3.935 \(\int \frac{(a+b x)^n (c+d x)^3}{x^3} \, dx\)
Optimal. Leaf size=176 \[ -\frac{c (a+b x)^{n+1} \left (6 a^2 d^2+6 a b c d n-b^2 c^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{2 a^3 (n+1)}-\frac{c (a+b x)^{n+1} \left (x \left (4 a^2 d^2+6 a b c d (n+1)-b^2 c^2 \left (1-n^2\right )\right )+a c (2 a d+b c (n+1))\right )}{2 a^2 b (n+1) x^2}+\frac{d (c+d x)^2 (a+b x)^{n+1}}{b (n+1) x^2} \]
[Out]
(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(1 + n)*x^2) - (c*(a + b*x)^(1 + n)*(a*c*(2*a*d + b*c*(1 + n)) + (4*a^2*d
^2 + 6*a*b*c*d*(1 + n) - b^2*c^2*(1 - n^2))*x))/(2*a^2*b*(1 + n)*x^2) - (c*(6*a^2*d^2 + 6*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))
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Rubi [A] time = 0.208896, antiderivative size = 176, 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 =
{100, 145, 65} \[ -\frac{c (a+b x)^{n+1} \left (6 a^2 d^2+6 a b c d n-b^2 c^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{2 a^3 (n+1)}-\frac{c (a+b x)^{n+1} \left (x \left (4 a^2 d^2+6 a b c d (n+1)-b^2 c^2 \left (1-n^2\right )\right )+a c (2 a d+b c (n+1))\right )}{2 a^2 b (n+1) x^2}+\frac{d (c+d x)^2 (a+b x)^{n+1}}{b (n+1) x^2} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^n*(c + d*x)^3)/x^3,x]
[Out]
(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(1 + n)*x^2) - (c*(a + b*x)^(1 + n)*(a*c*(2*a*d + b*c*(1 + n)) + (4*a^2*d
^2 + 6*a*b*c*d*(1 + n) - b^2*c^2*(1 - n^2))*x))/(2*a^2*b*(1 + n)*x^2) - (c*(6*a^2*d^2 + 6*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 100
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Rule 145
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] && !L
tQ[n, -2]))
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)^3}{x^3} \, dx &=\frac{d (a+b x)^{1+n} (c+d x)^2}{b (1+n) x^2}+\frac{\int \frac{(a+b x)^n (c+d x) (c (2 a d+b c (1+n))+b c d (3+n) x)}{x^3} \, dx}{b (1+n)}\\ &=\frac{d (a+b x)^{1+n} (c+d x)^2}{b (1+n) x^2}-\frac{c (a+b x)^{1+n} \left (a c (2 a d+b c (1+n))+\left (4 a^2 d^2+6 a b c d (1+n)-b^2 c^2 \left (1-n^2\right )\right ) x\right )}{2 a^2 b (1+n) x^2}+\frac{\left (c \left (6 a^2 d^2+6 a b c d n-b^2 c^2 (1-n) n\right )\right ) \int \frac{(a+b x)^n}{x} \, dx}{2 a^2}\\ &=\frac{d (a+b x)^{1+n} (c+d x)^2}{b (1+n) x^2}-\frac{c (a+b x)^{1+n} \left (a c (2 a d+b c (1+n))+\left (4 a^2 d^2+6 a b c d (1+n)-b^2 c^2 \left (1-n^2\right )\right ) x\right )}{2 a^2 b (1+n) x^2}-\frac{c \left (6 a^2 d^2+6 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 = 0.062349, size = 122, normalized size = 0.69 \[ \frac{(a+b x)^{n+1} \left (a \left (2 a^2 d^3 x^2-a b c^2 (n+1) (c+6 d x)-b^2 c^3 \left (n^2-1\right ) x\right )-b c x^2 \left (6 a^2 d^2+6 a b c d n+b^2 c^2 (n-1) n\right ) \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )\right )}{2 a^3 b (n+1) x^2} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^n*(c + d*x)^3)/x^3,x]
[Out]
((a + b*x)^(1 + n)*(a*(-(b^2*c^3*(-1 + n^2)*x) + 2*a^2*d^3*x^2 - a*b*c^2*(1 + n)*(c + 6*d*x)) - b*c*(6*a^2*d^2
+ 6*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*b*(1 + n)*x^
2)
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{3}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x+c)^3/x^3,x)
[Out]
int((b*x+a)^n*(d*x+c)^3/x^3,x)
________________________________________________________________________________________
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)^3/x^3,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^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\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)^3/x^3,x, algorithm="fricas")
[Out]
integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*(b*x + a)^n/x^3, x)
________________________________________________________________________________________
Sympy [A] time = 9.16492, size = 1868, normalized size = 10.61 \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)**3/x**3,x)
[Out]
-a**3*b**2*b**n*c**3*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**3*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**3*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**3*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**3*(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**3*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**3*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**3*
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**3*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**3*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**3*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**3*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**3*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**3*(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**3*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**3*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)) + 3*b**n*c**2*d*n**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(n
+ 2)) + 3*b**n*c**2*d*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(x*gamma(n + 2)) - 3*b**n*c**2
*d*n*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 3*b**n*c**2*d*(a/b + x)**n*gamma(n + 1)/(x*gamma(n + 2)) - 3
*b**n*c*d**2*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - 3*b**n*c*d**2*(a/b + x)*
*n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + d**3*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)) + 3*b*b**n*c**2*d*n**2*(a/b + x)**n*lerc
hphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) + 3*b*b**n*c**2*d*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1
, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c**2*d*n*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b*
*n*c**2*d*(a/b + x)**n*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c*d**2*n*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1,
n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b*b**n*c*d**2*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n +
1)/(a*gamma(n + 2)) - 3*b**2*b**n*c**2*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)) - 3*b**2*b**n*c**2*d*n*(a/b + x)**2*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gam
ma(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 )}^{3}{\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)^3/x^3,x, algorithm="giac")
[Out]
integrate((d*x + c)^3*(b*x + a)^n/x^3, x)