3.965 \(\int \frac{(a+b x)^n (c+d x)^{-n}}{x^4} \, dx\)

Optimal. Leaf size=194 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n} (a d (n+2)+b c (2-n))}{6 a^2 c^2 x^2}+\frac{(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \left (a^2 d^2 \left (n^2+3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac{c (a+b x)}{a (c+d x)}\right )}{6 a^4 c^2 (n+1)}-\frac{(a+b x)^{n+1} (c+d x)^{1-n}}{3 a c x^3} \]

[Out]

-((a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(3*a*c*x^3) + ((b*c*(2 - n) + a*d*(2 + n)
)*(a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(6*a^2*c^2*x^2) + ((b*c - a*d)*(2*a*b*c*d
*(1 - n^2) + b^2*c^2*(2 - 3*n + n^2) + a^2*d^2*(2 + 3*n + n^2))*(a + b*x)^(1 + n
)*(c + d*x)^(-1 - n)*Hypergeometric2F1[2, 1 + n, 2 + n, (c*(a + b*x))/(a*(c + d*
x))])/(6*a^4*c^2*(1 + n))

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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]

[Out]

-((a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(3*a*c*x^3) + ((b*c*(2 - n) + a*d*(2 + n)
)*(a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(6*a^2*c^2*x^2) + ((b*c - a*d)*(2*a*b*c*d
*(1 - n^2) + b^2*c^2*(2 - 3*n + n^2) + a^2*d^2*(2 + 3*n + n^2))*(a + b*x)^(1 + n
)*(c + d*x)^(-1 - n)*Hypergeometric2F1[2, 1 + n, 2 + n, (c*(a + b*x))/(a*(c + d*
x))])/(6*a^4*c^2*(1 + n))

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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)

[Out]

-(a + b*x)**(n + 1)*(c + d*x)**(-n + 1)/(3*a*c*x**3) + (a + b*x)**(n + 1)*(c + d
*x)**(-n + 1)*(a*d*(n + 2) + b*c*(-n + 2))/(6*a**2*c**2*x**2) + (a + b*x)**(n -
1)*(c + d*x)**(-n + 1)*(a*d - b*c)*(-2*a*b*c*d + (2*a*d + 2*b*c)*(a*d*(n + 2) +
b*c*(-n + 2)) - (a*d*(-n + 1) + b*c*(n + 1))*(a*d*(n + 2) + b*c*(-n + 2)))*hyper
((-n + 1, 2), (-n + 2,), a*(c + d*x)/(c*(a + b*x)))/(6*a**2*c**4*(-n + 1))

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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]

[Out]

(-4*b*d*(a + b*x)^n*AppellF1[3, -n, n, 4, -(a/(b*x)), -(c/(d*x))])/(3*x^2*(c + d
*x)^n*(4*b*d*x*AppellF1[3, -n, n, 4, -(a/(b*x)), -(c/(d*x))] + a*d*n*AppellF1[4,
1 - n, n, 5, -(a/(b*x)), -(c/(d*x))] - b*c*n*AppellF1[4, -n, 1 + n, 5, -(a/(b*x
)), -(c/(d*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)

[Out]

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")

[Out]

integrate((b*x + a)^n*(d*x + c)^(-n)/x^4, x)

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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")

[Out]

integral((b*x + a)^n/((d*x + c)^n*x^4), x)

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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)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{4}}\,{d x} \]

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")

[Out]

integrate((b*x + a)^n/((d*x + c)^n*x^4), x)