Optimal. Leaf size=206 \[ -\frac {c x^2 \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (\frac {2}{n};1,-q;\frac {2+n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2 \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (\frac {2}{n};1,-q;\frac {2+n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}} \]
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Rubi [A]
time = 0.13, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1570, 525, 524}
\begin {gather*} -\frac {c x^2 \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (\frac {2}{n};1,-q;\frac {n+2}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {c x^2 \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (\frac {2}{n};1,-q;\frac {n+2}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 1570
Rubi steps
\begin {align*} \int \frac {x \left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx &=\frac {(2 c) \int \frac {x \left (d+e x^n\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {x \left (d+e x^n\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2 c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q}\right ) \int \frac {x \left (1+\frac {e x^n}{d}\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (2 c \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q}\right ) \int \frac {x \left (1+\frac {e x^n}{d}\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^n} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {c x^2 \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (\frac {2}{n};1,-q;\frac {2+n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {c x^2 \left (d+e x^n\right )^q \left (1+\frac {e x^n}{d}\right )^{-q} F_1\left (\frac {2}{n};1,-q;\frac {2+n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [F]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (d +e \,x^{n}\right )^{q}}{a +b \,x^{n}+c \,x^{2 n}}\, dx\]
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 = -0.00 \begin {gather*} \int \frac {x\,{\left (d+e\,x^n\right )}^q}{a+b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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