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

Optimal. Leaf size=56 \[ \frac{d (a+b x)^{n+1}}{b (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0496364, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{d (a+b x)^{n+1}}{b (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.34175, size = 41, normalized size = 0.73 \[ \frac{d \left (a + b x\right )^{n + 1}}{b \left (n + 1\right )} - \frac{c \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0931202, size = 67, normalized size = 1.2 \[ \frac{c (a+b x)^n \left (\frac{a}{b x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n}+\frac{d (a+b x)^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) }{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)/x,x)

[Out]

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

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x + c)*(b*x + a)^n/x, x)

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Sympy [A]  time = 7.67598, size = 170, normalized size = 3.04 \[ - \frac{b^{n} c n \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac{b^{n} c \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + d \left (\begin{cases} a^{n} x & \text{for}\: b = 0 \\\frac{\begin{cases} \frac{\left (a + b x\right )^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (a + b x \right )} & \text{otherwise} \end{cases}}{b} & \text{otherwise} \end{cases}\right ) - \frac{b b^{n} c n x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac{b b^{n} c x \left (\frac{a}{b} + x\right )^{n} \Phi \left (1 + \frac{b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)*(b*x + a)^n/x, x)