Optimal. Leaf size=216 \[ \frac {a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac {(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac {c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac {e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac {(e+f x)^{n+2}}{b d f^2 (n+2)} \]
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Rubi [A] time = 0.15, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {180, 43, 68} \[ \frac {a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac {(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac {c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac {e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac {(e+f x)^{n+2}}{b d f^2 (n+2)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 68
Rule 180
Rubi steps
\begin {align*} \int \frac {x^3 (e+f x)^n}{(a+b x) (c+d x)} \, dx &=\int \left (\frac {(-b c-a d) (e+f x)^n}{b^2 d^2}+\frac {x (e+f x)^n}{b d}-\frac {a^3 (e+f x)^n}{b^2 (b c-a d) (a+b x)}-\frac {c^3 (e+f x)^n}{d^2 (-b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac {(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac {\int x (e+f x)^n \, dx}{b d}-\frac {a^3 \int \frac {(e+f x)^n}{a+b x} \, dx}{b^2 (b c-a d)}+\frac {c^3 \int \frac {(e+f x)^n}{c+d x} \, dx}{d^2 (b c-a d)}\\ &=-\frac {(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac {a^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{b^2 (b c-a d) (b e-a f) (1+n)}-\frac {c^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{d^2 (b c-a d) (d e-c f) (1+n)}+\frac {\int \left (-\frac {e (e+f x)^n}{f}+\frac {(e+f x)^{1+n}}{f}\right ) \, dx}{b d}\\ &=-\frac {e (e+f x)^{1+n}}{b d f^2 (1+n)}-\frac {(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac {(e+f x)^{2+n}}{b d f^2 (2+n)}+\frac {a^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{b^2 (b c-a d) (b e-a f) (1+n)}-\frac {c^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{d^2 (b c-a d) (d e-c f) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 174, normalized size = 0.81 \[ \frac {(e+f x)^{n+1} \left (\frac {a^3 \, _2F_1\left (1,n+1;n+2;\frac {b (e+f x)}{b e-a f}\right )}{b e-a f}+\frac {(b c-a d) (c f-d e) (a d f (n+2)+b c f (n+2)+b d (e-f (n+1) x))-b^2 c^3 f^2 (n+2) \, _2F_1\left (1,n+1;n+2;\frac {d (e+f x)}{d e-c f}\right )}{d^2 f^2 (n+2) (d e-c f)}\right )}{b^2 (n+1) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{n} x^{3}}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )}^{n} x^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )}^{n} x^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {x^{3} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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