3.908 \(\int (a+b x)^n (c+d x) \, dx\)

Optimal. Leaf size=46 \[ \frac{(b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.0464321, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d (a+b x)^{n+2}}{b^2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n*(c + d*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.054, size = 37, normalized size = 0.8 \[ \frac{d \left (a + b x\right )^{n + 2}}{b^{2} \left (n + 2\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a d - b c\right )}{b^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.038749, size = 41, normalized size = 0.89 \[ \frac{(a+b x)^{n+1} (-a d+b c (n+2)+b d (n+1) x)}{b^2 (n+1) (n+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 49, normalized size = 1.1 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -bdnx-bcn-bdx+ad-2\,bc \right ) }{{b}^{2} \left ({n}^{2}+3\,n+2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223474, size = 112, normalized size = 2.43 \[ \frac{{\left (a b c n + 2 \, a b c - a^{2} d +{\left (b^{2} d n + b^{2} d\right )} x^{2} +{\left (2 \, b^{2} c +{\left (b^{2} c + a b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.33721, size = 377, normalized size = 8.2 \[ \begin{cases} a^{n} \left (c x + \frac{d x^{2}}{2}\right ) & \text{for}\: b = 0 \\\frac{a d \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a d}{a b^{2} + b^{3} x} - \frac{b c}{a b^{2} + b^{3} x} + \frac{b d x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: n = -2 \\- \frac{a d \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{c \log{\left (\frac{a}{b} + x \right )}}{b} + \frac{d x}{b} & \text{for}\: n = -1 \\- \frac{a^{2} d \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b c n \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{2 a b c \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b d n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} c n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{2 b^{2} c x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} d n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} d x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n*(d*x+c),x)

[Out]

Piecewise((a**n*(c*x + d*x**2/2), Eq(b, 0)), (a*d*log(a/b + x)/(a*b**2 + b**3*x)
+ a*d/(a*b**2 + b**3*x) - b*c/(a*b**2 + b**3*x) + b*d*x*log(a/b + x)/(a*b**2 +
b**3*x), Eq(n, -2)), (-a*d*log(a/b + x)/b**2 + c*log(a/b + x)/b + d*x/b, Eq(n, -
1)), (-a**2*d*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + a*b*c*n*(a + b*x)**
n/(b**2*n**2 + 3*b**2*n + 2*b**2) + 2*a*b*c*(a + b*x)**n/(b**2*n**2 + 3*b**2*n +
2*b**2) + a*b*d*n*x*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*c*n*x*(
a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + 2*b**2*c*x*(a + b*x)**n/(b**2*n**2
+ 3*b**2*n + 2*b**2) + b**2*d*n*x**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**
2) + b**2*d*x**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2), True))

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GIAC/XCAS [A]  time = 0.234427, size = 200, normalized size = 4.35 \[ \frac{b^{2} d n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + b^{2} c n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b d n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + b^{2} d x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b c n e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, b^{2} c x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a b c e^{\left (n{\rm ln}\left (b x + a\right )\right )} - a^{2} d e^{\left (n{\rm ln}\left (b x + a\right )\right )}}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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