Optimal. Leaf size=887 \[ \frac {d \sqrt [6]{\frac {1-b x}{c+x}} (c+x)}{b c}-\frac {\left (b c^2+7 c+6 b\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c^2}-\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{c^2 (c-b) \sqrt [6]{1-c^2}}-\frac {\sqrt [6]{2} \sqrt {3} \left (b^2+d\right ) \tan ^{-1}\left (\frac {2^{5/6} \sqrt [6]{b c-1} \sqrt [6]{\frac {1-b x}{c+x}}-\sqrt [6]{b}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{b c-1}}-\frac {\sqrt [6]{2} \sqrt {3} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{b}+2^{5/6} \sqrt [6]{b c-1} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{b c-1}}+\frac {\left (b c^2+7 c+6 b\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{\frac {1-b x}{c+x}}-\sqrt [3]{b}}\right )}{6 b^{11/6} c^2}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b+c}-\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{c^2 (c-b) \sqrt [6]{1-c^2}}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{b c-1} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{b c-1}}-\frac {\left (b c^2+7 c+6 b\right ) d \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b}+\sqrt [3]{\frac {1-b x}{c+x}}}\right )}{2 \sqrt {3} b^{11/6} c^2}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tanh ^{-1}\left (\frac {2^{5/6} \sqrt [6]{b} \sqrt [6]{b c-1} \sqrt [6]{\frac {1-b x}{c+x}}}{2 \sqrt [3]{b}+2^{2/3} \sqrt [3]{b c-1} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{b^{11/6} (b-c) \sqrt [6]{b c-1}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt [3]{b+c}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}\right )}{c^2 (c-b) \sqrt [6]{1-c^2}} \]
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Rubi [A] time = 5.28, antiderivative size = 1549, normalized size of antiderivative = 1.75, number of steps used = 45, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6, 6725, 199, 209, 634, 618, 204, 628, 203, 205}
result too large to display
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 199
Rule 203
Rule 204
Rule 205
Rule 209
Rule 618
Rule 628
Rule 634
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [6]{\frac {1-b x}{c+x}} \left (1+d x^2\right )}{(1+b x) (1+c x)} \, dx &=(6 (1+b c)) \operatorname {Subst}\left (\int \frac {x^6 \left (b^2+2 b x^6+x^{12}+d \left (-1+c x^6\right )^2\right )}{\left (b+x^6\right )^2 \left (b+c+x^6-c^2 x^6\right ) \left (-x^6+b \left (-2+c x^6\right )\right )} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )\\ &=(6 (1+b c)) \operatorname {Subst}\left (\int \frac {x^6 \left (b^2+2 b x^6+x^{12}+d \left (-1+c x^6\right )^2\right )}{\left (b+x^6\right )^2 \left (b+c+\left (1-c^2\right ) x^6\right ) \left (-x^6+b \left (-2+c x^6\right )\right )} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )\\ &=(6 (1+b c)) \operatorname {Subst}\left (\int \left (\frac {d}{c \left (b+x^6\right )^2}-\frac {\left (b+2 c+b c^2\right ) d}{b c^2 (1+b c) \left (b+x^6\right )}+\frac {2 \left (-b^2-d\right )}{b (b-c) (1+b c) \left (2 b+(1-b c) x^6\right )}+\frac {(b+c) \left (c^2+d\right )}{(b-c) c^2 (1+b c) \left (b+c+\left (1-c^2\right ) x^6\right )}\right ) \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )\\ &=\frac {(6 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{\left (b+x^6\right )^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{c}-\frac {\left (6 \left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{b+x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b c^2}-\frac {\left (12 \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 b+(1-b c) x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b (b-c)}+\frac {\left (6 (b+c) \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b+c+\left (1-c^2\right ) x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{b+x^6} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b c}-\frac {\left (2 \left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}-\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} c^2}-\frac {\left (2 \left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}+\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} c^2}-\frac {\left (2 \left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{5/3} c^2}-\frac {\left (2 \sqrt [6]{2} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{2} \sqrt [6]{b}-\frac {1}{2} \sqrt {3} \sqrt [6]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} (b-c)}-\frac {\left (2 \sqrt [6]{2} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{2} \sqrt [6]{b}+\frac {1}{2} \sqrt {3} \sqrt [6]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{11/6} (b-c)}-\frac {\left (2 \sqrt [3]{2} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{b^{5/3} (b-c)}+\frac {\left (2 \sqrt [6]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b+c}-\frac {1}{2} \sqrt {3} \sqrt [6]{1-c^2} x}{\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}+\frac {\left (2 \sqrt [6]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b+c}+\frac {1}{2} \sqrt {3} \sqrt [6]{1-c^2} x}{\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}+\frac {\left (2 \sqrt [3]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b+c}+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{(b-c) c^2}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}-\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{3 b^{11/6} c}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {\sqrt [6]{b}+\frac {\sqrt {3} x}{2}}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{3 b^{11/6} c}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{3 b^{5/3} c}+\frac {\left (\sqrt {3} \left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}-\frac {\left (\sqrt {3} \left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}-\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{5/3} c^2}-\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 b^{5/3} c^2}-\frac {\left (b^2+d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{2/3} b^{5/3} (b-c)}-\frac {\left (b^2+d\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{2/3} b^{5/3} (b-c)}+\frac {\left (\sqrt {3} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c}+2 \sqrt [3]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\left (\sqrt {3} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c}+2 \sqrt [3]{1-b c} x}{\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} x+\sqrt [3]{1-b c} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {\left (\sqrt [3]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2}+\frac {\left (\sqrt [3]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2}-\frac {\left (\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2}+2 \sqrt [3]{1-c^2} x}{\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\left (\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2}+2 \sqrt [3]{1-c^2} x}{\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} x+\sqrt [3]{1-c^2} x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}-\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}-\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{b}+2 x}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{12 b^{5/3} c}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} x+x^2} \, dx,x,\sqrt [6]{\frac {1-b x}{c+x}}\right )}{12 b^{5/3} c}-\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} c^2}+\frac {\left (\left (b+2 c+b c^2\right ) d\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} c^2}-\frac {\left (\sqrt [6]{2} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {\left (\sqrt [6]{2} \left (b^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {\left (\sqrt [6]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b+c}}\right )}{\sqrt {3} (b-c) c^2 \sqrt [6]{1-c^2}}-\frac {\left (\sqrt [6]{b+c} \left (c^2+d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b+c}}\right )}{\sqrt {3} (b-c) c^2 \sqrt [6]{1-c^2}}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}-\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}+\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (1+b c) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {5 (1+b c) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}-\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{6 \sqrt {3} b^{11/6} c}-\frac {(5 (1+b c) d) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{6 \sqrt {3} b^{11/6} c}\\ &=\frac {d (c+x) \sqrt [6]{\frac {1-b x}{c+x}}}{b c}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c}-\frac {2 \left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{b^{11/6} c^2}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}+\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (1+b c) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{6 b^{11/6} c}+\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}+\frac {5 (1+b c) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{6 b^{11/6} c}-\frac {\left (b+2 c+b c^2\right ) d \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} c^2}+\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {3\ 2^{5/6} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )\right )}{b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )\right )}{(b-c) c^2 \sqrt [6]{1-c^2}}-\frac {5 (1+b c) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}+\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {5 (1+b c) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{4 \sqrt {3} b^{11/6} c}-\frac {\sqrt {3} \left (b+2 c+b c^2\right ) d \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 b^{11/6} c^2}+\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}-\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \left (b^2+d\right ) \log \left (\sqrt [3]{2} \sqrt [3]{b}+\sqrt [6]{2} \sqrt {3} \sqrt [6]{b} \sqrt [6]{1-b c} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-b c} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2^{5/6} b^{11/6} (b-c) \sqrt [6]{1-b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}-\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}+\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \log \left (\sqrt [3]{b+c}+\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}\right )}{2 (b-c) c^2 \sqrt [6]{1-c^2}}\\ \end {align*}
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Mathematica [C] time = 0.48, size = 322, normalized size = 0.36 \begin {gather*} \frac {6}{5} (c+x) \sqrt [6]{\frac {1-b x}{c+x}} \left (\frac {\left (b^2+d\right ) \left (\frac {1-b x}{b c+1}\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {b (c+x)}{b c+1}\right )}{b (b-c) (b x-1)}+\frac {2 \left (b^2+d\right ) \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-\frac {2 b (c+x)}{(b c-1) (b x-1)}\right )}{b (b-c) (b c-1) (b x-1)}+\frac {b \left (c^2+d\right ) \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {b (c+x)}{b c+1}\right )}{c^2 (b-c) (b c+1) \sqrt [6]{\frac {1-b x}{b c+1}}}-\frac {(b+c) \left (c^2+d\right ) \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-\frac {(b+c) (c+x)}{\left (c^2-1\right ) (b x-1)}\right )}{c^2 \left (c^2-1\right ) (b-c) (b x-1)}+\frac {d \, _2F_1\left (-\frac {1}{6},\frac {5}{6};\frac {11}{6};\frac {b (c+x)}{b c+1}\right )}{b c \sqrt [6]{\frac {1-b x}{b c+1}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.48, size = 898, normalized size = 1.01 \begin {gather*} \frac {(1+b c) d \sqrt [6]{\frac {1-b x}{c+x}}}{b c \left (b+\frac {1-b x}{c+x}\right )}-\frac {\left (6 b+7 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b}}\right )}{3 b^{11/6} c^2}-\frac {2 \sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c}}\right )}{c^2 (-b+c) \sqrt [6]{1-c^2}}+\frac {\sqrt [6]{2} \sqrt {3} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}-\frac {\sqrt [6]{2} \sqrt {3} \left (b^2+d\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt {3} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}+\frac {\left (6 b+7 c+b c^2\right ) d \tan ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{\frac {1-b x}{c+x}}}{\sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}\right )}{6 b^{11/6} c^2}+\frac {\sqrt [6]{b+c} \left (c^2+d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{b+c}-\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}{\sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}\right )}{c^2 (-b+c) \sqrt [6]{1-c^2}}-\frac {2 \sqrt [6]{2} \left (b^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [6]{2} \sqrt [6]{b}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}-\frac {\left (6 b+7 c+b c^2\right ) d \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b}+\sqrt [3]{\frac {1-b x}{c+x}}}\right )}{2 \sqrt {3} b^{11/6} c^2}-\frac {\sqrt [6]{2} \left (b^2+d\right ) \tanh ^{-1}\left (\frac {2^{5/6} \sqrt [6]{b} \sqrt [6]{-1+b c} \sqrt [6]{\frac {1-b x}{c+x}}}{2 \sqrt [3]{b}+2^{2/3} \sqrt [3]{-1+b c} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{b^{11/6} (b-c) \sqrt [6]{-1+b c}}-\frac {\sqrt {3} \sqrt [6]{b+c} \left (c^2+d\right ) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b+c} \sqrt [6]{1-c^2} \sqrt [6]{\frac {1-b x}{c+x}}}{\sqrt [3]{b+c}+\sqrt [3]{1-c^2} \sqrt [3]{\frac {1-b x}{c+x}}}\right )}{c^2 (-b+c) \sqrt [6]{1-c^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (\frac {-b x +1}{c +x}\right )^{\frac {1}{6}} \left (d \,x^{2}+1\right )}{\left (b x +1\right ) \left (c x +1\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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