Optimal. Leaf size=97 \[ -\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}+\frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c} \]
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Rubi [A]
time = 0.02, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {47, 37}
\begin {gather*} \frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c}-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int (a+b x)^{-1-\frac {b c}{b c-a d}} (c+d x)^{-1+\frac {a d}{b c-a d}} \, dx &=-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}-\frac {d \int (a+b x)^{\frac {b c}{-b c+a d}} (c+d x)^{-1+\frac {a d}{b c-a d}} \, dx}{b c}\\ &=-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}+\frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 46, normalized size = 0.47 \begin {gather*} \frac {x (a+b x)^{\frac {b c}{-b c+a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a c} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.20, size = 66, normalized size = 0.68
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{1-\frac {a d -2 b c}{a d -b c}} \left (d x +c \right )^{1-\frac {2 a d -b c}{a d -b c}} x}{a c}\) | \(66\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 84, normalized size = 0.87 \begin {gather*} \frac {b d x^{3} + a c x + {\left (b c + a d\right )} x^{2}}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}} a c} \end {gather*}
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 \begin {gather*} \int \left (a + b x\right )^{- \frac {b c}{- a d + b c} - 1} \left (c + d x\right )^{\frac {a d}{- a d + b c} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 = 2.14, size = 119, normalized size = 1.23 \begin {gather*} \frac {x\,{\left (a+b\,x\right )}^{\frac {b\,c}{a\,d-b\,c}-1}+\frac {x^2\,\left (a\,d+b\,c\right )\,{\left (a+b\,x\right )}^{\frac {b\,c}{a\,d-b\,c}-1}}{a\,c}+\frac {b\,d\,x^3\,{\left (a+b\,x\right )}^{\frac {b\,c}{a\,d-b\,c}-1}}{a\,c}}{{\left (c+d\,x\right )}^{\frac {a\,d}{a\,d-b\,c}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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