Optimal. Leaf size=203 \[ -\frac {(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{b^3 (n+1) (b c-a d)}+\frac {(c+d x)^{n+1} \left (a^2 d^2 D-a b d (C d-c D)-\left (b^2 \left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3 (n+1)}+\frac {(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac {D (c+d x)^{n+3}}{b d^3 (n+3)} \]
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Rubi [A] time = 0.18, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1620, 68} \[ -\frac {(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{b^3 (n+1) (b c-a d)}+\frac {(c+d x)^{n+1} \left (a^2 d^2 D-a b d (C d-c D)+b^2 \left (-\left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3 (n+1)}+\frac {(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac {D (c+d x)^{n+3}}{b d^3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 1620
Rubi steps
\begin {align*} \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{a+b x} \, dx &=\int \left (\frac {\left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) (c+d x)^n}{b^3 d^2}+\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^n}{b^3 (a+b x)}+\frac {(b C d-2 b c D-a d D) (c+d x)^{1+n}}{b^2 d^2}+\frac {D (c+d x)^{2+n}}{b d^2}\right ) \, dx\\ &=\frac {\left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) (c+d x)^{1+n}}{b^3 d^3 (1+n)}+\frac {(b C d-2 b c D-a d D) (c+d x)^{2+n}}{b^2 d^3 (2+n)}+\frac {D (c+d x)^{3+n}}{b d^3 (3+n)}+\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) \int \frac {(c+d x)^n}{a+b x} \, dx\\ &=\frac {\left (a^2 d^2 D-a b d (C d-c D)-b^2 \left (c C d-B d^2-c^2 D\right )\right ) (c+d x)^{1+n}}{b^3 d^3 (1+n)}+\frac {(b C d-2 b c D-a d D) (c+d x)^{2+n}}{b^2 d^3 (2+n)}+\frac {D (c+d x)^{3+n}}{b d^3 (3+n)}-\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) (c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 181, normalized size = 0.89 \[ \frac {(c+d x)^{n+1} \left (-\frac {\left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac {b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)}+\frac {a^2 d^2 D+a b d (c D-C d)+b^2 \left (B d^2+c^2 D-c C d\right )}{d^3 (n+1)}+\frac {b (c+d x) (-a d D-2 b c D+b C d)}{d^3 (n+2)}+\frac {b^2 D (c+d x)^2}{d^3 (n+3)}\right )}{b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a}, x\right ) \]
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 \[ \int \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (D x^{3}+C \,x^{2}+B x +A \right ) \left (d x +c \right )^{n}}{b x +a}\, dx \]
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 \[ \int \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{a+b\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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