3.362 \(\int \frac {(a+b x)^n (c+d x^2)^3}{x} \, dx\)
Optimal. Leaf size=246 \[ -\frac {a d^2 \left (10 a^2 d+9 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {d^2 \left (10 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac {a d \left (a^4 d^2+3 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+1}}{b^6 (n+1)}+\frac {d \left (5 a^4 d^2+9 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac {5 a d^3 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^3 (a+b x)^{n+6}}{b^6 (n+6)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
[Out]
-a*d*(a^4*d^2+3*a^2*b^2*c*d+3*b^4*c^2)*(b*x+a)^(1+n)/b^6/(1+n)+d*(5*a^4*d^2+9*a^2*b^2*c*d+3*b^4*c^2)*(b*x+a)^(
2+n)/b^6/(2+n)-a*d^2*(10*a^2*d+9*b^2*c)*(b*x+a)^(3+n)/b^6/(3+n)+d^2*(10*a^2*d+3*b^2*c)*(b*x+a)^(4+n)/b^6/(4+n)
-5*a*d^3*(b*x+a)^(5+n)/b^6/(5+n)+d^3*(b*x+a)^(6+n)/b^6/(6+n)-c^3*(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],1+b*x/
a)/a/(1+n)
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Rubi [A] time = 0.34, antiderivative size = 246, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used =
{952, 1620, 65} \[ -\frac {a d \left (3 a^2 b^2 c d+a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+1}}{b^6 (n+1)}+\frac {d \left (9 a^2 b^2 c d+5 a^4 d^2+3 b^4 c^2\right ) (a+b x)^{n+2}}{b^6 (n+2)}-\frac {a d^2 \left (10 a^2 d+9 b^2 c\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {d^2 \left (10 a^2 d+3 b^2 c\right ) (a+b x)^{n+4}}{b^6 (n+4)}-\frac {5 a d^3 (a+b x)^{n+5}}{b^6 (n+5)}+\frac {d^3 (a+b x)^{n+6}}{b^6 (n+6)}-\frac {c^3 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*x)^n*(c + d*x^2)^3)/x,x]
[Out]
-((a*d*(3*b^4*c^2 + 3*a^2*b^2*c*d + a^4*d^2)*(a + b*x)^(1 + n))/(b^6*(1 + n))) + (d*(3*b^4*c^2 + 9*a^2*b^2*c*d
+ 5*a^4*d^2)*(a + b*x)^(2 + n))/(b^6*(2 + n)) - (a*d^2*(9*b^2*c + 10*a^2*d)*(a + b*x)^(3 + n))/(b^6*(3 + n))
+ (d^2*(3*b^2*c + 10*a^2*d)*(a + b*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d^3*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^
3*(a + b*x)^(6 + n))/(b^6*(6 + n)) - (c^3*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(
a*(1 + 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])
Rule 952
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(c^p*(d
+ e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m + n + 2*p + 1)),
Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c
^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0]
&& NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] || !IntegerQ[m])
Rule 1620
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]
Rubi steps
\begin {align*} \int \frac {(a+b x)^n \left (c+d x^2\right )^3}{x} \, dx &=\frac {d^3 (a+b x)^{6+n}}{b^6 (6+n)}+\frac {\int \frac {(a+b x)^n \left (b^6 c^3 (6+n)-a^5 b d^3 (6+n) x+b^2 d \left (3 b^4 c^2-5 a^4 d^2\right ) (6+n) x^2-10 a^3 b^3 d^3 (6+n) x^3+b^4 d^2 \left (3 b^2 c-10 a^2 d\right ) (6+n) x^4-5 a b^5 d^3 (6+n) x^5\right )}{x} \, dx}{b^6 (6+n)}\\ &=\frac {d^3 (a+b x)^{6+n}}{b^6 (6+n)}+\frac {\int \left (-a b d \left (3 b^4 c^2+3 a^2 b^2 c d+a^4 d^2\right ) (6+n) (a+b x)^n+\frac {\left (6 b^6 c^3+b^6 c^3 n\right ) (a+b x)^n}{x}+b d \left (3 b^4 c^2+9 a^2 b^2 c d+5 a^4 d^2\right ) (6+n) (a+b x)^{1+n}-a b d^2 \left (9 b^2 c+10 a^2 d\right ) (6+n) (a+b x)^{2+n}+b d^2 \left (3 b^2 c+10 a^2 d\right ) (6+n) (a+b x)^{3+n}-5 a b d^3 (6+n) (a+b x)^{4+n}\right ) \, dx}{b^6 (6+n)}\\ &=-\frac {a d \left (3 b^4 c^2+3 a^2 b^2 c d+a^4 d^2\right ) (a+b x)^{1+n}}{b^6 (1+n)}+\frac {d \left (3 b^4 c^2+9 a^2 b^2 c d+5 a^4 d^2\right ) (a+b x)^{2+n}}{b^6 (2+n)}-\frac {a d^2 \left (9 b^2 c+10 a^2 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {d^2 \left (3 b^2 c+10 a^2 d\right ) (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^3 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^3 (a+b x)^{6+n}}{b^6 (6+n)}+c^3 \int \frac {(a+b x)^n}{x} \, dx\\ &=-\frac {a d \left (3 b^4 c^2+3 a^2 b^2 c d+a^4 d^2\right ) (a+b x)^{1+n}}{b^6 (1+n)}+\frac {d \left (3 b^4 c^2+9 a^2 b^2 c d+5 a^4 d^2\right ) (a+b x)^{2+n}}{b^6 (2+n)}-\frac {a d^2 \left (9 b^2 c+10 a^2 d\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {d^2 \left (3 b^2 c+10 a^2 d\right ) (a+b x)^{4+n}}{b^6 (4+n)}-\frac {5 a d^3 (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^3 (a+b x)^{6+n}}{b^6 (6+n)}-\frac {c^3 (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 = 0.20, size = 226, normalized size = 0.92 \[ (a+b x)^{n+1} \left (\frac {d^2 (a+b x)^3 \left (10 a^2 d+3 b^2 c\right )}{b^6 (n+4)}-\frac {a d^2 (a+b x)^2 \left (10 a^2 d+9 b^2 c\right )}{b^6 (n+3)}+\frac {d (a+b x) \left (5 a^4 d^2+9 a^2 b^2 c d+3 b^4 c^2\right )}{b^6 (n+2)}-\frac {a d \left (a^4 d^2+3 a^2 b^2 c d+3 b^4 c^2\right )}{b^6 (n+1)}+\frac {d^3 (a+b x)^5}{b^6 (n+6)}-\frac {5 a d^3 (a+b x)^4}{b^6 (n+5)}-\frac {c^3 \, _2F_1\left (1,n+1;n+2;\frac {a+b x}{a}\right )}{a n+a}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*x)^n*(c + d*x^2)^3)/x,x]
[Out]
(a + b*x)^(1 + n)*(-((a*d*(3*b^4*c^2 + 3*a^2*b^2*c*d + a^4*d^2))/(b^6*(1 + n))) + (d*(3*b^4*c^2 + 9*a^2*b^2*c*
d + 5*a^4*d^2)*(a + b*x))/(b^6*(2 + n)) - (a*d^2*(9*b^2*c + 10*a^2*d)*(a + b*x)^2)/(b^6*(3 + n)) + (d^2*(3*b^2
*c + 10*a^2*d)*(a + b*x)^3)/(b^6*(4 + n)) - (5*a*d^3*(a + b*x)^4)/(b^6*(5 + n)) + (d^3*(a + b*x)^5)/(b^6*(6 +
n)) - (c^3*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*x)/a])/(a + a*n))
________________________________________________________________________________________
fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )} {\left (b x + a\right )}^{n}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3/x,x, algorithm="fricas")
[Out]
integral((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*(b*x + a)^n/x, x)
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{2} + c\right )}^{3} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3/x,x, algorithm="giac")
[Out]
integrate((d*x^2 + c)^3*(b*x + a)^n/x, x)
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{2}+c \right )^{3} \left (b x +a \right )^{n}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x^2+c)^3/x,x)
[Out]
int((b*x+a)^n*(d*x^2+c)^3/x,x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{2} + c\right )}^{3} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x^2+c)^3/x,x, algorithm="maxima")
[Out]
integrate((d*x^2 + c)^3*(b*x + a)^n/x, x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d\,x^2+c\right )}^3\,{\left (a+b\,x\right )}^n}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((c + d*x^2)^3*(a + b*x)^n)/x,x)
[Out]
int(((c + d*x^2)^3*(a + b*x)^n)/x, x)
________________________________________________________________________________________
sympy [B] time = 17.73, size = 5692, normalized size = 23.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**n*(d*x**2+c)**3/x,x)
[Out]
-b**n*c**3*n*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) - b**n*c**3*(a/b + x)**n*ler
chphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/gamma(n + 2) + 3*c**2*d*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)) + 3*c*d**2*Piecewise((a**n*x**4/4, Eq(b, 0)), (6*a**3*log(a/b + x)/(6*a**
3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 11*a**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x*
*2 + 6*b**7*x**3) + 18*a**2*b*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 2
7*a**2*b*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*x**2*log(a/b + x)/(6*a**3
*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b
**6*x**2 + 6*b**7*x**3) + 6*b**3*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**
3), Eq(n, -4)), (-6*a**3*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3/(2*a**2*b**4 + 4*a*b**
5*x + 2*b**6*x**2) - 12*a**2*b*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*x/(2*a**2*b
**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*a*b**2*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3
*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**
3/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*x**2/(2*a*b**4 + 2*b**5*x)
+ b**3*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*log(a/b + x)/b**4 + a**2*x/b**3 - a*x**2/(2*b**2) + x**3
/(3*b), Eq(n, -1)), (-6*a**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*
a**3*b*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*n**2*x**
2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*n*x**2*(a + b*x)*
*n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*n**3*x**3*(a + b*x)**n/(b**4*n**4
+ 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*n**2*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n*
*3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**3*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n*
*2 + 50*b**4*n + 24*b**4) + b**4*n**3*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n +
24*b**4) + 6*b**4*n**2*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 11
*b**4*n*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*x**4*(a + b
*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4), True)) + d**3*Piecewise((a**n*x**6/6,
Eq(b, 0)), (60*a**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 3
00*a*b**10*x**4 + 60*b**11*x**5) + 137*a**5/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b*
*9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a**4*b*x*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*
a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 625*a**4*b*x/(60*a**5*b**6 + 300*a**
4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**3*b**2*x**2*lo
g(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b
**11*x**5) + 1100*a**3*b**2*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 3
00*a*b**10*x**4 + 60*b**11*x**5) + 600*a**2*b**3*x**3*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*
b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 900*a**2*b**3*x**3/(60*a**5*b**6 + 300*a*
*4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*x**4*log(
a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**
11*x**5) + 300*a*b**4*x**4/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b
**10*x**4 + 60*b**11*x**5) + 60*b**5*x**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 +
600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5), Eq(n, -6)), (-60*a**5*log(a/b + x)/(12*a**4*b**6 + 48*
a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 125*a**5/(12*a**4*b**6 + 48*a**3*b**7*x +
72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**4*b*x*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x
+ 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 440*a**4*b*x/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2
*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 360*a**3*b**2*x**2*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x
+ 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 540*a**3*b**2*x**2/(12*a**4*b**6 + 48*a**3*b**7*x + 72
*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*x**3*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b*
*7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*x**3/(12*a**4*b**6 + 48*a**3*b**7*x
+ 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 60*a*b**4*x**4*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b
**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) + 12*b**5*x**5/(12*a**4*b**6 + 48*a**3*b**7*x + 72
*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4), Eq(n, -5)), (60*a**5*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b*
*7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 110*a**5/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3)
+ 180*a**4*b*x*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 270*a**4*b*x/(6*a*
*3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 180*a**3*b**2*x**2*log(a/b + x)/(6*a**3*b**6 + 18*a
**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 180*a**3*b**2*x**2/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2
+ 6*b**9*x**3) + 60*a**2*b**3*x**3*log(a/b + x)/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3)
- 15*a*b**4*x**4/(6*a**3*b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 3*b**5*x**5/(6*a**3*b**6 + 1
8*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3), Eq(n, -4)), (-60*a**5*log(a/b + x)/(6*a**2*b**6 + 12*a*b**7*x +
6*b**8*x**2) - 90*a**5/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 120*a**4*b*x*log(a/b + x)/(6*a**2*b**6 + 1
2*a*b**7*x + 6*b**8*x**2) - 120*a**4*b*x/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) - 60*a**3*b**2*x**2*log(a/b
+ x)/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) + 20*a**2*b**3*x**3/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2)
- 5*a*b**4*x**4/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x**2) + 2*b**5*x**5/(6*a**2*b**6 + 12*a*b**7*x + 6*b**8*x*
*2), Eq(n, -3)), (60*a**5*log(a/b + x)/(12*a*b**6 + 12*b**7*x) + 60*a**5/(12*a*b**6 + 12*b**7*x) + 60*a**4*b*x
*log(a/b + x)/(12*a*b**6 + 12*b**7*x) - 30*a**3*b**2*x**2/(12*a*b**6 + 12*b**7*x) + 10*a**2*b**3*x**3/(12*a*b*
*6 + 12*b**7*x) - 5*a*b**4*x**4/(12*a*b**6 + 12*b**7*x) + 3*b**5*x**5/(12*a*b**6 + 12*b**7*x), Eq(n, -2)), (-a
**5*log(a/b + x)/b**6 + a**4*x/b**5 - a**3*x**2/(2*b**4) + a**2*x**3/(3*b**3) - a*x**4/(4*b**2) + x**5/(5*b),
Eq(n, -1)), (-120*a**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2
+ 1764*b**6*n + 720*b**6) + 120*a**5*b*n*x*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*
n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 60*a**4*b**2*n**2*x**2*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**
5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 60*a**4*b**2*n*x**2*(a + b*x)**
n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 20*a*
*3*b**3*n**3*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 17
64*b**6*n + 720*b**6) + 60*a**3*b**3*n**2*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b*
*6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 40*a**3*b**3*n*x**3*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**
5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 5*a**2*b**4*n**4*x**4*(a + b*x)
**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 30*
a**2*b**4*n**3*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 +
1764*b**6*n + 720*b**6) - 55*a**2*b**4*n**2*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*
b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) - 30*a**2*b**4*n*x**4*(a + b*x)**n/(b**6*n**6 + 21*b**6*n
**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + a*b**5*n**5*x**5*(a + b*x)**n
/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 10*a*b
**5*n**4*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b
**6*n + 720*b**6) + 35*a*b**5*n**3*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3
+ 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 50*a*b**5*n**2*x**5*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175
*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 24*a*b**5*n*x**5*(a + b*x)**n/(b**6*n*
*6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + b**6*n**5*x**6*
(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b*
*6) + 15*b**6*n**4*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**
2 + 1764*b**6*n + 720*b**6) + 85*b**6*n**3*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b
**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 225*b**6*n**2*x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**
5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 274*b**6*n*x**6*(a + b*x)**n/(b
**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 720*b**6) + 120*b**6*
x**6*(a + b*x)**n/(b**6*n**6 + 21*b**6*n**5 + 175*b**6*n**4 + 735*b**6*n**3 + 1624*b**6*n**2 + 1764*b**6*n + 7
20*b**6), True)) - b*b**n*c**3*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**3*x*(a/b + x)**n*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2))
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