Optimal. Leaf size=88 \[ \frac {b d (2+m+n) (a+b x)^{1+m} (c+d x)^{1+n} \left (\frac {f (a d (1+m)+b c (1+n))}{b d (2+m+n)}+f x\right )^{-2-m-n}}{(b c-a d)^2 f (1+m) (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {97}
\begin {gather*} \frac {b d (m+n+2) (a+b x)^{m+1} (c+d x)^{n+1} \left (\frac {f (a d (m+1)+b c (n+1))}{b d (m+n+2)}+f x\right )^{-m-n-2}}{f (m+1) (n+1) (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 97
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^n \left (\frac {b c f+a d f+a d f m+b c f n}{b d (2+m+n)}+f x\right )^{-3-m-n} \, dx &=\frac {b d (2+m+n) (a+b x)^{1+m} (c+d x)^{1+n} \left (\frac {f (a d (1+m)+b c (1+n))}{b d (2+m+n)}+f x\right )^{-2-m-n}}{(b c-a d)^2 f (1+m) (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 120, normalized size = 1.36 \begin {gather*} \frac {b^3 d^3 (2+m+n)^3 (a+b x)^{1+m} (c+d x)^{1+n} \left (\frac {f (a d (1+m)+b c (1+n)+b d (2+m+n) x)}{b d (2+m+n)}\right )^{-m-n}}{(b c-a d)^2 f^3 (1+m) (1+n) (a d (1+m)+b c (1+n)+b d (2+m+n) x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(197\) vs.
\(2(88)=176\).
time = 0.24, size = 198, normalized size = 2.25
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{1+m} \left (d x +c \right )^{1+n} \left (b d x m +b d x n +a d m +b c n +2 b d x +a d +b c \right ) \left (\frac {f \left (b d x m +b d x n +a d m +b c n +2 b d x +a d +b c \right )}{b d \left (2+m +n \right )}\right )^{-3-m -n}}{a^{2} d^{2} m n -2 a b c d m n +b^{2} c^{2} m n +a^{2} d^{2} m +a^{2} d^{2} n -2 a b c d m -2 a b c d n +b^{2} c^{2} m +b^{2} c^{2} n +a^{2} d^{2}-2 a b c d +b^{2} c^{2}}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1022 vs.
\(2 (88) = 176\).
time = 2.44, size = 1022, normalized size = 11.61 \begin {gather*} \frac {{\left ({\left (m^{3} + 3 \, m^{2} {\left (n + 2\right )} + n^{3} + 3 \, {\left (n^{2} + 4 \, n + 4\right )} m + 6 \, n^{2} + 12 \, n + 8\right )} a b^{m + n + 3} c d^{m + n + 3} {\left (m + n + 2\right )}^{m + n} + {\left (m^{3} + 3 \, m^{2} {\left (n + 2\right )} + n^{3} + 3 \, {\left (n^{2} + 4 \, n + 4\right )} m + 6 \, n^{2} + 12 \, n + 8\right )} b^{m + n + 4} d^{m + n + 4} {\left (m + n + 2\right )}^{m + n} x^{2} + {\left ({\left (m^{3} + 3 \, m^{2} {\left (n + 2\right )} + n^{3} + 3 \, {\left (n^{2} + 4 \, n + 4\right )} m + 6 \, n^{2} + 12 \, n + 8\right )} a b^{m + n + 3} d^{m + n + 4} + {\left (m^{3} + 3 \, m^{2} {\left (n + 2\right )} + n^{3} + 3 \, {\left (n^{2} + 4 \, n + 4\right )} m + 6 \, n^{2} + 12 \, n + 8\right )} b^{m + n + 4} c d^{m + n + 3}\right )} {\left (m + n + 2\right )}^{m + n} x\right )} e^{\left (-m \log \left (a d m + b c n + b c + a d + {\left (b d m + b d n + 2 \, b d\right )} x\right ) - n \log \left (a d m + b c n + b c + a d + {\left (b d m + b d n + 2 \, b d\right )} x\right ) + m \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}{{\left (n^{3} + {\left (n^{3} + 3 \, n^{2} + 3 \, n + 1\right )} m + 3 \, n^{2} + 3 \, n + 1\right )} b^{4} c^{4} f^{m + n + 3} + 2 \, {\left ({\left (n^{2} + 2 \, n + 1\right )} m^{2} - n^{3} - {\left (n^{3} + n^{2} - n - 1\right )} m - 2 \, n^{2} - n\right )} a b^{3} c^{3} d f^{m + n + 3} + {\left (m^{3} {\left (n + 1\right )} - {\left (4 \, n^{2} + 5 \, n + 1\right )} m^{2} + n^{3} + {\left (n^{3} - 5 \, n^{2} - 10 \, n - 4\right )} m - n^{2} - 4 \, n - 2\right )} a^{2} b^{2} c^{2} d^{2} f^{m + n + 3} - 2 \, {\left (m^{3} {\left (n + 1\right )} - {\left (n^{2} - n - 2\right )} m^{2} - {\left (2 \, n^{2} + n - 1\right )} m - n^{2} - n\right )} a^{3} b c d^{3} f^{m + n + 3} + {\left (m^{3} {\left (n + 1\right )} + 3 \, m^{2} {\left (n + 1\right )} + 3 \, m {\left (n + 1\right )} + n + 1\right )} a^{4} d^{4} f^{m + n + 3} + {\left ({\left (m^{3} {\left (n + 1\right )} + {\left (2 \, n^{2} + 7 \, n + 5\right )} m^{2} + n^{3} + {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} m + 5 \, n^{2} + 8 \, n + 4\right )} b^{4} c^{2} d^{2} f^{m + n + 3} - 2 \, {\left (m^{3} {\left (n + 1\right )} + {\left (2 \, n^{2} + 7 \, n + 5\right )} m^{2} + n^{3} + {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} m + 5 \, n^{2} + 8 \, n + 4\right )} a b^{3} c d^{3} f^{m + n + 3} + {\left (m^{3} {\left (n + 1\right )} + {\left (2 \, n^{2} + 7 \, n + 5\right )} m^{2} + n^{3} + {\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} m + 5 \, n^{2} + 8 \, n + 4\right )} a^{2} b^{2} d^{4} f^{m + n + 3}\right )} x^{2} + 2 \, {\left ({\left ({\left (n^{2} + 2 \, n + 1\right )} m^{2} + n^{3} + {\left (n^{3} + 5 \, n^{2} + 7 \, n + 3\right )} m + 4 \, n^{2} + 5 \, n + 2\right )} b^{4} c^{3} d f^{m + n + 3} + {\left (m^{3} {\left (n + 1\right )} - {\left (n^{2} - n - 2\right )} m^{2} - 2 \, n^{3} - {\left (2 \, n^{3} + 8 \, n^{2} + 7 \, n + 1\right )} m - 7 \, n^{2} - 7 \, n - 2\right )} a b^{3} c^{2} d^{2} f^{m + n + 3} - {\left (2 \, m^{3} {\left (n + 1\right )} + {\left (n^{2} + 8 \, n + 7\right )} m^{2} - n^{3} - {\left (n^{3} + n^{2} - 7 \, n - 7\right )} m - 2 \, n^{2} + n + 2\right )} a^{2} b^{2} c d^{3} f^{m + n + 3} + {\left (m^{3} {\left (n + 1\right )} + {\left (n^{2} + 5 \, n + 4\right )} m^{2} + {\left (2 \, n^{2} + 7 \, n + 5\right )} m + n^{2} + 3 \, n + 2\right )} a^{3} b d^{4} f^{m + n + 3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs.
\(2 (88) = 176\).
time = 0.94, size = 332, normalized size = 3.77 \begin {gather*} \frac {{\left (a^{2} c d m + a b c^{2} n + a b c^{2} + a^{2} c d + {\left (b^{2} d^{2} m + b^{2} d^{2} n + 2 \, b^{2} d^{2}\right )} x^{3} + {\left (3 \, b^{2} c d + 3 \, a b d^{2} + {\left (b^{2} c d + 2 \, a b d^{2}\right )} m + {\left (2 \, b^{2} c d + a b d^{2}\right )} n\right )} x^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + {\left (2 \, a b c d + a^{2} d^{2}\right )} m + {\left (b^{2} c^{2} + 2 \, a b c d\right )} n\right )} x\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \left (\frac {a d f m + b c f n + {\left (b c + a d\right )} f + {\left (b d f m + b d f n + 2 \, b d f\right )} x}{b d m + b d n + 2 \, b d}\right )^{-m - n - 3}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.26, size = 299, normalized size = 3.40 \begin {gather*} \frac {\frac {x\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2+b^2\,c^2+a^2\,d^2\,m+b^2\,c^2\,n+4\,a\,b\,c\,d+2\,a\,b\,c\,d\,m+2\,a\,b\,c\,d\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (m+1\right )\,\left (n+1\right )}+\frac {a\,c\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n\,\left (a\,d+b\,c+a\,d\,m+b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (m+1\right )\,\left (n+1\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n\,\left (3\,a\,d+3\,b\,c+2\,a\,d\,m+b\,c\,m+a\,d\,n+2\,b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^2\,\left (m+1\right )\,\left (n+1\right )}+\frac {b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n\,\left (m+n+2\right )}{{\left (a\,d-b\,c\right )}^2\,\left (m+1\right )\,\left (n+1\right )}}{{\left (f\,x+\frac {a\,d\,f+b\,c\,f+a\,d\,f\,m+b\,c\,f\,n}{b\,d\,\left (m+n+2\right )}\right )}^{m+n+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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