Optimal. Leaf size=263 \[ \frac{c \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^n\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^n+d\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{a n (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^n\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^n+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{a n (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{\left (d+e x^n\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{e x^n}{d}+1\right )}{a d n (q+1)} \]
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Rubi [A] time = 0.733565, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1474, 960, 65, 830, 68} \[ \frac{c \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \left (d+e x^n\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^n+d\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{a n (q+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^n\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^n+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{a n (q+1) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{\left (d+e x^n\right )^{q+1} \, _2F_1\left (1,q+1;q+2;\frac{e x^n}{d}+1\right )}{a d n (q+1)} \]
Antiderivative was successfully verified.
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Rule 1474
Rule 960
Rule 65
Rule 830
Rule 68
Rubi steps
\begin{align*} \int \frac{\left (d+e x^n\right )^q}{x \left (a+b x^n+c x^{2 n}\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(d+e x)^q}{x \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{(d+e x)^q}{a x}+\frac{(-b-c x) (d+e x)^q}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(d+e x)^q}{x} \, dx,x,x^n\right )}{a n}+\frac{\operatorname{Subst}\left (\int \frac{(-b-c x) (d+e x)^q}{a+b x+c x^2} \, dx,x,x^n\right )}{a n}\\ &=-\frac{\left (d+e x^n\right )^{1+q} \, _2F_1\left (1,1+q;2+q;1+\frac{e x^n}{d}\right )}{a d n (1+q)}+\frac{\operatorname{Subst}\left (\int \left (\frac{\left (-c-\frac{b c}{\sqrt{b^2-4 a c}}\right ) (d+e x)^q}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (-c+\frac{b c}{\sqrt{b^2-4 a c}}\right ) (d+e x)^q}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx,x,x^n\right )}{a n}\\ &=-\frac{\left (d+e x^n\right )^{1+q} \, _2F_1\left (1,1+q;2+q;1+\frac{e x^n}{d}\right )}{a d n (1+q)}-\frac{\left (c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^q}{b+\sqrt{b^2-4 a c}+2 c x} \, dx,x,x^n\right )}{a n}-\frac{\left (c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^q}{b-\sqrt{b^2-4 a c}+2 c x} \, dx,x,x^n\right )}{a n}\\ &=\frac{c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^n\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac{2 c \left (d+e x^n\right )}{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}\right )}{a \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) n (1+q)}+\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \left (d+e x^n\right )^{1+q} \, _2F_1\left (1,1+q;2+q;\frac{2 c \left (d+e x^n\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{a \left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) n (1+q)}-\frac{\left (d+e x^n\right )^{1+q} \, _2F_1\left (1,1+q;2+q;1+\frac{e x^n}{d}\right )}{a d n (1+q)}\\ \end{align*}
Mathematica [A] time = 0.727833, size = 218, normalized size = 0.83 \[ \frac{\left (d+e x^n\right )^{q+1} \left (\frac{c \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^n+d\right )}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}+\frac{c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,q+1;q+2;\frac{2 c \left (e x^n+d\right )}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}-\frac{\, _2F_1\left (1,q+1;q+2;\frac{e x^n}{d}+1\right )}{d}\right )}{a n (q+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{q}}{x \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x^{n} + d\right )}^{q}}{c x x^{2 \, n} + b x x^{n} + a x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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