Optimal. Leaf size=332 \[ \frac {(d+e x)^{n+1} \left (3 \sqrt {-a} c d^2+a \sqrt {c} d e n+\sqrt {-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 c^2 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}-\frac {(d+e x)^{n+1} \left (3 \sqrt {-a} c d^2-a \sqrt {c} d e n+\sqrt {-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 c^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}+\frac {a (d+e x)^{n+1} (a e+c d x)}{2 c^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{n+1}}{c^2 e (n+1)} \]
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Rubi [A] time = 0.45, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1649, 1629, 68} \[ \frac {(d+e x)^{n+1} \left (3 \sqrt {-a} c d^2+a \sqrt {c} d e n+\sqrt {-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{4 c^2 (n+1) \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (a e^2+c d^2\right )}-\frac {(d+e x)^{n+1} \left (3 \sqrt {-a} c d^2-a \sqrt {c} d e n+\sqrt {-a} a e^2 (n+3)\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{4 c^2 (n+1) \left (\sqrt {-a} e+\sqrt {c} d\right ) \left (a e^2+c d^2\right )}+\frac {a (d+e x)^{n+1} (a e+c d x)}{2 c^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {(d+e x)^{n+1}}{c^2 e (n+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 1629
Rule 1649
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)^n}{\left (a+c x^2\right )^2} \, dx &=\frac {a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \frac {(d+e x)^n \left (\frac {a^2 \left (c d^2+a e^2 (1+n)\right )}{c^2}+\frac {a^2 d e n x}{c}-2 a \left (d^2+\frac {a e^2}{c}\right ) x^2\right )}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\int \left (-\frac {2 a \left (c d^2+a e^2\right ) (d+e x)^n}{c^2}+\frac {\left (-\frac {a^3 d e n}{c^{3/2}}+\sqrt {-a} \left (\frac {3 a^2 d^2}{c}+\frac {3 a^3 e^2}{c^2}+\frac {a^3 e^2 n}{c^2}\right )\right ) (d+e x)^n}{2 a \left (\sqrt {-a}-\sqrt {c} x\right )}+\frac {\left (\frac {a^3 d e n}{c^{3/2}}+\sqrt {-a} \left (\frac {3 a^2 d^2}{c}+\frac {3 a^3 e^2}{c^2}+\frac {a^3 e^2 n}{c^2}\right )\right ) (d+e x)^n}{2 a \left (\sqrt {-a}+\sqrt {c} x\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {(d+e x)^{1+n}}{c^2 e (1+n)}+\frac {a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (3 \sqrt {-a} c d^2-a \sqrt {c} d e n+\sqrt {-a} a e^2 (3+n)\right ) \int \frac {(d+e x)^n}{\sqrt {-a}-\sqrt {c} x} \, dx}{4 c^2 \left (c d^2+a e^2\right )}-\frac {\left (3 \sqrt {-a} c d^2+a \sqrt {c} d e n+\sqrt {-a} a e^2 (3+n)\right ) \int \frac {(d+e x)^n}{\sqrt {-a}+\sqrt {c} x} \, dx}{4 c^2 \left (c d^2+a e^2\right )}\\ &=\frac {(d+e x)^{1+n}}{c^2 e (1+n)}+\frac {a (a e+c d x) (d+e x)^{1+n}}{2 c^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\left (3 \sqrt {-a} c d^2+a \sqrt {c} d e n+\sqrt {-a} a e^2 (3+n)\right ) (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 )}{4 c^2 \left (\sqrt {c} d-\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}-\frac {\left (3 \sqrt {-a} c d^2-a \sqrt {c} d e n+\sqrt {-a} a e^2 (3+n)\right ) (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 )}{4 c^2 \left (\sqrt {c} d+\sqrt {-a} e\right ) \left (c d^2+a e^2\right ) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.83, size = 413, normalized size = 1.24 \[ \frac {(d+e x)^{n+1} \left (\frac {a \left (\frac {\left (\sqrt {-a} \sqrt {c} d e n-a e^2 (n-1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {c} d-\sqrt {-a} e}-\frac {\left (-\sqrt {-a} \sqrt {c} d e n-a e^2 (n-1)+c d^2\right ) \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} e+\sqrt {c} d}\right )}{\sqrt {-a} (n+1) \left (a e^2+c d^2\right )}+\frac {2 a (a e+c d x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac {4 \sqrt {-a} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{(n+1) \left (\sqrt {c} d-\sqrt {-a} e\right )}-\frac {4 \sqrt {-a} \, _2F_1\left (1,n+1;n+2;\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{(n+1) \left (\sqrt {-a} e+\sqrt {c} d\right )}+\frac {4}{e n+e}\right )}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{n} x^{4}}{c^{2} x^{4} + 2 \, a c x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{n} x^{4}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (e x +d \right )^{n}}{\left (c \,x^{2}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{n} x^{4}}{{\left (c x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,{\left (d+e\,x\right )}^n}{{\left (c\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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