Optimal. Leaf size=295 \[ \frac {(a+b x)^{n+1} (c+d x)^{1-n} \left (a^2 d^2 \left (n^2-5 n+6\right )+2 a b c d \left (3-n^2\right )-2 b d x (a d (3-n)+b c (n+3))+b^2 c^2 \left (n^2+5 n+6\right )\right )}{24 b^3 d^3}-\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (a^3 d^3 \left (-n^3+6 n^2-11 n+6\right )+3 a^2 b c d^2 \left (n^3-2 n^2-n+2\right )+3 a b^2 c^2 d \left (-n^3-2 n^2+n+2\right )+b^3 c^3 \left (n^3+6 n^2+11 n+6\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (n+1)}+\frac {x^2 (a+b x)^{n+1} (c+d x)^{1-n}}{4 b d} \]
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Rubi [A] time = 0.25, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {100, 147, 70, 69} \[ -\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (3 a^2 b c d^2 \left (n^3-2 n^2-n+2\right )+a^3 d^3 \left (-n^3+6 n^2-11 n+6\right )+3 a b^2 c^2 d \left (-n^3-2 n^2+n+2\right )+b^3 c^3 \left (n^3+6 n^2+11 n+6\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (n+1)}+\frac {(a+b x)^{n+1} (c+d x)^{1-n} \left (a^2 d^2 \left (n^2-5 n+6\right )+2 a b c d \left (3-n^2\right )-2 b d x (a d (3-n)+b c (n+3))+b^2 c^2 \left (n^2+5 n+6\right )\right )}{24 b^3 d^3}+\frac {x^2 (a+b x)^{n+1} (c+d x)^{1-n}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 100
Rule 147
Rubi steps
\begin {align*} \int x^3 (a+b x)^n (c+d x)^{-n} \, dx &=\frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {\int x (a+b x)^n (c+d x)^{-n} (-2 a c+(-a d (3-n)-b c (3+n)) x) \, dx}{4 b d}\\ &=\frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) \int (a+b x)^n (c+d x)^{-n} \, dx}{24 b^3 d^3}\\ &=\frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx}{24 b^3 d^3}\\ &=\frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 262, normalized size = 0.89 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n} \left (-b^2 c^2 (b c (n+3)-a d (n-1)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )-(b c-a d)^2 (b c (n+3)-a d (n-3)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n-2,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )+2 b c (b c-a d) (b c (n+3)-a d (n-2)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )+b^3 d^2 (n+1) x^2 (c+d x)\right )}{4 b^4 d^3 (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^{3} \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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