Optimal. Leaf size=61 \[ -\frac{(a+b x)^{m+1} (c+d x)^{n+1} \, _2F_1\left (1,m+n+2;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0740607, antiderivative size = 74, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 14.4892, size = 56, normalized size = 0.92 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**n,x)
[Out]
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Mathematica [A] time = 0.0828212, size = 73, normalized size = 1.2 \[ \frac{(a+b x)^m (c+d x)^{n+1} \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^n,x]
[Out]
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Maple [F] time = 0.137, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^n,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^n,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right )^{m} \left (c + d x\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**m*(d*x+c)**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^n,x, algorithm="giac")
[Out]