Optimal. Leaf size=85 \[ \frac{b (a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a^2 (n+1)} \]
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Rubi [A] time = 0.121609, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{b (a+b x)^{n+1} (c+d x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^{-p} F_1\left (n+1;-p,2;n+2;-\frac{d (a+b x)}{b c-a d},\frac{a+b x}{a}\right )}{a^2 (n+1)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^n*(c + d*x)^p)/x^2,x]
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Rubi in Sympy [A] time = 17.4961, size = 65, normalized size = 0.76 \[ \frac{b \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- p} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{p} \operatorname{appellf_{1}}{\left (n + 1,2,- p,n + 2,\frac{a + b x}{a},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{a^{2} \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x+c)**p/x**2,x)
[Out]
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Mathematica [B] time = 0.427185, size = 216, normalized size = 2.54 \[ \frac{b d (n+p-2) (a+b x)^n (c+d x)^p F_1\left (-n-p+1;-n,-p;-n-p+2;-\frac{a}{b x},-\frac{c}{d x}\right )}{(n+p-1) \left (b d x (n+p-2) F_1\left (-n-p+1;-n,-p;-n-p+2;-\frac{a}{b x},-\frac{c}{d x}\right )-a d n F_1\left (-n-p+2;1-n,-p;-n-p+3;-\frac{a}{b x},-\frac{c}{d x}\right )-b c p F_1\left (-n-p+2;-n,1-p;-n-p+3;-\frac{a}{b x},-\frac{c}{d x}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((a + b*x)^n*(c + d*x)^p)/x^2,x]
[Out]
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Maple [F] time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}}{{x}^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x+c)^p/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^p/x^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^p/x^2,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**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^p/x^2,x, algorithm="giac")
[Out]