Optimal. Leaf size=194 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n} (a d (n+2)+b c (2-n))}{6 a^2 c^2 x^2}+\frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \left (a^2 d^2 \left (n^2+3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{6 a^4 c^2 (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n}}{3 a c x^3} \]
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Rubi [A] time = 0.37606, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n} (a d (n+2)+b c (2-n))}{6 a^2 c^2 x^2}+\frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \left (a^2 d^2 \left (n^2+3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{6 a^4 c^2 (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n}}{3 a c x^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n/(x^4*(c + d*x)^n),x]
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Rubi in Sympy [A] time = 48.8883, size = 180, normalized size = 0.93 \[ - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1}}{3 a c x^{3}} + \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1} \left (a d \left (n + 2\right ) + b c \left (- n + 2\right )\right )}{6 a^{2} c^{2} x^{2}} + \frac{\left (a + b x\right )^{n - 1} \left (c + d x\right )^{- n + 1} \left (a d - b c\right ) \left (- 2 a b c d + \left (2 a d + 2 b c\right ) \left (a d \left (n + 2\right ) + b c \left (- n + 2\right )\right ) - \left (a d \left (- n + 1\right ) + b c \left (n + 1\right )\right ) \left (a d \left (n + 2\right ) + b c \left (- n + 2\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n + 1, 2 \\ - n + 2 \end{matrix}\middle |{\frac{a \left (c + d x\right )}{c \left (a + b x\right )}} \right )}}{6 a^{2} c^{4} \left (- n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n/x**4/((d*x+c)**n),x)
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Mathematica [C] time = 0.469868, size = 146, normalized size = 0.75 \[ -\frac{4 b d (a+b x)^n (c+d x)^{-n} F_1\left (3;-n,n;4;-\frac{a}{b x},-\frac{c}{d x}\right )}{3 x^2 \left (4 b d x F_1\left (3;-n,n;4;-\frac{a}{b x},-\frac{c}{d x}\right )+a d n F_1\left (4;1-n,n;5;-\frac{a}{b x},-\frac{c}{d x}\right )-b c n F_1\left (4;-n,n+1;5;-\frac{a}{b x},-\frac{c}{d x}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^n/(x^4*(c + d*x)^n),x]
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Maple [F] time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{4} \left ( dx+c \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n/x^4/((d*x+c)^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)^n*x^4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)^n*x^4),x, algorithm="fricas")
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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/x**4/((d*x+c)**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)^n*x^4),x, algorithm="giac")
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