Integrand size = 22, antiderivative size = 349 \[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt [4]{-a} d\right ) (2+n)} \]
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Time = 0.53 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6857, 845, 70} \[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right )}-\frac {(c+d x)^{n+2} \operatorname {Hypergeometric2F1}\left (1,n+2,n+3,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (n+2) \left (\sqrt [4]{-a} d+\sqrt [4]{b} c\right )} \]
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Rule 70
Rule 845
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x (c+d x)^{1+n}}{2 \left (-\sqrt {-a} \sqrt {b}+b x^2\right )}+\frac {x (c+d x)^{1+n}}{2 \left (\sqrt {-a} \sqrt {b}+b x^2\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {x (c+d x)^{1+n}}{-\sqrt {-a} \sqrt {b}+b x^2} \, dx+\frac {1}{2} \int \frac {x (c+d x)^{1+n}}{\sqrt {-a} \sqrt {b}+b x^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {(c+d x)^{1+n}}{2 b^{3/4} \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}+\frac {(c+d x)^{1+n}}{2 b^{3/4} \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}\right ) \, dx+\frac {1}{2} \int \left (-\frac {(c+d x)^{1+n}}{2 b^{3/4} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac {(c+d x)^{1+n}}{2 b^{3/4} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {(c+d x)^{1+n}}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}-\frac {\int \frac {(c+d x)^{1+n}}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac {\int \frac {(c+d x)^{1+n}}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}}+\frac {\int \frac {(c+d x)^{1+n}}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{3/4}} \\ & = -\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right ) (2+n)}-\frac {(c+d x)^{2+n} \, _2F_1\left (1,2+n;3+n;\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt [4]{-a} d\right ) (2+n)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=\frac {(c+d x)^{2+n} \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+n,3+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (2+n)} \]
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\[\int \frac {x^{3} \left (d x +c \right )^{1+n}}{b \,x^{4}+a}d x\]
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\[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=\int { \frac {{\left (d x + c\right )}^{n + 1} x^{3}}{b x^{4} + a} \,d x } \]
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Timed out. \[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=\int { \frac {{\left (d x + c\right )}^{n + 1} x^{3}}{b x^{4} + a} \,d x } \]
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\[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=\int { \frac {{\left (d x + c\right )}^{n + 1} x^{3}}{b x^{4} + a} \,d x } \]
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Timed out. \[ \int \frac {x^3 (c+d x)^{1+n}}{a+b x^4} \, dx=\int \frac {x^3\,{\left (c+d\,x\right )}^{n+1}}{b\,x^4+a} \,d x \]
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