3.928 \(\int \frac{(a+b x)^n (c+d x)^2}{x^3} \, dx\)
Optimal. Leaf size=124 \[ -\frac{(a+b x)^{n+1} \left (2 a^2 d^2+4 a b c d n-b^2 c^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{2 a^3 (n+1)}-\frac{c (a+b x)^{n+1} (4 a d-b c (1-n))}{2 a^2 x}-\frac{c^2 (a+b x)^{n+1}}{2 a x^2} \]
[Out]
-(c^2*(a + b*x)^(1 + n))/(2*a*x^2) - (c*(4*a*d - b*c*(1 - n))*(a + b*x)^(1 + n))/(2*a^2*x) - ((2*a^2*d^2 + 4*a
*b*c*d*n - b^2*c^2*(1 - n)*n)*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(2*a^3*(1 + n
))
________________________________________________________________________________________
Rubi [A] time = 0.0721881, antiderivative size = 124, 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, 78, 65} \[ -\frac{(a+b x)^{n+1} \left (2 a^2 d^2+4 a b c d n-b^2 c^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{2 a^3 (n+1)}-\frac{c (a+b x)^{n+1} (4 a d-b c (1-n))}{2 a^2 x}-\frac{c^2 (a+b x)^{n+1}}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^n*(c + d*x)^2)/x^3,x]
[Out]
-(c^2*(a + b*x)^(1 + n))/(2*a*x^2) - (c*(4*a*d - b*c*(1 - n))*(a + b*x)^(1 + n))/(2*a^2*x) - ((2*a^2*d^2 + 4*a
*b*c*d*n - b^2*c^2*(1 - n)*n)*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(2*a^3*(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 78
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ
[p, n]))))
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^3} \, dx &=-\frac{c^2 (a+b x)^{1+n}}{2 a x^2}+\frac{\int \frac{(a+b x)^n \left (c (4 a d-b c (1-n))+2 a d^2 x\right )}{x^2} \, dx}{2 a}\\ &=-\frac{c^2 (a+b x)^{1+n}}{2 a x^2}-\frac{c (4 a d-b c (1-n)) (a+b x)^{1+n}}{2 a^2 x}+\frac{\left (2 a^2 d^2+b c (4 a d-b c (1-n)) n\right ) \int \frac{(a+b x)^n}{x} \, dx}{2 a^2}\\ &=-\frac{c^2 (a+b x)^{1+n}}{2 a x^2}-\frac{c (4 a d-b c (1-n)) (a+b x)^{1+n}}{2 a^2 x}-\frac{\left (2 a^2 d^2+4 a b c d n-b^2 c^2 (1-n) n\right ) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{2 a^3 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0474343, size = 94, normalized size = 0.76 \[ -\frac{(a+b x)^{n+1} \left (x^2 \left (2 a^2 d^2+4 a b c d n+b^2 c^2 (n-1) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )+a c (n+1) (a (c+4 d x)+b c (n-1) x)\right )}{2 a^3 (n+1) x^2} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^n*(c + d*x)^2)/x^3,x]
[Out]
-((a + b*x)^(1 + n)*(a*c*(1 + n)*(b*c*(-1 + n)*x + a*(c + 4*d*x)) + (2*a^2*d^2 + 4*a*b*c*d*n + b^2*c^2*(-1 + n
)*n)*x^2*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(2*a^3*(1 + n)*x^2)
________________________________________________________________________________________
Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x+c)^2/x^3,x)
[Out]
int((b*x+a)^n*(d*x+c)^2/x^3,x)
________________________________________________________________________________________
Maxima [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^{3}}\,{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^3,x, algorithm="maxima")
[Out]
integrate((d*x + c)^2*(b*x + a)^n/x^3, x)
________________________________________________________________________________________
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^{3}}, 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^3,x, algorithm="fricas")
[Out]
integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^n/x^3, x)
________________________________________________________________________________________
Sympy [B] time = 11.396, size = 1807, normalized size = 14.57 \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**3,x)
[Out]
-a**3*b**2*b**n*c**2*n**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a*
*4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**3*b**2*b**n*c**2*n**2*(a/b + x)**n*gamma(n +
1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**3*b**2*b**n*
c**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2)
+ 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**3*b**2*b**n*c**2*n*(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n +
2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 2*a**3*b**2*b**n*c**2*(a/b + x)**n*ga
mma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**
3*b**n*c**2*n**3*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*
gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**2*b**3*b**n*c**2*n**2*x*(a/b + x)**n*gamma(n + 1)/(
-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**2*b**3*b**n*c**2*
n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) +
2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**3*b**n*c**2*n*x*(a/b + x)**n*gamma(n + 1)/(-2*a**5*gamma(n +
2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 2*a**2*b**3*b**n*c**2*x*(a/b + x)**n*g
amma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 2*a*b**
4*b**n*c**2*n**3*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) -
4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a*b**4*b**n*c**2*n**2*(a/b + x)**2*(a/b + x
)**n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 2
*a*b**4*b**n*c**2*n*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2)
- 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a*b**4*b**n*c**2*(a/b + x)**2*(a/b + x)*
*n*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - b**
5*b**n*c**2*n**3*(a/b + x)**3*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) -
4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + b**5*b**n*c**2*n*(a/b + x)**3*(a/b + x)**n*
lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b
+ x)**2*gamma(n + 2)) + 2*b**n*c*d*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*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)/(x*gamma(n + 2)) - 2*b**n*c*d*n*(a/b
+ x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 2*b**n*c*d*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - b**n*d**2*n*(
a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - b**n*d**2*(a/b + x)**n*lerchphi(1 + b*x/
a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 2*b*b**n*c*d*n**2*(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*n*(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*n*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - 2*b*b**n*c*d*(a/b + x)**n*gamma(n + 1)/(a*gamm
a(n + 2)) - b*b**n*d**2*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b*b**n*
d**2*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 2*b**2*b**n*c*d*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)) - 2*b**2*b**n*c*d*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))
________________________________________________________________________________________
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^{3}}\,{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^3,x, algorithm="giac")
[Out]
integrate((d*x + c)^2*(b*x + a)^n/x^3, x)