Optimal. Leaf size=79 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.0702366, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (b c-a d)}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-3 - m),x]
[Out]
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Rubi in Sympy [A] time = 13.6061, size = 63, normalized size = 0.8 \[ \frac{b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1}}{\left (m + 1\right ) \left (m + 2\right ) \left (a d - b c\right )^{2}} - \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 2}}{\left (m + 2\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-3-m),x)
[Out]
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Mathematica [A] time = 0.10147, size = 59, normalized size = 0.75 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} (-a d (m+1)+b c (m+2)+b d x)}{(m+1) (m+2) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-3 - m),x]
[Out]
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Maple [A] time = 0.006, size = 124, normalized size = 1.6 \[ -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-2-m} \left ( adm-bcm-bdx+ad-2\,bc \right ) }{{a}^{2}{d}^{2}{m}^{2}-2\,abcd{m}^{2}+{b}^{2}{c}^{2}{m}^{2}+3\,{a}^{2}{d}^{2}m-6\,abcdm+3\,{b}^{2}{c}^{2}m+2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-3-m),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229824, size = 277, normalized size = 3.51 \[ \frac{{\left (b^{2} d^{2} x^{3} + 2 \, a b c^{2} - a^{2} c d +{\left (3 \, b^{2} c d +{\left (b^{2} c d - a b d^{2}\right )} m\right )} x^{2} +{\left (a b c^{2} - a^{2} c d\right )} m +{\left (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} m\right )} x\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m^{2} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**m*(d*x+c)**(-3-m),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 3),x, algorithm="giac")
[Out]