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 = 0.73, antiderivative size = 263, normalized size of antiderivative = 1.00, 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 65
Rule 68
Rule 830
Rule 960
Rule 1474
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*}
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Mathematica [A] time = 0.67, 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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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{n} + d\right )}^{q}}{c x x^{2 \, n} + b x x^{n} + a x}, x\right ) \]
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 \[ \int \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+d \right )^{q}}{\left (b \,x^{n}+c \,x^{2 n}+a \right ) x}\, dx \]
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 \[ \int \frac {{\left (e x^{n} + d\right )}^{q}}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x^n\right )}^q}{x\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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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