3.958 \(\int x^3 (a+b x)^n (c+d x)^{-n} \, dx\)

Optimal. Leaf size=295 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n} \left (a^2 d^2 \left (n^2-5 n+6\right )+2 a b c d \left (3-n^2\right )-2 b d x (a d (3-n)+b c (n+3))+b^2 c^2 \left (n^2+5 n+6\right )\right )}{24 b^3 d^3}-\frac{(a+b x)^{n+1} (c+d x)^{-n} \left (a^3 d^3 \left (-n^3+6 n^2-11 n+6\right )+3 a^2 b c d^2 \left (n^3-2 n^2-n+2\right )+3 a b^2 c^2 d \left (-n^3-2 n^2+n+2\right )+b^3 c^3 \left (n^3+6 n^2+11 n+6\right )\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (n+1)}+\frac{x^2 (a+b x)^{n+1} (c+d x)^{1-n}}{4 b d} \]

[Out]

(x^2*(a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(4*b*d) + ((a + b*x)^(1 + n)*(c + d*x)
^(1 - n)*(2*a*b*c*d*(3 - n^2) + a^2*d^2*(6 - 5*n + n^2) + b^2*c^2*(6 + 5*n + n^2
) - 2*b*d*(a*d*(3 - n) + b*c*(3 + n))*x))/(24*b^3*d^3) - ((3*a*b^2*c^2*d*(2 + n
- 2*n^2 - n^3) + a^3*d^3*(6 - 11*n + 6*n^2 - n^3) + 3*a^2*b*c*d^2*(2 - n - 2*n^2
+ n^3) + b^3*c^3*(6 + 11*n + 6*n^2 + n^3))*(a + b*x)^(1 + n)*((b*(c + d*x))/(b*
c - a*d))^n*Hypergeometric2F1[n, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(2
4*b^4*d^3*(1 + n)*(c + d*x)^n)

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Rubi [A]  time = 0.604959, antiderivative size = 295, 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 \[ \frac{(a+b x)^{n+1} (c+d x)^{1-n} \left (a^2 d^2 \left (n^2-5 n+6\right )+2 a b c d \left (3-n^2\right )-2 b d x (a d (3-n)+b c (n+3))+b^2 c^2 \left (n^2+5 n+6\right )\right )}{24 b^3 d^3}-\frac{(a+b x)^{n+1} (c+d x)^{-n} \left (a^3 d^3 \left (-n^3+6 n^2-11 n+6\right )+3 a^2 b c d^2 \left (n^3-2 n^2-n+2\right )+3 a b^2 c^2 d \left (-n^3-2 n^2+n+2\right )+b^3 c^3 \left (n^3+6 n^2+11 n+6\right )\right ) \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (n+1)}+\frac{x^2 (a+b x)^{n+1} (c+d x)^{1-n}}{4 b d} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(a + b*x)^n)/(c + d*x)^n,x]

[Out]

(x^2*(a + b*x)^(1 + n)*(c + d*x)^(1 - n))/(4*b*d) + ((a + b*x)^(1 + n)*(c + d*x)
^(1 - n)*(2*a*b*c*d*(3 - n^2) + a^2*d^2*(6 - 5*n + n^2) + b^2*c^2*(6 + 5*n + n^2
) - 2*b*d*(a*d*(3 - n) + b*c*(3 + n))*x))/(24*b^3*d^3) - ((3*a*b^2*c^2*d*(2 + n
- 2*n^2 - n^3) + a^3*d^3*(6 - 11*n + 6*n^2 - n^3) + 3*a^2*b*c*d^2*(2 - n - 2*n^2
+ n^3) + b^3*c^3*(6 + 11*n + 6*n^2 + n^3))*(a + b*x)^(1 + n)*((b*(c + d*x))/(b*
c - a*d))^n*Hypergeometric2F1[n, 1 + n, 2 + n, -((d*(a + b*x))/(b*c - a*d))])/(2
4*b^4*d^3*(1 + n)*(c + d*x)^n)

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Rubi in Sympy [A]  time = 51.7568, size = 275, normalized size = 0.93 \[ \frac{x^{2} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1}}{4 b d} - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n + 1} \left (6 a b c d - a d \left (- n + 2\right ) \left (a d \left (- n + 3\right ) + b c \left (n + 3\right )\right ) - b c \left (n + 2\right ) \left (a d \left (- n + 3\right ) + b c \left (n + 3\right )\right ) + 2 b d x \left (a d \left (- n + 3\right ) + b c \left (n + 3\right )\right )\right )}{24 b^{3} d^{3}} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{n} \left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n} \left (- a^{2} d^{2} \left (- n + 1\right ) \left (- n + 2\right ) \left (a d \left (- n + 3\right ) + b c \left (n + 3\right )\right ) + 2 a b c d \left (- n + 1\right ) \left (3 a d - \left (n + 1\right ) \left (a d \left (- n + 3\right ) + b c \left (n + 3\right )\right )\right ) + b^{2} c^{2} \left (n + 1\right ) \left (6 a d - \left (n + 2\right ) \left (a d \left (- n + 3\right ) + b c \left (n + 3\right )\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{24 b^{4} d^{3} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**n/((d*x+c)**n),x)

[Out]

x**2*(a + b*x)**(n + 1)*(c + d*x)**(-n + 1)/(4*b*d) - (a + b*x)**(n + 1)*(c + d*
x)**(-n + 1)*(6*a*b*c*d - a*d*(-n + 2)*(a*d*(-n + 3) + b*c*(n + 3)) - b*c*(n + 2
)*(a*d*(-n + 3) + b*c*(n + 3)) + 2*b*d*x*(a*d*(-n + 3) + b*c*(n + 3)))/(24*b**3*
d**3) + (b*(-c - d*x)/(a*d - b*c))**n*(a + b*x)**(n + 1)*(c + d*x)**(-n)*(-a**2*
d**2*(-n + 1)*(-n + 2)*(a*d*(-n + 3) + b*c*(n + 3)) + 2*a*b*c*d*(-n + 1)*(3*a*d
- (n + 1)*(a*d*(-n + 3) + b*c*(n + 3))) + b**2*c**2*(n + 1)*(6*a*d - (n + 2)*(a*
d*(-n + 3) + b*c*(n + 3))))*hyper((n, n + 1), (n + 2,), d*(a + b*x)/(a*d - b*c))
/(24*b**4*d**3*(n + 1))

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Mathematica [C]  time = 0.363504, size = 130, normalized size = 0.44 \[ \frac{5 a c x^4 (a+b x)^n (c+d x)^{-n} F_1\left (4;-n,n;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{20 a c F_1\left (4;-n,n;5;-\frac{b x}{a},-\frac{d x}{c}\right )+4 b c n x F_1\left (5;1-n,n;6;-\frac{b x}{a},-\frac{d x}{c}\right )-4 a d n x F_1\left (5;-n,n+1;6;-\frac{b x}{a},-\frac{d x}{c}\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(a + b*x)^n)/(c + d*x)^n,x]

[Out]

(5*a*c*x^4*(a + b*x)^n*AppellF1[4, -n, n, 5, -((b*x)/a), -((d*x)/c)])/((c + d*x)
^n*(20*a*c*AppellF1[4, -n, n, 5, -((b*x)/a), -((d*x)/c)] + 4*b*c*n*x*AppellF1[5,
1 - n, n, 6, -((b*x)/a), -((d*x)/c)] - 4*a*d*n*x*AppellF1[5, -n, 1 + n, 6, -((b
*x)/a), -((d*x)/c)]))

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^n/((d*x+c)^n),x)

[Out]

int(x^3*(b*x+a)^n/((d*x+c)^n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^3/(d*x + c)^n,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*(d*x + c)^(-n)*x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^3/(d*x + c)^n,x, algorithm="fricas")

[Out]

integral((b*x + a)^n*x^3/(d*x + c)^n, 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(x**3*(b*x+a)**n/((d*x+c)**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^3/(d*x + c)^n,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^3/(d*x + c)^n, x)