Optimal. Leaf size=61 \[ -\frac{(a+b x)^{n+1} (c+d x)^{p+1} \, _2F_1\left (1,n+p+2;p+2;\frac{b (c+d x)}{b c-a d}\right )}{(p+1) (b c-a d)} \]
[Out]
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Rubi [A] time = 0.0651789, 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 \[ \frac{(a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n*(c + d*x)^p,x]
[Out]
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Rubi in Sympy [A] time = 13.9201, size = 56, normalized size = 0.92 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x+c)**p,x)
[Out]
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Mathematica [A] time = 0.0946055, size = 73, normalized size = 1.2 \[ \frac{(a+b x)^n (c+d x)^{p+1} \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (-n,p+1;p+2;\frac{b (c+d x)}{b c-a d}\right )}{d (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n*(c + d*x)^p,x]
[Out]
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Maple [F] time = 0.121, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x+c)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^p,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)**n*(d*x+c)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^p,x, algorithm="giac")
[Out]