Optimal. Leaf size=135 \[ -\frac {d x \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1}{n};\frac {1}{2},1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) \sqrt {a+b x^n}}+\frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2187, 251, 441,
440} \begin {gather*} \frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2}-\frac {d x \sqrt {\frac {b x^n}{a}+1} F_1\left (\frac {1}{n};\frac {1}{2},1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) \sqrt {a+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 440
Rule 441
Rule 2187
Rubi steps
\begin {align*} \int \frac {1}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx &=(a c) \int \frac {1}{a^2 c^2-a d^2+a b c^2 x^n} \, dx-(a d) \int \frac {1}{\sqrt {a+b x^n} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx\\ &=\frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2}-\frac {\left (a d \sqrt {1+\frac {b x^n}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b x^n}{a}} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx}{\sqrt {a+b x^n}}\\ &=-\frac {d x \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1}{n};\frac {1}{2},1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) \sqrt {a+b x^n}}+\frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(320\) vs. \(2(135)=270\).
time = 0.49, size = 320, normalized size = 2.37 \begin {gather*} -\frac {2 a d \left (a c^2-d^2\right ) (1+n) x F_1\left (\frac {1}{n};\frac {1}{2},1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\sqrt {a+b x^n} \left (a c^2-d^2+b c^2 x^n\right ) \left (-2 a b c^2 n x^n F_1\left (1+\frac {1}{n};\frac {1}{2},2;2+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )+\left (a c^2-d^2\right ) \left (-b n x^n F_1\left (1+\frac {1}{n};\frac {3}{2},1;2+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )+2 a (1+n) F_1\left (\frac {1}{n};\frac {1}{2},1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )\right )\right )}+\frac {c x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{a c^2-d^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{a c +b c \,x^{n}+d \sqrt {a +b \,x^{n}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{a c + b c x^{n} + d \sqrt {a + b x^{n}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{a\,c+d\,\sqrt {a+b\,x^n}+b\,c\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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