Optimal. Leaf size=97 \[ \frac{(a+b x)^{-\frac{a d}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{a b c}-\frac{(a+b x)^{-\frac{b c}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{b c} \]
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Rubi [A] time = 0.0221451, antiderivative size = 97, normalized size of antiderivative = 1., 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 = {45, 37} \[ \frac{(a+b x)^{-\frac{a d}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{a b c}-\frac{(a+b x)^{-\frac{b c}{b c-a d}} (c+d x)^{\frac{a d}{b c-a d}}}{b c} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
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*}
Mathematica [A] time = 0.0285467, size = 46, normalized size = 0.47 \[ \frac{x (a+b x)^{\frac{b c}{a d-b c}} (c+d x)^{\frac{a d}{b c-a d}}}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 66, normalized size = 0.7 \begin{align*}{\frac{x}{ac} \left ( bx+a \right ) ^{1-{\frac{ad-2\,bc}{ad-bc}}} \left ( dx+c \right ) ^{1-{\frac{2\,ad-bc}{ad-bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{-\frac{b c}{b c - a d} - 1}{\left (d x + c\right )}^{\frac{a d}{b c - a d} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52583, size = 161, normalized size = 1.66 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{-\frac{b c}{b c - a d} - 1}{\left (d x + c\right )}^{\frac{a d}{b c - a d} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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