3.2.47 \(\int (a+b x)^4 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\) [147]

Optimal. Leaf size=171 \[ \frac {B (b c-a d)^4 n x}{5 d^4}-\frac {B (b c-a d)^3 n (a+b x)^2}{10 b d^3}+\frac {B (b c-a d)^2 n (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) n (a+b x)^4}{20 b d}-\frac {B (b c-a d)^5 n \log (c+d x)}{5 b d^5}+\frac {(a+b x)^5 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{5 b} \]

[Out]

1/5*B*(-a*d+b*c)^4*n*x/d^4-1/10*B*(-a*d+b*c)^3*n*(b*x+a)^2/b/d^3+1/15*B*(-a*d+b*c)^2*n*(b*x+a)^3/b/d^2-1/20*B*
(-a*d+b*c)*n*(b*x+a)^4/b/d-1/5*B*(-a*d+b*c)^5*n*ln(d*x+c)/b/d^5+1/5*(b*x+a)^5*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))
)/b

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Rubi [A]
time = 0.06, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2548, 45} \begin {gather*} \frac {(a+b x)^5 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{5 b}-\frac {B n (b c-a d)^5 \log (c+d x)}{5 b d^5}+\frac {B n x (b c-a d)^4}{5 d^4}-\frac {B n (a+b x)^2 (b c-a d)^3}{10 b d^3}+\frac {B n (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac {B n (a+b x)^4 (b c-a d)}{20 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(B*(b*c - a*d)^4*n*x)/(5*d^4) - (B*(b*c - a*d)^3*n*(a + b*x)^2)/(10*b*d^3) + (B*(b*c - a*d)^2*n*(a + b*x)^3)/(
15*b*d^2) - (B*(b*c - a*d)*n*(a + b*x)^4)/(20*b*d) - (B*(b*c - a*d)^5*n*Log[c + d*x])/(5*b*d^5) + ((a + b*x)^5
*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(5*b)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps

\begin {align*} \int (a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (a+b x)^4+B (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac {A (a+b x)^5}{5 b}+B \int (a+b x)^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac {A (a+b x)^5}{5 b}+\frac {B (a+b x)^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{5 b}-\frac {(B (b c-a d) n) \int \frac {(a+b x)^4}{c+d x} \, dx}{5 b}\\ &=\frac {A (a+b x)^5}{5 b}+\frac {B (a+b x)^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{5 b}-\frac {(B (b c-a d) n) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b}\\ &=\frac {B (b c-a d)^4 n x}{5 d^4}-\frac {B (b c-a d)^3 n (a+b x)^2}{10 b d^3}+\frac {B (b c-a d)^2 n (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) n (a+b x)^4}{20 b d}+\frac {A (a+b x)^5}{5 b}-\frac {B (b c-a d)^5 n \log (c+d x)}{5 b d^5}+\frac {B (a+b x)^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{5 b}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 338, normalized size = 1.98 \begin {gather*} \frac {1}{60} \left (\frac {12 a^5 B n \log (a+b x)}{b}-\frac {12 B c \left (b^4 c^4-5 a b^3 c^3 d+10 a^2 b^2 c^2 d^2-10 a^3 b c d^3+5 a^4 d^4\right ) n \log (c+d x)}{d^5}+\frac {x \left (12 a^4 d^4 (5 A+4 B n)+12 a^3 b d^3 (-10 B c n+10 A d x+3 B d n x)+4 a^2 b^2 d^2 \left (30 A d^2 x^2+B n \left (30 c^2-15 c d x+4 d^2 x^2\right )\right )+b^4 \left (12 A d^4 x^4+B c n \left (12 c^3-6 c^2 d x+4 c d^2 x^2-3 d^3 x^3\right )\right )+a b^3 d \left (60 A d^3 x^3+B n \left (-60 c^3+30 c^2 d x-20 c d^2 x^2+3 d^3 x^3\right )\right )+12 B d^4 \left (5 a^4+10 a^3 b x+10 a^2 b^2 x^2+5 a b^3 x^3+b^4 x^4\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{d^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

((12*a^5*B*n*Log[a + b*x])/b - (12*B*c*(b^4*c^4 - 5*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 + 5*a^4*
d^4)*n*Log[c + d*x])/d^5 + (x*(12*a^4*d^4*(5*A + 4*B*n) + 12*a^3*b*d^3*(-10*B*c*n + 10*A*d*x + 3*B*d*n*x) + 4*
a^2*b^2*d^2*(30*A*d^2*x^2 + B*n*(30*c^2 - 15*c*d*x + 4*d^2*x^2)) + b^4*(12*A*d^4*x^4 + B*c*n*(12*c^3 - 6*c^2*d
*x + 4*c*d^2*x^2 - 3*d^3*x^3)) + a*b^3*d*(60*A*d^3*x^3 + B*n*(-60*c^3 + 30*c^2*d*x - 20*c*d^2*x^2 + 3*d^3*x^3)
) + 12*B*d^4*(5*a^4 + 10*a^3*b*x + 10*a^2*b^2*x^2 + 5*a*b^3*x^3 + b^4*x^4)*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/
d^4)/60

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.46, size = 2372, normalized size = 13.87

method	result	size
risch	\(\text {Expression too large to display}\)	\(2372\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x,method=_RETURNVERBOSE)

[Out]

1/5*A*b^4*x^5+x*A*a^4-1/d^3*b^3*B*a*c^3*n*x+2/d^2*b*B*ln(d*x+c)*a^3*c^2*n-2/d^3*b^2*B*ln(d*x+c)*a^2*c^3*n+1/d^
4*b^3*B*ln(d*x+c)*a*c^4*n+1/10*I*b^4*B*Pi*x^5*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/10*I*b^4*B*P
i*x^5*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/10*I*b^4*B*Pi*x^5*csgn(I*(b*x+a)^n/((d*x+c)^n))*cs
gn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/10*I*b^4*B*Pi*x^5*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+4/15*b^2*B*a^2
*n*x^3+1/15/d^2*b^4*B*c^2*n*x^3+3/5*b*B*a^3*n*x^2-1/10/d^3*b^4*B*c^3*n*x^2+4/5*B*a^4*n*x+1/5/d^4*b^4*B*c^4*n*x
-1/5/d^5*b^4*B*ln(d*x+c)*c^5*n-1/5*(b*x+a)^5*B/b*ln((d*x+c)^n)-1/d*B*ln(d*x+c)*a^4*c*n-1/2*I*B*Pi*a^4*x*csgn(I
*(b*x+a)^n/((d*x+c)^n))^3-1/2*I*B*Pi*a^4*x*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/10*I*b^4*B*Pi*x^5*csgn(I*(b*x+a
)^n/((d*x+c)^n))^3-1/10*I*b^4*B*Pi*x^5*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/3/d*b^3*B*a*c*n*x^3-1/d*b^2*B*a^2*c
*n*x^2+1/2/d^2*b^3*B*a*c^2*n*x^2-2/d*b*B*a^3*c*n*x+2/d^2*b^2*B*a^2*c^2*n*x+I*b^2*B*Pi*a^2*x^3*csgn(I*(b*x+a)^n
)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*b^2*B*Pi*a^2*x^3*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*b^2
*B*Pi*a^2*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I*b^2*B*Pi*a^2*x^3*csgn(I*e)*csg
n(I*e/((d*x+c)^n)*(b*x+a)^n)^2+I*b*B*Pi*a^3*x^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*b*B*Pi*a^3
*x^2*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*b*B*Pi*a^3*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I
*e/((d*x+c)^n)*(b*x+a)^n)^2+I*b*B*Pi*a^3*x^2*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-1/2*I*B*Pi*a^4*x*csgn
(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*I*B*Pi*a^4*x*csgn(I*(b*x+a)^n)*csgn(I/
((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-1/10*I*b^4*B*Pi*x^5*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*
e/((d*x+c)^n)*(b*x+a)^n)-1/10*I*b^4*B*Pi*x^5*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n
))+1/2*I*b^3*B*Pi*a*x^4*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*b^3*B*Pi*a*x^4*csgn(I/((d*x+c)
^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*b^3*B*Pi*a*x^4*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(
b*x+a)^n)^2+1/2*I*b^3*B*Pi*a*x^4*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*a^4*x*csgn(I*(b*x+a)^n
)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*B*Pi*a^4*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+b^3*A*a
*x^4+2*b^2*A*a^2*x^3+2*b*A*a^3*x^2+2*b^2*B*a^2*x^3*ln((b*x+a)^n)+2*b*B*ln(e)*a^3*x^2+2*b*B*a^3*x^2*ln((b*x+a)^
n)+1/5/b*B*ln(d*x+c)*a^5*n+b^3*B*ln(e)*a*x^4+b^3*B*a*x^4*ln((b*x+a)^n)+2*b^2*B*ln(e)*a^2*x^3-1/2*I*b^3*B*Pi*a*
x^4*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-1/2*I*b^3*B*Pi*a*x^4*csgn(I*(b*x+a
)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))-I*b^2*B*Pi*a^2*x^3*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^
n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-I*b^2*B*Pi*a^2*x^3*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/
((d*x+c)^n))-I*b*B*Pi*a^3*x^2*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)-I*b*B*Pi
*a^3*x^2*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))+B*ln(e)*a^4*x+B*a^4*x*ln((b*x+a)^
n)+1/5*b^4*B*ln(e)*x^5+1/5*b^4*B*x^5*ln((b*x+a)^n)+1/5/b*B*a^5*ln((b*x+a)^n)+1/20*b^3*B*a*n*x^4-1/20/d*b^4*B*c
*n*x^4-1/2*I*b^3*B*Pi*a*x^4*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/2*I*b^3*B*Pi*a*x^4*csgn(I*e/((d*x+c)^n)*(b*x+a)^
n)^3-I*b^2*B*Pi*a^2*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-I*b^2*B*Pi*a^2*x^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-I
*b*B*Pi*a^3*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-I*b*B*Pi*a^3*x^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+1/2*I*B*Pi*
a^4*x*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*a^4*x*csgn(I*e)*csgn(I*e/((d*
x+c)^n)*(b*x+a)^n)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 681 vs. \(2 (160) = 320\).
time = 0.32, size = 681, normalized size = 3.98 \begin {gather*} \frac {1}{5} \, B b^{4} x^{5} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{5} \, A b^{4} x^{5} + B a b^{3} x^{4} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a b^{3} x^{4} + 2 \, B a^{2} b^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + 2 \, A a^{2} b^{2} x^{3} + 2 \, B a^{3} b x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + 2 \, A a^{3} b x^{2} + {\left (\frac {a n e \log \left (b x + a\right )}{b} - \frac {c n e \log \left (d x + c\right )}{d}\right )} B a^{4} e^{\left (-1\right )} - 2 \, {\left (\frac {a^{2} n e \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} n e \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c n - a d n\right )} x e}{b d}\right )} B a^{3} b e^{\left (-1\right )} + {\left (\frac {2 \, a^{3} n e \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} n e \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d n - a b d^{2} n\right )} x^{2} e - 2 \, {\left (b^{2} c^{2} n - a^{2} d^{2} n\right )} x e}{b^{2} d^{2}}\right )} B a^{2} b^{2} e^{\left (-1\right )} - \frac {1}{6} \, {\left (\frac {6 \, a^{4} n e \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} n e \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} n - a b^{2} d^{3} n\right )} x^{3} e - 3 \, {\left (b^{3} c^{2} d n - a^{2} b d^{3} n\right )} x^{2} e + 6 \, {\left (b^{3} c^{3} n - a^{3} d^{3} n\right )} x e}{b^{3} d^{3}}\right )} B a b^{3} e^{\left (-1\right )} + \frac {1}{60} \, {\left (\frac {12 \, a^{5} n e \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} n e \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} n - a b^{3} d^{4} n\right )} x^{4} e - 4 \, {\left (b^{4} c^{2} d^{2} n - a^{2} b^{2} d^{4} n\right )} x^{3} e + 6 \, {\left (b^{4} c^{3} d n - a^{3} b d^{4} n\right )} x^{2} e - 12 \, {\left (b^{4} c^{4} n - a^{4} d^{4} n\right )} x e}{b^{4} d^{4}}\right )} B b^{4} e^{\left (-1\right )} + B a^{4} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

1/5*B*b^4*x^5*log((b*x + a)^n*e/(d*x + c)^n) + 1/5*A*b^4*x^5 + B*a*b^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + A*
a*b^3*x^4 + 2*B*a^2*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 2*A*a^2*b^2*x^3 + 2*B*a^3*b*x^2*log((b*x + a)^n*e
/(d*x + c)^n) + 2*A*a^3*b*x^2 + (a*n*e*log(b*x + a)/b - c*n*e*log(d*x + c)/d)*B*a^4*e^(-1) - 2*(a^2*n*e*log(b*
x + a)/b^2 - c^2*n*e*log(d*x + c)/d^2 + (b*c*n - a*d*n)*x*e/(b*d))*B*a^3*b*e^(-1) + (2*a^3*n*e*log(b*x + a)/b^
3 - 2*c^3*n*e*log(d*x + c)/d^3 - ((b^2*c*d*n - a*b*d^2*n)*x^2*e - 2*(b^2*c^2*n - a^2*d^2*n)*x*e)/(b^2*d^2))*B*
a^2*b^2*e^(-1) - 1/6*(6*a^4*n*e*log(b*x + a)/b^4 - 6*c^4*n*e*log(d*x + c)/d^4 + (2*(b^3*c*d^2*n - a*b^2*d^3*n)
*x^3*e - 3*(b^3*c^2*d*n - a^2*b*d^3*n)*x^2*e + 6*(b^3*c^3*n - a^3*d^3*n)*x*e)/(b^3*d^3))*B*a*b^3*e^(-1) + 1/60
*(12*a^5*n*e*log(b*x + a)/b^5 - 12*c^5*n*e*log(d*x + c)/d^5 - (3*(b^4*c*d^3*n - a*b^3*d^4*n)*x^4*e - 4*(b^4*c^
2*d^2*n - a^2*b^2*d^4*n)*x^3*e + 6*(b^4*c^3*d*n - a^3*b*d^4*n)*x^2*e - 12*(b^4*c^4*n - a^4*d^4*n)*x*e)/(b^4*d^
4))*B*b^4*e^(-1) + B*a^4*x*log((b*x + a)^n*e/(d*x + c)^n) + A*a^4*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (160) = 320\).
time = 0.37, size = 503, normalized size = 2.94 \begin {gather*} \frac {12 \, {\left (A + B\right )} b^{5} d^{5} x^{5} + 3 \, {\left (20 \, {\left (A + B\right )} a b^{4} d^{5} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} n\right )} x^{4} + 4 \, {\left (30 \, {\left (A + B\right )} a^{2} b^{3} d^{5} + {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + 4 \, B a^{2} b^{3} d^{5}\right )} n\right )} x^{3} + 6 \, {\left (20 \, {\left (A + B\right )} a^{3} b^{2} d^{5} - {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 6 \, B a^{3} b^{2} d^{5}\right )} n\right )} x^{2} + 12 \, {\left (5 \, {\left (A + B\right )} a^{4} b d^{5} + {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 10 \, B a^{3} b^{2} c d^{4} + 4 \, B a^{4} b d^{5}\right )} n\right )} x + 12 \, {\left (B b^{5} d^{5} n x^{5} + 5 \, B a b^{4} d^{5} n x^{4} + 10 \, B a^{2} b^{3} d^{5} n x^{3} + 10 \, B a^{3} b^{2} d^{5} n x^{2} + 5 \, B a^{4} b d^{5} n x + B a^{5} d^{5} n\right )} \log \left (b x + a\right ) - 12 \, {\left (B b^{5} d^{5} n x^{5} + 5 \, B a b^{4} d^{5} n x^{4} + 10 \, B a^{2} b^{3} d^{5} n x^{3} + 10 \, B a^{3} b^{2} d^{5} n x^{2} + 5 \, B a^{4} b d^{5} n x + {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} n\right )} \log \left (d x + c\right )}{60 \, b d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

1/60*(12*(A + B)*b^5*d^5*x^5 + 3*(20*(A + B)*a*b^4*d^5 - (B*b^5*c*d^4 - B*a*b^4*d^5)*n)*x^4 + 4*(30*(A + B)*a^
2*b^3*d^5 + (B*b^5*c^2*d^3 - 5*B*a*b^4*c*d^4 + 4*B*a^2*b^3*d^5)*n)*x^3 + 6*(20*(A + B)*a^3*b^2*d^5 - (B*b^5*c^
3*d^2 - 5*B*a*b^4*c^2*d^3 + 10*B*a^2*b^3*c*d^4 - 6*B*a^3*b^2*d^5)*n)*x^2 + 12*(5*(A + B)*a^4*b*d^5 + (B*b^5*c^
4*d - 5*B*a*b^4*c^3*d^2 + 10*B*a^2*b^3*c^2*d^3 - 10*B*a^3*b^2*c*d^4 + 4*B*a^4*b*d^5)*n)*x + 12*(B*b^5*d^5*n*x^
5 + 5*B*a*b^4*d^5*n*x^4 + 10*B*a^2*b^3*d^5*n*x^3 + 10*B*a^3*b^2*d^5*n*x^2 + 5*B*a^4*b*d^5*n*x + B*a^5*d^5*n)*l
og(b*x + a) - 12*(B*b^5*d^5*n*x^5 + 5*B*a*b^4*d^5*n*x^4 + 10*B*a^2*b^3*d^5*n*x^3 + 10*B*a^3*b^2*d^5*n*x^2 + 5*
B*a^4*b*d^5*n*x + (B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2*d^3 + 5*B*a^4*b*c*d^4
)*n)*log(d*x + c))/(b*d^5)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n))),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (160) = 320\).
time = 39.24, size = 497, normalized size = 2.91 \begin {gather*} \frac {B a^{5} n \log \left (b x + a\right )}{5 \, b} + \frac {1}{5} \, {\left (A b^{4} + B b^{4}\right )} x^{5} - \frac {{\left (B b^{4} c n - B a b^{3} d n - 20 \, A a b^{3} d - 20 \, B a b^{3} d\right )} x^{4}}{20 \, d} + \frac {{\left (B b^{4} c^{2} n - 5 \, B a b^{3} c d n + 4 \, B a^{2} b^{2} d^{2} n + 30 \, A a^{2} b^{2} d^{2} + 30 \, B a^{2} b^{2} d^{2}\right )} x^{3}}{15 \, d^{2}} + \frac {1}{5} \, {\left (B b^{4} n x^{5} + 5 \, B a b^{3} n x^{4} + 10 \, B a^{2} b^{2} n x^{3} + 10 \, B a^{3} b n x^{2} + 5 \, B a^{4} n x\right )} \log \left (b x + a\right ) - \frac {1}{5} \, {\left (B b^{4} n x^{5} + 5 \, B a b^{3} n x^{4} + 10 \, B a^{2} b^{2} n x^{3} + 10 \, B a^{3} b n x^{2} + 5 \, B a^{4} n x\right )} \log \left (d x + c\right ) - \frac {{\left (B b^{4} c^{3} n - 5 \, B a b^{3} c^{2} d n + 10 \, B a^{2} b^{2} c d^{2} n - 6 \, B a^{3} b d^{3} n - 20 \, A a^{3} b d^{3} - 20 \, B a^{3} b d^{3}\right )} x^{2}}{10 \, d^{3}} + \frac {{\left (B b^{4} c^{4} n - 5 \, B a b^{3} c^{3} d n + 10 \, B a^{2} b^{2} c^{2} d^{2} n - 10 \, B a^{3} b c d^{3} n + 4 \, B a^{4} d^{4} n + 5 \, A a^{4} d^{4} + 5 \, B a^{4} d^{4}\right )} x}{5 \, d^{4}} - \frac {{\left (B b^{4} c^{5} n - 5 \, B a b^{3} c^{4} d n + 10 \, B a^{2} b^{2} c^{3} d^{2} n - 10 \, B a^{3} b c^{2} d^{3} n + 5 \, B a^{4} c d^{4} n\right )} \log \left (-d x - c\right )}{5 \, d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

1/5*B*a^5*n*log(b*x + a)/b + 1/5*(A*b^4 + B*b^4)*x^5 - 1/20*(B*b^4*c*n - B*a*b^3*d*n - 20*A*a*b^3*d - 20*B*a*b
^3*d)*x^4/d + 1/15*(B*b^4*c^2*n - 5*B*a*b^3*c*d*n + 4*B*a^2*b^2*d^2*n + 30*A*a^2*b^2*d^2 + 30*B*a^2*b^2*d^2)*x
^3/d^2 + 1/5*(B*b^4*n*x^5 + 5*B*a*b^3*n*x^4 + 10*B*a^2*b^2*n*x^3 + 10*B*a^3*b*n*x^2 + 5*B*a^4*n*x)*log(b*x + a
) - 1/5*(B*b^4*n*x^5 + 5*B*a*b^3*n*x^4 + 10*B*a^2*b^2*n*x^3 + 10*B*a^3*b*n*x^2 + 5*B*a^4*n*x)*log(d*x + c) - 1
/10*(B*b^4*c^3*n - 5*B*a*b^3*c^2*d*n + 10*B*a^2*b^2*c*d^2*n - 6*B*a^3*b*d^3*n - 20*A*a^3*b*d^3 - 20*B*a^3*b*d^
3)*x^2/d^3 + 1/5*(B*b^4*c^4*n - 5*B*a*b^3*c^3*d*n + 10*B*a^2*b^2*c^2*d^2*n - 10*B*a^3*b*c*d^3*n + 4*B*a^4*d^4*
n + 5*A*a^4*d^4 + 5*B*a^4*d^4)*x/d^4 - 1/5*(B*b^4*c^5*n - 5*B*a*b^3*c^4*d*n + 10*B*a^2*b^2*c^3*d^2*n - 10*B*a^
3*b*c^2*d^3*n + 5*B*a^4*c*d^4*n)*log(-d*x - c)/d^5

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Mupad [B]
time = 4.56, size = 936, normalized size = 5.47 \begin {gather*} x^4\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{20\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{20\,d}\right )-x^3\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{15\,b\,d}-\frac {a\,b^2\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3\,d}+\frac {A\,a\,b^3\,c}{3\,d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (B\,a^4\,x+2\,B\,a^3\,b\,x^2+2\,B\,a^2\,b^2\,x^3+B\,a\,b^3\,x^4+\frac {B\,b^4\,x^5}{5}\right )+x\,\left (\frac {a^3\,\left (5\,A\,a\,d+10\,A\,b\,c+2\,B\,a\,d\,n-2\,B\,b\,c\,n\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {2\,a^2\,b\,\left (5\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{5\,b\,d}-\frac {a\,b^2\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^3\,c}{d}\right )}{5\,b\,d}-\frac {a\,c\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{b\,d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{5\,b\,d}-\frac {a\,b^2\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^3\,c}{d}\right )}{b\,d}\right )+x^2\,\left (\frac {a^2\,b\,\left (5\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{5\,b\,d}-\frac {a\,b^2\,\left (10\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b^3\,c}{d}\right )}{10\,b\,d}-\frac {a\,c\,\left (\frac {b^3\,\left (25\,A\,a\,d+5\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5\,d}-\frac {A\,b^3\,\left (5\,a\,d+5\,b\,c\right )}{5\,d}\right )}{2\,b\,d}\right )+\frac {A\,b^4\,x^5}{5}-\frac {\ln \left (c+d\,x\right )\,\left (5\,B\,n\,a^4\,c\,d^4-10\,B\,n\,a^3\,b\,c^2\,d^3+10\,B\,n\,a^2\,b^2\,c^3\,d^2-5\,B\,n\,a\,b^3\,c^4\,d+B\,n\,b^4\,c^5\right )}{5\,d^5}+\frac {B\,a^5\,n\,\ln \left (a+b\,x\right )}{5\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))*(a + b*x)^4,x)

[Out]

x^4*((b^3*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(20*d) - (A*b^3*(5*a*d + 5*b*c))/(20*d)) - x^3*(((5*a*d +
5*b*c)*((b^3*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*(5*a*d + 5*b*c))/(5*d)))/(15*b*d) - (a*b
^2*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(3*d) + (A*a*b^3*c)/(3*d)) + log((e*(a + b*x)^n)/(c + d*x)^n)*((B
*b^4*x^5)/5 + B*a^4*x + 2*B*a^3*b*x^2 + B*a*b^3*x^4 + 2*B*a^2*b^2*x^3) + x*((a^3*(5*A*a*d + 10*A*b*c + 2*B*a*d
*n - 2*B*b*c*n))/d - ((5*a*d + 5*b*c)*((2*a^2*b*(5*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/d + ((5*a*d + 5*b*c)*
(((5*a*d + 5*b*c)*((b^3*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*(5*a*d + 5*b*c))/(5*d)))/(5*b
*d) - (a*b^2*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b^3*c)/d))/(5*b*d) - (a*c*((b^3*(25*A*a*d + 5*
A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*(5*a*d + 5*b*c))/(5*d)))/(b*d)))/(5*b*d) + (a*c*(((5*a*d + 5*b*c)*(
(b^3*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*(5*a*d + 5*b*c))/(5*d)))/(5*b*d) - (a*b^2*(10*A*
a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b^3*c)/d))/(b*d)) + x^2*((a^2*b*(5*A*a*d + 5*A*b*c + B*a*d*n - B*
b*c*n))/d + ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((b^3*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*
(5*a*d + 5*b*c))/(5*d)))/(5*b*d) - (a*b^2*(10*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b^3*c)/d))/(10*b*
d) - (a*c*((b^3*(25*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/(5*d) - (A*b^3*(5*a*d + 5*b*c))/(5*d)))/(2*b*d)) + (
A*b^4*x^5)/5 - (log(c + d*x)*(B*b^4*c^5*n + 5*B*a^4*c*d^4*n + 10*B*a^2*b^2*c^3*d^2*n - 5*B*a*b^3*c^4*d*n - 10*
B*a^3*b*c^2*d^3*n))/(5*d^5) + (B*a^5*n*log(a + b*x))/(5*b)

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