Optimal. Leaf size=59 \[ \frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} F_1\left (\frac {1}{n};-p,1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440}
\begin {gather*} \frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} F_1\left (\frac {1}{n};-p,1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^p}{c+d x^n} \, dx &=\left (\left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^n}{a}\right )^p}{c+d x^n} \, dx\\ &=\frac {x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} F_1\left (\frac {1}{n};-p,1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(59)=118\).
time = 0.21, size = 180, normalized size = 3.05 \begin {gather*} \frac {a c (1+n) x \left (a+b x^n\right )^p F_1\left (\frac {1}{n};-p,1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{\left (c+d x^n\right ) \left (b c n p x^n F_1\left (1+\frac {1}{n};1-p,1;2+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )-a d n x^n F_1\left (1+\frac {1}{n};-p,2;2+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) F_1\left (\frac {1}{n};-p,1;1+\frac {1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,x^{n}\right )^{p}}{c +d \,x^{n}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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 = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x^n\right )}^p}{c+d\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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