Optimal. Leaf size=51 \[ \frac{(a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)} \]
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Rubi [A] time = 0.034466, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(1 + 2*n - 2*(1 + n)),x]
[Out]
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Rubi in Sympy [A] time = 5.27879, size = 37, normalized size = 0.73 \[ - \frac{\left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (m + 1\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m/(d*x+c),x)
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Mathematica [A] time = 0.0377666, size = 66, normalized size = 1.29 \[ \frac{(a+b x)^m \left (\frac{d (a+b x)}{b (c+d x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b c-a d}{b c+b d x}\right )}{d m} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^(1 + 2*n - 2*(1 + n)),x]
[Out]
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Maple [F] time = 0., size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{dx+c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m/(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m/(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}}{d x + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m/(d*x + c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{m}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**m/(d*x+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m/(d*x + c),x, algorithm="giac")
[Out]