Optimal. Leaf size=177 \[ -\frac {(A b-a B) (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} F_1\left (n+1;1,-p;n+2;\frac {b (c+d x)}{b c-a d},-\frac {f (c+d x)}{d e-c f}\right )}{b (n+1) (b c-a d)}-\frac {B (c+d x)^{n+1} (e+f x)^{p+1} \, _2F_1\left (1,n+p+2;p+2;\frac {d (e+f x)}{d e-c f}\right )}{b (p+1) (d e-c f)} \]
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Rubi [A] time = 0.12, antiderivative size = 190, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {157, 70, 69, 137, 136} \[ \frac {B (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac {f (c+d x)}{d e-c f}\right )}{b d (n+1)}-\frac {(A b-a B) (c+d x)^{n+1} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac {f (c+d x)}{d e-c f},\frac {b (c+d x)}{b c-a d}\right )}{b (n+1) (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Rule 69
Rule 70
Rule 136
Rule 137
Rule 157
Rubi steps
\begin {align*} \int \frac {(A+B x) (c+d x)^n (e+f x)^p}{a+b x} \, dx &=\frac {B \int (c+d x)^n (e+f x)^p \, dx}{b}+\frac {(A b-a B) \int \frac {(c+d x)^n (e+f x)^p}{a+b x} \, dx}{b}\\ &=\frac {\left (B (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p}\right ) \int (c+d x)^n \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^p \, dx}{b}+\frac {\left ((A b-a B) (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p}\right ) \int \frac {(c+d x)^n \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^p}{a+b x} \, dx}{b}\\ &=-\frac {(A b-a B) (c+d x)^{1+n} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} F_1\left (1+n;-p,1;2+n;-\frac {f (c+d x)}{d e-c f},\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d) (1+n)}+\frac {B (c+d x)^{1+n} (e+f x)^p \left (\frac {d (e+f x)}{d e-c f}\right )^{-p} \, _2F_1\left (1+n,-p;2+n;-\frac {f (c+d x)}{d e-c f}\right )}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 199, normalized size = 1.12 \[ \frac {(c+d x)^n (e+f x)^p \left (\frac {(A b-a B) \left (\frac {b (c+d x)}{d (a+b x)}\right )^{-n} \left (\frac {b (e+f x)}{f (a+b x)}\right )^{-p} F_1\left (-n-p;-n,-p;-n-p+1;\frac {a d-b c}{d (a+b x)},\frac {a f-b e}{f (a+b x)}\right )}{n+p}+\frac {b B (e+f x) \left (\frac {f (c+d x)}{c f-d e}\right )^{-n} \, _2F_1\left (-n,p+1;p+2;\frac {d (e+f x)}{d e-c f}\right )}{f (p+1)}\right )}{b^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{b x + a}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (d x +c \right )^{n} \left (f x +e \right )^{p}}{b x +a}\, dx \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e+f\,x\right )}^p\,\left (A+B\,x\right )\,{\left (c+d\,x\right )}^n}{a+b\,x} \,d x \]
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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