Optimal. Leaf size=367 \[ \frac {\sqrt [3]{c-d} (b c-a d) \log \left (-\sqrt [3]{d} \sqrt [3]{x^3+x} x \sqrt [3]{c-d}+x^2 (c-d)^{2/3}+d^{2/3} \left (x^3+x\right )^{2/3}\right )}{4 c^2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^3+x}-x\right ) (-a c+3 a d-3 b c)}{6 c^2}-\frac {\sqrt [3]{c-d} (b c-a d) \log \left (x \sqrt [3]{c-d}+\sqrt [3]{d} \sqrt [3]{x^3+x}\right )}{2 c^2 \sqrt [3]{d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right ) (a c-3 a d+3 b c)}{2 \sqrt {3} c^2}-\frac {\sqrt {3} \sqrt [3]{c-d} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{c-d}}{x \sqrt [3]{c-d}-2 \sqrt [3]{d} \sqrt [3]{x^3+x}}\right )}{2 c^2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) (a c-3 a d+3 b c)}{12 c^2}+\frac {a \sqrt [3]{x^3+x} x}{2 c} \]
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Rubi [A] time = 0.86, antiderivative size = 523, normalized size of antiderivative = 1.43, number of steps used = 22, number of rules used = 16, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2056, 581, 584, 329, 275, 331, 292, 31, 634, 618, 204, 628, 466, 465, 494, 617} \begin {gather*} \frac {\sqrt [3]{x^3+x} \sqrt [3]{c-d} (b c-a d) \log \left (\frac {x^{4/3} (c-d)^{2/3}}{\left (x^2+1\right )^{2/3}}-\frac {\sqrt [3]{d} x^{2/3} \sqrt [3]{c-d}}{\sqrt [3]{x^2+1}}+d^{2/3}\right )}{4 c^2 \sqrt [3]{d} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{x^3+x} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right ) (a (c-3 d)+3 b c)}{6 c^2 \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{x^3+x} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right ) (a (c-3 d)+3 b c)}{12 c^2 \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{x^3+x} \sqrt [3]{c-d} (b c-a d) \log \left (\frac {x^{2/3} \sqrt [3]{c-d}}{\sqrt [3]{x^2+1}}+\sqrt [3]{d}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{x^3+x} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right ) (a (c-3 d)+3 b c)}{2 \sqrt {3} c^2 \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt {3} \sqrt [3]{x^3+x} \sqrt [3]{c-d} (b c-a d) \tan ^{-1}\left (\frac {\sqrt [3]{d}-\frac {2 x^{2/3} \sqrt [3]{c-d}}{\sqrt [3]{x^2+1}}}{\sqrt {3} \sqrt [3]{d}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {a \sqrt [3]{x^3+x} x}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 465
Rule 466
Rule 494
Rule 581
Rule 584
Rule 617
Rule 618
Rule 628
Rule 634
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (b+a x^2\right ) \sqrt [3]{x+x^3}}{d+c x^2} \, dx &=\frac {\sqrt [3]{x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a x^2\right )}{d+c x^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\sqrt [3]{x+x^3} \int \frac {\sqrt [3]{x} \left (\frac {2}{3} (3 b c-2 a d)+\frac {2}{3} (3 b c+a (c-3 d)) x^2\right )}{\left (1+x^2\right )^{2/3} \left (d+c x^2\right )} \, dx}{2 c \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\sqrt [3]{x+x^3} \int \left (\frac {2 (3 b c+a (c-3 d)) \sqrt [3]{x}}{3 c \left (1+x^2\right )^{2/3}}+\frac {\left (-\frac {2}{3} (3 b c+a (c-3 d)) d+\frac {2}{3} c (3 b c-2 a d)\right ) \sqrt [3]{x}}{c \left (1+x^2\right )^{2/3} \left (d+c x^2\right )}\right ) \, dx}{2 c \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{3 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((c-d) (b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3} \left (d+c x^2\right )} \, dx}{c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3 (c-d) (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3} \left (d+c x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3 (c-d) (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3} \left (d+c x^3\right )} \, dx,x,x^{2/3}\right )}{2 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3 (c-d) (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{d-(-c+d) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((c-d)^{2/3} (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{d}+\sqrt [3]{c-d} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((c-d)^{2/3} (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{d}+\sqrt [3]{c-d} x}{d^{2/3}-\sqrt [3]{c-d} \sqrt [3]{d} x+(c-d)^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {(3 b c+a (c-3 d)) \sqrt [3]{x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3} \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{c-d} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3 (c-d)^{2/3} (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{d^{2/3}-\sqrt [3]{c-d} \sqrt [3]{d} x+(c-d)^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (\sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {-\sqrt [3]{c-d} \sqrt [3]{d}+2 (c-d)^{2/3} x}{d^{2/3}-\sqrt [3]{c-d} \sqrt [3]{d} x+(c-d)^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {(3 b c+a (c-3 d)) \sqrt [3]{x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {(3 b c+a (c-3 d)) \sqrt [3]{x+x^3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3} \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{c-d} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3} \log \left (d^{2/3}+\frac {(c-d)^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{c-d} \sqrt [3]{d} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((3 b c+a (c-3 d)) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (3 \sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c-d} x^{2/3}}{\sqrt [3]{d} \sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {(3 b c+a (c-3 d)) \sqrt [3]{x+x^3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\sqrt {3} \sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{c-d} x^{2/3}}{\sqrt [3]{d} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {(3 b c+a (c-3 d)) \sqrt [3]{x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {(3 b c+a (c-3 d)) \sqrt [3]{x+x^3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 c^2 \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3} \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{c-d} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\sqrt [3]{c-d} (b c-a d) \sqrt [3]{x+x^3} \log \left (d^{2/3}+\frac {(c-d)^{2/3} x^{4/3}}{\left (1+x^2\right )^{2/3}}-\frac {\sqrt [3]{c-d} \sqrt [3]{d} x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 c^2 \sqrt [3]{d} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 168, normalized size = 0.46 \begin {gather*} \frac {x \sqrt [3]{x^3+x} \left (2 x^2 \left (\frac {c x^2}{d}+1\right )^{2/3} (a (c-3 d)+3 b c) F_1\left (\frac {5}{3};\frac {2}{3},1;\frac {8}{3};-x^2,-\frac {c x^2}{d}\right )+5 \left ((3 b c-2 a d) \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {(c-d) x^2}{c x^2+d}\right )+2 a d \sqrt [3]{x^2+1} \left (\frac {c x^2}{d}+1\right )^{2/3}\right )\right )}{20 c d \sqrt [3]{x^2+1} \left (\frac {c x^2}{d}+1\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.20, size = 367, normalized size = 1.00 \begin {gather*} \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {(a c+3 b c-3 a d) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 \sqrt {3} c^2}-\frac {\sqrt {3} \sqrt [3]{c-d} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c-d} x}{\sqrt [3]{c-d} x-2 \sqrt [3]{d} \sqrt [3]{x+x^3}}\right )}{2 c^2 \sqrt [3]{d}}+\frac {(-a c-3 b c+3 a d) \log \left (-x+\sqrt [3]{x+x^3}\right )}{6 c^2}-\frac {\sqrt [3]{c-d} (b c-a d) \log \left (\sqrt [3]{c-d} x+\sqrt [3]{d} \sqrt [3]{x+x^3}\right )}{2 c^2 \sqrt [3]{d}}+\frac {(a c+3 b c-3 a d) \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{12 c^2}+\frac {\sqrt [3]{c-d} (b c-a d) \log \left ((c-d)^{2/3} x^2-\sqrt [3]{c-d} \sqrt [3]{d} x \sqrt [3]{x+x^3}+d^{2/3} \left (x+x^3\right )^{2/3}\right )}{4 c^2 \sqrt [3]{d}} \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 [A] time = 0.89, size = 354, normalized size = 0.96 \begin {gather*} \frac {a x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}}{2 \, c} + \frac {{\left (b c^{2} - a c d - b c d + a d^{2}\right )} \left (-\frac {c - d}{d}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {c - d}{d}\right )^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right )}{2 \, {\left (c^{3} - c^{2} d\right )}} + \frac {\sqrt {3} {\left (a c + 3 \, b c - 3 \, a d\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right )}{6 \, c^{2}} + \frac {{\left (a c + 3 \, b c - 3 \, a d\right )} \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}{12 \, c^{2}} - \frac {{\left (a c + 3 \, b c - 3 \, a d\right )} \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right )}{6 \, c^{2}} - \frac {{\left (\sqrt {3} {\left (-c d^{2} + d^{3}\right )}^{\frac {1}{3}} b c - \sqrt {3} {\left (-c d^{2} + d^{3}\right )}^{\frac {1}{3}} a d\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {c - d}{d}\right )^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c - d}{d}\right )^{\frac {1}{3}}}\right )}{2 \, c^{2} d} - \frac {{\left ({\left (-c d^{2} + d^{3}\right )}^{\frac {1}{3}} b c - {\left (-c d^{2} + d^{3}\right )}^{\frac {1}{3}} a d\right )} \log \left (\left (-\frac {c - d}{d}\right )^{\frac {2}{3}} + \left (-\frac {c - d}{d}\right )^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )}{4 \, c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b \right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{c \,x^{2}+d}\, dx\]
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 \begin {gather*} \int \frac {{\left (a x^{2} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{c x^{2} + d}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (a\,x^2+b\right )\,{\left (x^3+x\right )}^{1/3}}{c\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{2} + b\right )}{c x^{2} + d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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