, dx}{2 a b}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}+\frac {\int \frac {12 a b e x-10 b^2 c x^3+6 b^2 e x^5}{x^3 \sqrt {a+b x^4}} \, dx}{12 a^2 b}+\frac {\int \frac {8 a b f x+8 b^2 f x^5}{x^2 \sqrt {a+b x^4}} \, dx}{8 a^2 b}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}+\frac {\int \frac {12 a b e-10 b^2 c x^2+6 b^2 e x^4}{x^2 \sqrt {a+b x^4}} \, dx}{12 a^2 b}+\frac {\int \frac {8 a b f+8 b^2 f x^4}{x \sqrt {a+b x^4}} \, dx}{8 a^2 b}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}-\frac {\int \frac {20 a b^2 c x-36 a b^2 e x^3}{x \sqrt {a+b x^4}} \, dx}{24 a^3 b}+\frac {f \int \frac {\sqrt {a+b x^4}}{x} \, dx}{a^2}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}-\frac {\int \frac {20 a b^2 c-36 a b^2 e x^2}{\sqrt {a+b x^4}} \, dx}{24 a^3 b}+\frac {f \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^4\right )}{4 a^2}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {f \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}-\frac {\left (3 \sqrt {b} e\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{2 a^{3/2}}-\frac {\left (\sqrt {b} \left (5 \sqrt {b} c-9 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{6 a^2}+\frac {f \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{4 a}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {f \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} e x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 \sqrt [4]{b} 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 )}{2 a^{7/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (5 \sqrt {b} c-9 \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 )}{12 a^{9/4} \sqrt {a+b x^4}}+\frac {f \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{2 a b}\\ &=-\frac {x \left (b c+b d x+b e x^2+b f x^3\right )}{2 a^2 \sqrt {a+b x^4}}+\frac {f \sqrt {a+b x^4}}{2 a^2}-\frac {c \sqrt {a+b x^4}}{3 a^2 x^3}-\frac {d \sqrt {a+b x^4}}{2 a^2 x^2}-\frac {e \sqrt {a+b x^4}}{a^2 x}+\frac {3 \sqrt {b} e x \sqrt {a+b x^4}}{2 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{2 a^{3/2}}-\frac {3 \sqrt [4]{b} 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 )}{2 a^{7/4} \sqrt {a+b x^4}}-\frac {\sqrt [4]{b} \left (5 \sqrt {b} c-9 \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 )}{12 a^{9/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 136, normalized size = 0.35 \[ \frac {-2 a c \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};-\frac {b x^4}{a}\right )-3 x \left (2 a e x \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {b x^4}{a}\right )-a f x^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^4}{a}+1\right )+a d+2 b d x^4\right )}{6 a^2 x^3 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{b^{2} x^{12} + 2 \, a b x^{8} + a^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 383, normalized size = 0.99 \[ -\frac {b e \,x^{3}}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, a^{2}}-\frac {3 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {b}\, e \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {3}{2}}}+\frac {3 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {b}\, e \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{\frac {3}{2}}}-\frac {b c x}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, a^{2}}-\frac {5 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b c \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{6 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a^{2}}-\frac {f \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{2 a^{\frac {3}{2}}}+\frac {f}{2 \sqrt {b \,x^{4}+a}\, a}-\frac {\sqrt {b \,x^{4}+a}\, e}{a^{2} x}-\frac {\left (2 b \,x^{4}+a \right ) d}{2 \sqrt {b \,x^{4}+a}\, a^{2} x^{2}}-\frac {\sqrt {b \,x^{4}+a}\, c}{3 a^{2} x^{3}} \]
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 \[ \int \frac {f x^{3} + e x^{2} + d x + c}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {f\,x^3+e\,x^2+d\,x+c}{x^4\,{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 33.73, size = 321, normalized size = 0.83 \[ d \left (- \frac {1}{2 a \sqrt {b} x^{4} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {\sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{4}} + 1}}\right ) + f \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{4}}{a}}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{3} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} + \frac {a^{2} b x^{4} \log {\left (\frac {b x^{4}}{a} \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}} - \frac {2 a^{2} b x^{4} \log {\left (\sqrt {1 + \frac {b x^{4}}{a}} + 1 \right )}}{4 a^{\frac {9}{2}} + 4 a^{\frac {7}{2}} b x^{4}}\right ) + \frac {c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {e \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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