3.1869 \(\int (a+b x)^{-3-n} (c+d x)^n \, dx\)

Optimal. Leaf size=80 \[ \frac{d (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (b c-a d)^2}-\frac{(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)} \]

[Out]

-(((a + b*x)^(-2 - n)*(c + d*x)^(1 + n))/((b*c - a*d)*(2 + n))) + (d*(a + b*x)^(
-1 - n)*(c + d*x)^(1 + n))/((b*c - a*d)^2*(1 + n)*(2 + n))

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Rubi [A]  time = 0.0679673, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{d (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (b c-a d)^2}-\frac{(a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(((a + b*x)^(-2 - n)*(c + d*x)^(1 + n))/((b*c - a*d)*(2 + n))) + (d*(a + b*x)^(
-1 - n)*(c + d*x)^(1 + n))/((b*c - a*d)^2*(1 + n)*(2 + n))

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Rubi in Sympy [A]  time = 12.1437, size = 63, normalized size = 0.79 \[ \frac{d \left (a + b x\right )^{- n - 1} \left (c + d x\right )^{n + 1}}{\left (n + 1\right ) \left (n + 2\right ) \left (a d - b c\right )^{2}} + \frac{\left (a + b x\right )^{- n - 2} \left (c + d x\right )^{n + 1}}{\left (n + 2\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112332, size = 60, normalized size = 0.75 \[ \frac{(a+b x)^{-n-2} (c+d x)^{n+1} (a d (n+2)-b (c n+c-d x))}{(n+1) (n+2) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(-2 - n)*(c + d*x)^(1 + n)*(a*d*(2 + n) - b*(c + c*n - d*x)))/((b*c -
a*d)^2*(1 + n)*(2 + n))

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Maple [A]  time = 0.007, size = 123, normalized size = 1.5 \[{\frac{ \left ( bx+a \right ) ^{-2-n} \left ( dx+c \right ) ^{1+n} \left ( adn-bcn+bdx+2\,ad-bc \right ) }{{a}^{2}{d}^{2}{n}^{2}-2\,abcd{n}^{2}+{b}^{2}{c}^{2}{n}^{2}+3\,{a}^{2}{d}^{2}n-6\,abcdn+3\,{b}^{2}{c}^{2}n+2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(-3-n)*(d*x+c)^n,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230733, size = 279, normalized size = 3.49 \[ \frac{{\left (b^{2} d^{2} x^{3} - a b c^{2} + 2 \, a^{2} c d +{\left (3 \, a b d^{2} -{\left (b^{2} c d - a b d^{2}\right )} n\right )} x^{2} -{\left (a b c^{2} - a^{2} c d\right )} n -{\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2} +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} n\right )} x\right )}{\left (b x + a\right )}^{-n - 3}{\left (d x + c\right )}^{n}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n^{2} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*d^2*x^3 - a*b*c^2 + 2*a^2*c*d + (3*a*b*d^2 - (b^2*c*d - a*b*d^2)*n)*x^2 - (
a*b*c^2 - a^2*c*d)*n - (b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2 + (b^2*c^2 - a^2*d^2)*n)
*x)*(b*x + a)^(-n - 3)*(d*x + c)^n/(2*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2 + (b^2*c^2
- 2*a*b*c*d + a^2*d^2)*n^2 + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*n)

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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)**(-3-n)*(d*x+c)**n,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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