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

Optimal. Leaf size=88 \[ \frac{d (2 b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d^2 (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

(d*(2*b*c - a*d)*(a + b*x)^(1 + n))/(b^2*(1 + n)) + (d^2*(a + b*x)^(2 + n))/(b^2*(2 + n)) - (c^2*(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 = 0.0448251, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {88, 65} \[ \frac{d (2 b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac{d^2 (a+b x)^{n+2}}{b^2 (n+2)}-\frac{c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^n*(c + d*x)^2)/x,x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^n (c+d x)^2}{x} \, dx &=\int \left (-\frac{d (-2 b c+a d) (a+b x)^n}{b}+\frac{c^2 (a+b x)^n}{x}+\frac{d^2 (a+b x)^{1+n}}{b}\right ) \, dx\\ &=\frac{d (2 b c-a d) (a+b x)^{1+n}}{b^2 (1+n)}+\frac{d^2 (a+b x)^{2+n}}{b^2 (2+n)}+c^2 \int \frac{(a+b x)^n}{x} \, dx\\ &=\frac{d (2 b c-a d) (a+b x)^{1+n}}{b^2 (1+n)}+\frac{d^2 (a+b x)^{2+n}}{b^2 (2+n)}-\frac{c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0554464, size = 77, normalized size = 0.88 \[ -\frac{(a+b x)^{n+1} \left (b^2 c^2 (n+2) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )+a d (a d-b (2 c (n+2)+d (n+1) x))\right )}{a b^2 (n+1) (n+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 7.63977, size = 386, normalized size = 4.39 \begin{align*} - \frac{b^{n} c^{2} 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^{2} \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 )} + 2 c 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 ) + d^{2} \left (\begin{cases} \frac{a^{n} x^{2}}{2} & \text{for}\: b = 0 \\\frac{a \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a}{a b^{2} + b^{3} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: n = -2 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: n = -1 \\- \frac{a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac{b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text{otherwise} \end{cases}\right ) - \frac{b b^{n} c^{2} 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^{2} 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - b**n*c**2*(a/b + x)**n*ler
chphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 2*c*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)) + d**2*Piecewise((a**n*x**2/2, Eq(b, 0)), (a*
log(a/b + x)/(a*b**2 + b**3*x) + a/(a*b**2 + b**3*x) + b*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(n, -2)), (-a*log
(a/b + x)/b**2 + x/b, Eq(n, -1)), (-a**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + a*b*n*x*(a + b*x)**n/(
b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*n*x**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*x**2*(a + b*x
)**n/(b**2*n**2 + 3*b**2*n + 2*b**2), True)) - b*b**n*c**2*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamm
a(n + 1)/(a*gamma(n + 2)) - b*b**n*c**2*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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}{\left (b x + a\right )}^{n}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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