Optimal. Leaf size=607 \[ -\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}+\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {5 \sqrt [3]{b} c \sqrt {a+b x^3}}{3 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 d \sqrt {a+b x^3}}{3 a^2} \]
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Rubi [A] time = 0.56, antiderivative size = 607, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1829, 1835, 1832, 266, 63, 208, 1886, 261, 1878, 218, 1877} \[ -\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {5 \sqrt [3]{b} c \sqrt {a+b x^3}}{3 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 d \sqrt {a+b x^3}}{3 a^2}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 218
Rule 261
Rule 266
Rule 1829
Rule 1832
Rule 1835
Rule 1877
Rule 1878
Rule 1886
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {2 \int \frac {-\frac {3 b c}{2}-\frac {3 b d x}{2}-\frac {1}{2} b e x^2-\frac {b^2 c x^3}{2 a}-\frac {3 b^2 d x^4}{2 a}}{x^2 \sqrt {a+b x^3}} \, dx}{3 a b}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\int \frac {3 a b d+a b e x+\frac {5}{2} b^2 c x^2+3 b^2 d x^3}{x \sqrt {a+b x^3}} \, dx}{3 a^2 b}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\int \frac {a b e+\frac {5}{2} b^2 c x+3 b^2 d x^2}{\sqrt {a+b x^3}} \, dx}{3 a^2 b}+\frac {d \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{a}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\int \frac {a b e+\frac {5}{2} b^2 c x}{\sqrt {a+b x^3}} \, dx}{3 a^2 b}+\frac {d \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}+\frac {(b d) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{a^2}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 d \sqrt {a+b x^3}}{3 a^2}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\left (5 b^{2/3} c\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{6 a^2}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a b}-\frac {\left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{6 a^{5/3}}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 d \sqrt {a+b x^3}}{3 a^2}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {5 \sqrt [3]{b} c \sqrt {a+b x^3}}{3 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {2+\sqrt {3}} \left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 121, normalized size = 0.20 \[ \frac {-3 c \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {1}{3},\frac {3}{2};\frac {2}{3};-\frac {b x^3}{a}\right )+2 d x \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^3}{a}+1\right )+e x^2 \left (\sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};-\frac {b x^3}{a}\right )+2\right )}{3 a x \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a} {\left (e x^{2} + d x + c\right )}}{b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 825, normalized size = 1.36 \[ \text {result too large to display} \]
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 \[ \int \frac {e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.80, size = 136, normalized size = 0.22 \[ \frac {2\,d}{3\,a\,\sqrt {b\,x^3+a}}+\frac {d\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,a^{3/2}}-\frac {2\,c\,{\left (\frac {a}{b\,x^3}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {11}{6};\ \frac {17}{6};\ -\frac {a}{b\,x^3}\right )}{11\,x\,{\left (b\,x^3+a\right )}^{3/2}}+\frac {e\,x\,{\left (\frac {b\,x^3}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {3}{2};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (b\,x^3+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.30, size = 267, normalized size = 0.44 \[ d \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{2} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{2} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}}\right ) + \frac {c \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x \Gamma \left (\frac {2}{3}\right )} + \frac {e x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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