Optimal. Leaf size=210 \[ \frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}+\frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
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Rubi [A] time = 0.48, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1556, 511, 510} \[ \frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}+\frac {c \left (d+e x^n\right )^q \left (\frac {e x^n}{d}+1\right )^{-q} F_1\left (-\frac {2}{n};1,-q;-\frac {2-n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},-\frac {e x^n}{d}\right )}{x^2 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 1556
Rubi steps
\begin {align*} \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx &=\frac {(2 c) \int \frac {\left (d+e x^n\right )^q}{x^3 \left (b-\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {\left (d+e x^n\right )^q}{x^3 \left (b+\sqrt {b^2-4 a c}+2 c x^n\right )} \, 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 {\left (1+\frac {e x^n}{d}\right )^q}{x^3 \left (b-\sqrt {b^2-4 a c}+2 c x^n\right )} \, 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 {\left (1+\frac {e x^n}{d}\right )^q}{x^3 \left (b+\sqrt {b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {c \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 )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) x^2}+\frac {c \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 )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) x^2}\\ \end {align*}
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Mathematica [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{n} + d\right )}^{q}}{c x^{3} x^{2 \, n} + b x^{3} x^{n} + a x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {\left (e \,x^{n}+d \right )^{q}}{\left (b \,x^{n}+c \,x^{2 n}+a \right ) x^{3}}\, dx \]
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 \[ \int \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x^n\right )}^q}{x^3\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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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