Optimal. Leaf size=125 \[ \frac {2 \sqrt {-c-d} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-c-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^2 \sqrt {d}}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) (a c+2 a d+2 b c)}{3 c^2}+\frac {a \sqrt {x^4+x} x}{3 c} \]
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Rubi [A] time = 0.35, antiderivative size = 144, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2056, 581, 584, 329, 275, 215, 466, 465, 377, 208} \begin {gather*} \frac {\sqrt {x^4+x} \sinh ^{-1}\left (x^{3/2}\right ) (a (c+2 d)+2 b c)}{3 c^2 \sqrt {x^3+1} \sqrt {x}}-\frac {2 \sqrt {x^4+x} \sqrt {c+d} (a d+b c) \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {c+d}}{\sqrt {d} \sqrt {x^3+1}}\right )}{3 c^2 \sqrt {d} \sqrt {x^3+1} \sqrt {x}}+\frac {a \sqrt {x^4+x} x}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 215
Rule 275
Rule 329
Rule 377
Rule 465
Rule 466
Rule 581
Rule 584
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx &=\frac {\sqrt {x+x^4} \int \frac {\sqrt {x} \sqrt {1+x^3} \left (b+a x^3\right )}{-d+c x^3} \, dx}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\sqrt {x+x^4} \int \frac {\sqrt {x} \left (\frac {3}{2} (2 b c+a d)+\frac {3}{2} (2 b c+a (c+2 d)) x^3\right )}{\sqrt {1+x^3} \left (-d+c x^3\right )} \, dx}{3 c \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\sqrt {x+x^4} \int \left (\frac {3 (2 b c+a (c+2 d)) \sqrt {x}}{2 c \sqrt {1+x^3}}+\frac {\left (\frac {3}{2} c (2 b c+a d)+\frac {3}{2} d (2 b c+a (c+2 d))\right ) \sqrt {x}}{c \sqrt {1+x^3} \left (-d+c x^3\right )}\right ) \, dx}{3 c \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\left ((c+d) (b c+a d) \sqrt {x+x^4}\right ) \int \frac {\sqrt {x}}{\sqrt {1+x^3} \left (-d+c x^3\right )} \, dx}{c^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left ((2 b c+a (c+2 d)) \sqrt {x+x^4}\right ) \int \frac {\sqrt {x}}{\sqrt {1+x^3}} \, dx}{2 c^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\left (2 (c+d) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^6} \left (-d+c x^6\right )} \, dx,x,\sqrt {x}\right )}{c^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left ((2 b c+a (c+2 d)) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^6}} \, dx,x,\sqrt {x}\right )}{c^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\left (2 (c+d) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (-d+c x^2\right )} \, dx,x,x^{3/2}\right )}{3 c^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left ((2 b c+a (c+2 d)) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 c^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {(2 b c+a (c+2 d)) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left (2 (c+d) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-d-(-c-d) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 c^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {(2 b c+a (c+2 d)) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c^2 \sqrt {x} \sqrt {1+x^3}}-\frac {2 \sqrt {c+d} (b c+a d) \sqrt {x+x^4} \tanh ^{-1}\left (\frac {\sqrt {c+d} x^{3/2}}{\sqrt {d} \sqrt {1+x^3}}\right )}{3 c^2 \sqrt {d} \sqrt {x} \sqrt {1+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.35, size = 170, normalized size = 1.36 \begin {gather*} -\frac {x \sqrt {x^4+x} \left (x^3 \sqrt {-\frac {x^3 (c+d)}{d}} (a (c+2 d)+2 b c) F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};-x^3,\frac {c x^3}{d}\right )+3 (a d+2 b c) \sin ^{-1}\left (\frac {\sqrt {-\frac {x^3 (c+d)}{d}}}{\sqrt {1-\frac {c x^3}{d}}}\right )-3 a d \sqrt {x^3+1} \sqrt {-\frac {x^3 (c+d)}{d}}\right )}{9 c d \sqrt {x^3+1} \sqrt {-\frac {x^3 (c+d)}{d}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.76, size = 125, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-c-d} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-c-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^2 \sqrt {d}}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) (a c+2 a d+2 b c)}{3 c^2}+\frac {a \sqrt {x^4+x} x}{3 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 4.06, size = 266, normalized size = 2.13 \begin {gather*} \left [\frac {2 \, \sqrt {x^{4} + x} a c x + {\left (b c + a d\right )} \sqrt {\frac {c + d}{d}} \log \left (-\frac {{\left (c^{2} + 8 \, c d + 8 \, d^{2}\right )} x^{6} + 2 \, {\left (3 \, c d + 4 \, d^{2}\right )} x^{3} + d^{2} - 4 \, {\left ({\left (c d + 2 \, d^{2}\right )} x^{4} + d^{2} x\right )} \sqrt {x^{4} + x} \sqrt {\frac {c + d}{d}}}{c^{2} x^{6} - 2 \, c d x^{3} + d^{2}}\right ) + {\left ({\left (a + 2 \, b\right )} c + 2 \, a d\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right )}{6 \, c^{2}}, \frac {2 \, \sqrt {x^{4} + x} a c x + 2 \, {\left (b c + a d\right )} \sqrt {-\frac {c + d}{d}} \arctan \left (\frac {2 \, \sqrt {x^{4} + x} d x \sqrt {-\frac {c + d}{d}}}{{\left (c + 2 \, d\right )} x^{3} + d}\right ) + {\left ({\left (a + 2 \, b\right )} c + 2 \, a d\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right )}{6 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 128, normalized size = 1.02 \begin {gather*} \frac {\sqrt {x^{4} + x} a x}{3 \, c} + \frac {{\left (a c + 2 \, b c + 2 \, a d\right )} \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, c^{2}} - \frac {{\left (a c + 2 \, b c + 2 \, a d\right )} \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, c^{2}} + \frac {2 \, {\left (b c^{2} + a c d + b c d + a d^{2}\right )} \arctan \left (\frac {d \sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-c d - d^{2}}}\right )}{3 \, \sqrt {-c d - d^{2}} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 970, normalized size = 7.76
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 \begin {gather*} \int \frac {{\left (a x^{3} + b\right )} \sqrt {x^{4} + x}}{c x^{3} - 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 = -0.01 \begin {gather*} \int -\frac {\left (a\,x^3+b\right )\,\sqrt {x^4+x}}{d-c\,x^3} \,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 {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{3} + b\right )}{c x^{3} - d}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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