Integrand size = 20, antiderivative size = 293 \[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\frac {(e+f x)^{1+n}}{b f (1+n)}+\frac {\sqrt [3]{a} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}+\frac {\sqrt [3]{a} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}-\frac {\sqrt [3]{a} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 b \left (\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f\right ) (1+n)} \]
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Time = 0.38 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6857, 70} \[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\frac {\sqrt [3]{a} (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac {\sqrt [3]{a} (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac {\sqrt [3]{a} (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 b (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )}+\frac {(e+f x)^{n+1}}{b f (n+1)} \]
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Rule 70
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(e+f x)^n}{b}-\frac {a (e+f x)^n}{b \left (a+b x^3\right )}\right ) \, dx \\ & = \frac {(e+f x)^{1+n}}{b f (1+n)}-\frac {a \int \frac {(e+f x)^n}{a+b x^3} \, dx}{b} \\ & = \frac {(e+f x)^{1+n}}{b f (1+n)}-\frac {a \int \left (-\frac {(e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {(e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {(e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{b} \\ & = \frac {(e+f x)^{1+n}}{b f (1+n)}+\frac {\sqrt [3]{a} \int \frac {(e+f x)^n}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 b}+\frac {\sqrt [3]{a} \int \frac {(e+f x)^n}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 b}+\frac {\sqrt [3]{a} \int \frac {(e+f x)^n}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 b} \\ & = \frac {(e+f x)^{1+n}}{b f (1+n)}+\frac {\sqrt [3]{a} (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}+\frac {\sqrt [3]{a} (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}-\frac {\sqrt [3]{a} (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 b \left (\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f\right ) (1+n)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.82 \[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\frac {(e+f x)^{1+n} \left (\frac {3}{f}+\frac {\sqrt [3]{a} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} f}+\frac {\sqrt [3]{a} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}-\frac {\sqrt [3]{a} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 b (1+n)} \]
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\[\int \frac {x^{3} \left (f x +e \right )^{n}}{b \,x^{3}+a}d x\]
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\[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{3}}{b x^{3} + a} \,d x } \]
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Timed out. \[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{3}}{b x^{3} + a} \,d x } \]
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\[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{3}}{b x^{3} + a} \,d x } \]
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Timed out. \[ \int \frac {x^3 (e+f x)^n}{a+b x^3} \, dx=\int \frac {x^3\,{\left (e+f\,x\right )}^n}{b\,x^3+a} \,d x \]
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