Optimal. Leaf size=171 \[ \frac {x \sqrt {\frac {c x^{2 n}}{a}+1} F_1\left (\frac {1}{2 n};\frac {1}{2},1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d \sqrt {a+c x^{2 n}}}-\frac {e x^{n+1} \sqrt {\frac {c x^{2 n}}{a}+1} F_1\left (\frac {n+1}{2 n};\frac {1}{2},1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1) \sqrt {a+c x^{2 n}}} \]
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Rubi [A] time = 0.17, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1438, 430, 429, 511, 510} \[ \frac {x \sqrt {\frac {c x^{2 n}}{a}+1} F_1\left (\frac {1}{2 n};\frac {1}{2},1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d \sqrt {a+c x^{2 n}}}-\frac {e x^{n+1} \sqrt {\frac {c x^{2 n}}{a}+1} F_1\left (\frac {n+1}{2 n};\frac {1}{2},1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1) \sqrt {a+c x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 510
Rule 511
Rule 1438
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^n\right ) \sqrt {a+c x^{2 n}}} \, dx &=\int \left (\frac {d}{\sqrt {a+c x^{2 n}} \left (d^2-e^2 x^{2 n}\right )}+\frac {e x^n}{\sqrt {a+c x^{2 n}} \left (-d^2+e^2 x^{2 n}\right )}\right ) \, dx\\ &=d \int \frac {1}{\sqrt {a+c x^{2 n}} \left (d^2-e^2 x^{2 n}\right )} \, dx+e \int \frac {x^n}{\sqrt {a+c x^{2 n}} \left (-d^2+e^2 x^{2 n}\right )} \, dx\\ &=\frac {\left (d \sqrt {1+\frac {c x^{2 n}}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {c x^{2 n}}{a}} \left (d^2-e^2 x^{2 n}\right )} \, dx}{\sqrt {a+c x^{2 n}}}+\frac {\left (e \sqrt {1+\frac {c x^{2 n}}{a}}\right ) \int \frac {x^n}{\sqrt {1+\frac {c x^{2 n}}{a}} \left (-d^2+e^2 x^{2 n}\right )} \, dx}{\sqrt {a+c x^{2 n}}}\\ &=\frac {x \sqrt {1+\frac {c x^{2 n}}{a}} F_1\left (\frac {1}{2 n};\frac {1}{2},1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d \sqrt {a+c x^{2 n}}}-\frac {e x^{1+n} \sqrt {1+\frac {c x^{2 n}}{a}} F_1\left (\frac {1+n}{2 n};\frac {1}{2},1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n) \sqrt {a+c x^{2 n}}}\\ \end {align*}
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Mathematica [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d+e x^n\right ) \sqrt {a+c x^{2 n}}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2 \, n} + a}}{a e x^{n} + a d + {\left (c e x^{n} + c d\right )} x^{2 \, n}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {c x^{2 \, n} + a} {\left (e x^{n} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{n}+d \right ) \sqrt {c \,x^{2 n}+a}}\, dx \]
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 \[ \int \frac {1}{\sqrt {c x^{2 \, n} + a} {\left (e x^{n} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+c\,x^{2\,n}}\,\left (d+e\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + c x^{2 n}} \left (d + e x^{n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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