3.927 \(\int \frac{(a+b x)^n (c+d x)^2}{x^2} \, dx\)
Optimal. Leaf size=87 \[ -\frac{c (a+b x)^{n+1} (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c^2 (a+b x)^{n+1}}{a x}+\frac{d^2 (a+b x)^{n+1}}{b (n+1)} \]
[Out]
(d^2*(a + b*x)^(1 + n))/(b*(1 + n)) - (c^2*(a + b*x)^(1 + n))/(a*x) - (c*(2*a*d + b*c*n)*(a + b*x)^(1 + n)*Hyp
ergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(1 + n))
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Rubi [A] time = 0.0383093, antiderivative size = 87, 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, Rules used =
{89, 80, 65} \[ -\frac{c (a+b x)^{n+1} (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)}-\frac{c^2 (a+b x)^{n+1}}{a x}+\frac{d^2 (a+b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^n*(c + d*x)^2)/x^2,x]
[Out]
(d^2*(a + b*x)^(1 + n))/(b*(1 + n)) - (c^2*(a + b*x)^(1 + n))/(a*x) - (c*(2*a*d + b*c*n)*(a + b*x)^(1 + n)*Hyp
ergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(1 + n))
Rule 89
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Rule 80
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]
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^2} \, dx &=-\frac{c^2 (a+b x)^{1+n}}{a x}+\frac{\int \frac{(a+b x)^n \left (c (2 a d+b c n)+a d^2 x\right )}{x} \, dx}{a}\\ &=\frac{d^2 (a+b x)^{1+n}}{b (1+n)}-\frac{c^2 (a+b x)^{1+n}}{a x}+\frac{(c (2 a d+b c n)) \int \frac{(a+b x)^n}{x} \, dx}{a}\\ &=\frac{d^2 (a+b x)^{1+n}}{b (1+n)}-\frac{c^2 (a+b x)^{1+n}}{a x}-\frac{c (2 a d+b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0304093, size = 73, normalized size = 0.84 \[ \frac{(a+b x)^{n+1} \left (a \left (a d^2 x-b c^2 (n+1)\right )-b c x (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )\right )}{a^2 b (n+1) x} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^n*(c + d*x)^2)/x^2,x]
[Out]
((a + b*x)^(1 + n)*(a*(-(b*c^2*(1 + n)) + a*d^2*x) - b*c*(2*a*d + b*c*n)*x*Hypergeometric2F1[1, 1 + n, 2 + n,
1 + (b*x)/a]))/(a^2*b*(1 + n)*x)
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x+c)^2/x^2,x)
[Out]
int((b*x+a)^n*(d*x+c)^2/x^2,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^2,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^{2}}, 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^2,x, algorithm="fricas")
[Out]
integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^n/x^2, x)
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Sympy [A] time = 9.43136, size = 554, normalized size = 6.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**n*(d*x+c)**2/x**2,x)
[Out]
b**n*c**2*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(n + 2)) + b**n*c**2*n*(a/b + x
)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(n + 2)) - b**n*c**2*n*(a/b + x)**n*gamma(n + 1)/(x*ga
mma(n + 2)) - b**n*c**2*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 2*b**n*c*d*n*(a/b + x)**n*lerchphi(1 + b*
x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - 2*b**n*c*d*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/
gamma(n + 2) + d**2*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**2*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n
+ 2)) + b*b**n*c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**2*n
*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*c**2*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - 2*b*b**
n*c*d*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 2*b*b**n*c*d*x*(a/b + x)*
*n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b**2*b**n*c**2*n**2*(a/b + x)**2*(a/b + x)**n
*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a**2*x*gamma(n + 2)) - b**2*b**n*c**2*n*(a/b + x)**2*(a/b + x)**n
*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a**2*x*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^{2}}\,{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^2,x, algorithm="giac")
[Out]
integrate((d*x + c)^2*(b*x + a)^n/x^2, x)