Integrand size = 28, antiderivative size = 125 \[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {2 \sqrt {-c-d} (b c+a d) \arctan \left (\frac {\sqrt {-c-d} x \sqrt {x+x^4}}{\sqrt {d} (1+x) \left (1-x+x^2\right )}\right )}{3 c^2 \sqrt {d}}+\frac {(a c+2 b c+2 a d) \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 c^2} \]
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Time = 0.22 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2081, 595, 598, 335, 281, 221, 477, 476, 385, 214} \[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\frac {\sqrt {x^4+x} \text {arcsinh}\left (x^{3/2}\right ) (a (c+2 d)+2 b c)}{3 c^2 \sqrt {x^3+1} \sqrt {x}}-\frac {2 \sqrt {x^4+x} \sqrt {c+d} (a d+b c) \text {arctanh}\left (\frac {x^{3/2} \sqrt {c+d}}{\sqrt {d} \sqrt {x^3+1}}\right )}{3 c^2 \sqrt {d} \sqrt {x^3+1} \sqrt {x}}+\frac {a \sqrt {x^4+x} x}{3 c} \]
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Rule 214
Rule 221
Rule 281
Rule 335
Rule 385
Rule 476
Rule 477
Rule 595
Rule 598
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \text {arcsinh}\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 ) \text {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} \text {arcsinh}\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} \text {arctanh}\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*}
Time = 0.67 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.15 \[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\frac {\sqrt {x+x^4} \left (2 \sqrt {c+d} (b c+a d) \text {arctanh}\left (\frac {d-c x^{3/2} \left (x^{3/2}+\sqrt {1+x^3}\right )}{\sqrt {d} \sqrt {c+d}}\right )+\sqrt {d} \left (a c x^{3/2} \sqrt {1+x^3}+(a c+2 b c+2 a d) \log \left (x^{3/2}+\sqrt {1+x^3}\right )\right )\right )}{3 c^2 \sqrt {d} \sqrt {x} \sqrt {1+x^3}} \]
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Time = 0.90 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {a \,x^{2} \left (x^{3}+1\right )}{3 c \sqrt {x \left (x^{3}+1\right )}}+\frac {-\frac {\left (a c +2 a d +2 b c \right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3 c}-\frac {4 \left (a c d +a \,d^{2}+b \,c^{2}+b c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x}\, d}{x^{2} \sqrt {\left (c +d \right ) d}}\right )}{3 c \sqrt {\left (c +d \right ) d}}}{2 c}\) | \(117\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {\sqrt {x^{4}+x}\, \sqrt {\left (c +d \right ) d}\, a c x}{2}+\frac {\left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right ) \left (\left (c +2 d \right ) a +2 b c \right ) \sqrt {\left (c +d \right ) d}}{4}+\left (c +d \right ) \left (a d +b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+x}\, d}{x^{2} \sqrt {\left (c +d \right ) d}}\right )\right )}{3 \sqrt {\left (c +d \right ) d}\, c^{2}}\) | \(122\) |
default | \(\frac {a \left (\frac {x \sqrt {x^{4}+x}}{3}+\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}\right )}{c}+\frac {\left (a d +b c \right ) \left (\ln \left (\frac {x^{2}+\sqrt {x \left (x^{3}+1\right )}}{x^{2}}\right )-\frac {2 \left (c +d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{3}+1\right )}\, d}{x^{2} \sqrt {\left (c +d \right ) d}}\right )}{\sqrt {\left (c +d \right ) d}}-\ln \left (\frac {-x^{2}+\sqrt {x \left (x^{3}+1\right )}}{x^{2}}\right )\right )}{3 c^{2}}\) | \(143\) |
elliptic | \(\text {Expression too large to display}\) | \(701\) |
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Time = 2.48 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.13 \[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\left [\frac {2 \, \sqrt {x^{4} + x} a c x + {\left (b c + a d\right )} \sqrt {\frac {c + d}{d}} \log \left (-\frac {{\left (c^{2} + 8 \, c d + 8 \, d^{2}\right )} x^{6} + 2 \, {\left (3 \, c d + 4 \, d^{2}\right )} x^{3} + d^{2} - 4 \, {\left ({\left (c d + 2 \, d^{2}\right )} x^{4} + d^{2} x\right )} \sqrt {x^{4} + x} \sqrt {\frac {c + d}{d}}}{c^{2} x^{6} - 2 \, c d x^{3} + d^{2}}\right ) + {\left ({\left (a + 2 \, b\right )} c + 2 \, a d\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right )}{6 \, c^{2}}, \frac {2 \, \sqrt {x^{4} + x} a c x + 2 \, {\left (b c + a d\right )} \sqrt {-\frac {c + d}{d}} \arctan \left (\frac {2 \, \sqrt {x^{4} + x} d x \sqrt {-\frac {c + d}{d}}}{{\left (c + 2 \, d\right )} x^{3} + d}\right ) + {\left ({\left (a + 2 \, b\right )} c + 2 \, a d\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right )}{6 \, c^{2}}\right ] \]
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\[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{3} + b\right )}{c x^{3} - d}\, dx \]
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\[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\int { \frac {{\left (a x^{3} + b\right )} \sqrt {x^{4} + x}}{c x^{3} - d} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\frac {\sqrt {x^{4} + x} a x}{3 \, c} + \frac {{\left (a c + 2 \, b c + 2 \, a d\right )} \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, c^{2}} - \frac {{\left (a c + 2 \, b c + 2 \, a d\right )} \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, c^{2}} + \frac {2 \, {\left (b c^{2} + a c d + b c d + a d^{2}\right )} \arctan \left (\frac {d \sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-c d - d^{2}}}\right )}{3 \, \sqrt {-c d - d^{2}} c^{2}} \]
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Timed out. \[ \int \frac {\left (b+a x^3\right ) \sqrt {x+x^4}}{-d+c x^3} \, dx=\int -\frac {\left (a\,x^3+b\right )\,\sqrt {x^4+x}}{d-c\,x^3} \,d x \]
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