Integrand size = 20, antiderivative size = 250 \[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {(-a)^{3/2} (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c^2 \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {(-a)^{3/2} (d+e x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c^2 \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)} \]
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Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1643, 726, 70} \[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\frac {\left (c d^2-a e^2\right ) (d+e x)^{n+1}}{c^2 e^3 (n+1)}+\frac {(-a)^{3/2} (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c^2 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {(-a)^{3/2} (d+e x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}-\frac {2 d (d+e x)^{n+2}}{c e^3 (n+2)}+\frac {(d+e x)^{n+3}}{c e^3 (n+3)} \]
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Rule 70
Rule 726
Rule 1643
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-a e^2\right ) (d+e x)^n}{c^2 e^2}-\frac {2 d (d+e x)^{1+n}}{c e^2}+\frac {(d+e x)^{2+n}}{c e^2}+\frac {a^2 (d+e x)^n}{c^2 \left (a+c x^2\right )}\right ) \, dx \\ & = \frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {a^2 \int \frac {(d+e x)^n}{a+c x^2} \, dx}{c^2} \\ & = \frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {a^2 \int \left (\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\sqrt {-a} (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{c^2} \\ & = \frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}-\frac {(-a)^{3/2} \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{2 c^2}-\frac {(-a)^{3/2} \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{2 c^2} \\ & = \frac {\left (c d^2-a e^2\right ) (d+e x)^{1+n}}{c^2 e^3 (1+n)}-\frac {2 d (d+e x)^{2+n}}{c e^3 (2+n)}+\frac {(d+e x)^{3+n}}{c e^3 (3+n)}+\frac {(-a)^{3/2} (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{2 c^2 \left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}-\frac {(-a)^{3/2} (d+e x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{2 c^2 \left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.87 \[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\frac {(d+e x)^{1+n} \left (\frac {2 \left (c d^2-a e^2\right )}{e^3 (1+n)}-\frac {4 c d (d+e x)}{e^3 (2+n)}+\frac {2 c (d+e x)^2}{e^3 (3+n)}+\frac {(-a)^{3/2} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) (1+n)}+\frac {\sqrt {-a} a \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) (1+n)}\right )}{2 c^2} \]
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\[\int \frac {x^{4} \left (e x +d \right )^{n}}{c \,x^{2}+a}d x\]
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\[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a} \,d x } \]
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\[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\int \frac {x^{4} \left (d + e x\right )^{n}}{a + c x^{2}}\, dx \]
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\[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a} \,d x } \]
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\[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{n} x^{4}}{c x^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {x^4 (d+e x)^n}{a+c x^2} \, dx=\int \frac {x^4\,{\left (d+e\,x\right )}^n}{c\,x^2+a} \,d x \]
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