3.930 \(\int \frac{x^3 (a+b x)^n}{(c+d x)^2} \, dx\)

Optimal. Leaf size=203 \[ -\frac{(a+b x)^{n+1} (d x (b c-a d) (a d+b c (n+3))+c (b c (n+2) (a d+b c (n+3))-a d (a d+b c (3 n+5))))}{b^2 d^3 (n+1) (n+2) (c+d x) (b c-a d)}-\frac{c^2 (a+b x)^{n+1} (3 a d-b c (n+3)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d^3 (n+1) (b c-a d)^2}+\frac{x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.480511, antiderivative size = 203, 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 \[ -\frac{(a+b x)^{n+1} (d x (b c-a d) (a d+b c (n+3))+c (b c (n+2) (a d+b c (n+3))-a d (a d+b c (3 n+5))))}{b^2 d^3 (n+1) (n+2) (c+d x) (b c-a d)}-\frac{c^2 (a+b x)^{n+1} (3 a d-b c (n+3)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d^3 (n+1) (b c-a d)^2}+\frac{x^2 (a+b x)^{n+1}}{b d (n+2) (c+d x)} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 43.5055, size = 178, normalized size = 0.88 \[ - \frac{c^{2} \left (a + b x\right )^{n + 1} \left (3 a d - b c n - 3 b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{d^{3} \left (n + 1\right ) \left (a d - b c\right )^{2}} + \frac{x^{2} \left (a + b x\right )^{n + 1}}{b d \left (c + d x\right ) \left (n + 2\right )} - \frac{\left (a + b x\right )^{n + 1} \left (c \left (a d \left (a d + 2 b c \left (n + 1\right ) + b c \left (n + 3\right )\right ) - b c \left (n + 2\right ) \left (a d + b c \left (n + 3\right )\right )\right ) + d x \left (a d - b c\right ) \left (a d + b c \left (n + 3\right )\right )\right )}{b^{2} d^{3} \left (c + d x\right ) \left (n + 1\right ) \left (n + 2\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**n/(d*x+c)**2,x)

[Out]

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

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Mathematica [C]  time = 0.376143, size = 126, normalized size = 0.62 \[ \frac{5 a c x^4 (a+b x)^n F_1\left (4;-n,2;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 (c+d x)^2 \left (5 a c F_1\left (4;-n,2;5;-\frac{b x}{a},-\frac{d x}{c}\right )+b c n x F_1\left (5;1-n,2;6;-\frac{b x}{a},-\frac{d x}{c}\right )-2 a d x F_1\left (5;-n,3;6;-\frac{b x}{a},-\frac{d x}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

(5*a*c*x^4*(a + b*x)^n*AppellF1[4, -n, 2, 5, -((b*x)/a), -((d*x)/c)])/(4*(c + d*
x)^2*(5*a*c*AppellF1[4, -n, 2, 5, -((b*x)/a), -((d*x)/c)] + b*c*n*x*AppellF1[5,
1 - n, 2, 6, -((b*x)/a), -((d*x)/c)] - 2*a*d*x*AppellF1[5, -n, 3, 6, -((b*x)/a),
-((d*x)/c)]))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^n/(d*x+c)^2,x)

[Out]

int(x^3*(b*x+a)^n/(d*x+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^3/(d*x + c)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x^3/(d*x + c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n} x^{3}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^3/(d*x + c)^2,x, algorithm="fricas")

[Out]

integral((b*x + a)^n*x^3/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (a + b x\right )^{n}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)**n/(d*x+c)**2,x)

[Out]

Integral(x**3*(a + b*x)**n/(c + d*x)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n*x^3/(d*x + c)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^3/(d*x + c)^2, x)