Optimal. Leaf size=95 \[ -\frac{d (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c (n+1)} \]
[Out]
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Rubi [A] time = 0.0874322, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{d (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n/(x*(c + d*x)),x]
[Out]
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Rubi in Sympy [A] time = 12.1695, size = 66, normalized size = 0.69 \[ \frac{d \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{c \left (n + 1\right ) \left (a d - b c\right )} - \frac{\left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a c \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n/x/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.107481, size = 105, normalized size = 1.11 \[ \frac{(a+b x)^n \left (\left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )-\left (\frac{d (a+b x)}{b (c+d x)}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b c-a d}{b c+b d x}\right )\right )}{c n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n/(x*(c + d*x)),x]
[Out]
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Maple [F] time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( dx+c \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n/x/(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{d x^{2} + c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{n}}{x \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n/x/(d*x+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)*x),x, algorithm="giac")
[Out]