3.2.49 \(\int (a+b x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\) [149]

Optimal. Leaf size=113 \[ \frac {B (b c-a d)^2 n x}{3 d^2}-\frac {B (b c-a d) n (a+b x)^2}{6 b d}-\frac {B (b c-a d)^3 n \log (c+d x)}{3 b d^3}+\frac {(a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b} \]

[Out]

1/3*B*(-a*d+b*c)^2*n*x/d^2-1/6*B*(-a*d+b*c)*n*(b*x+a)^2/b/d-1/3*B*(-a*d+b*c)^3*n*ln(d*x+c)/b/d^3+1/3*(b*x+a)^3
*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b

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Rubi [A]
time = 0.04, antiderivative size = 113, 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)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b}-\frac {B n (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac {B n x (b c-a d)^2}{3 d^2}-\frac {B n (a+b x)^2 (b c-a d)}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(B*(b*c - a*d)^2*n*x)/(3*d^2) - (B*(b*c - a*d)*n*(a + b*x)^2)/(6*b*d) - (B*(b*c - a*d)^3*n*Log[c + d*x])/(3*b*
d^3) + ((a + b*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*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)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=\int \left (A (a+b x)^2+B (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx\\ &=\frac {A (a+b x)^3}{3 b}+B \int (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=\frac {A (a+b x)^3}{3 b}+\frac {B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac {(B (b c-a d) n) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b}\\ &=\frac {A (a+b x)^3}{3 b}+\frac {B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}-\frac {(B (b c-a d) n) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b}\\ &=\frac {B (b c-a d)^2 n x}{3 d^2}-\frac {B (b c-a d) n (a+b x)^2}{6 b d}+\frac {A (a+b x)^3}{3 b}-\frac {B (b c-a d)^3 n \log (c+d x)}{3 b d^3}+\frac {B (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 171, normalized size = 1.51 \begin {gather*} \frac {2 a^3 B d^3 n \log (a+b x)-2 b B c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) n \log (c+d x)+b d x \left (2 a^2 d^2 (3 A+2 B n)+a b d (-6 B c n+6 A d x+B d n x)+b^2 \left (2 A d^2 x^2+B c n (2 c-d x)\right )+2 B d^2 \left (3 a^2+3 a b x+b^2 x^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{6 b d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]),x]

[Out]

(2*a^3*B*d^3*n*Log[a + b*x] - 2*b*B*c*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*n*Log[c + d*x] + b*d*x*(2*a^2*d^2*(3*A
+ 2*B*n) + a*b*d*(-6*B*c*n + 6*A*d*x + B*d*n*x) + b^2*(2*A*d^2*x^2 + B*c*n*(2*c - d*x)) + 2*B*d^2*(3*a^2 + 3*
a*b*x + b^2*x^2)*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(6*b*d^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.33, size = 1323, normalized size = 11.71

method	result	size
risch	\(\text {Expression too large to display}\)	\(1323\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x,method=_RETURNVERBOSE)

[Out]

x*A*a^2+B*a^2*x*ln((b*x+a)^n)+B*ln(e)*a^2*x+1/3*b^2*B*x^3*ln((b*x+a)^n)+1/3*b^2*B*ln(e)*x^3+1/3/b*B*a^3*ln((b*
x+a)^n)+1/3*A*b^2*x^3+1/6*b*B*a*n*x^2-1/6/d*b^2*B*c*n*x^2+2/3*B*a^2*n*x+1/3/d^2*b^2*B*c^2*n*x-1/3/d^3*b^2*B*ln
(d*x+c)*c^3*n-1/d*B*ln(d*x+c)*a^2*c*n-1/2*I*B*Pi*a^2*x*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/3*(b*x+a)^3*B/b*ln((d
*x+c)^n)+1/2*I*b*B*Pi*a*x^2*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*b*B*Pi*a*x^2*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^2*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^2*x*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+
c)^n))+b*A*a*x^2-1/d*b*B*a*c*n*x+1/d^2*b*B*ln(d*x+c)*a*c^2*n-1/2*I*B*Pi*a^2*x*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^
3-1/6*I*b^2*B*Pi*x^3*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/6*I*b^2*B*Pi*x^3*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3-1/6*
I*b^2*B*Pi*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)-1/6*I*b^2*B*Pi*x^3*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*B*Pi*a*x^2*csgn(I*e)*csgn(I*e/((d*x+c)
^n)*(b*x+a)^n)^2+1/2*I*b*B*Pi*a*x^2*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-1/2*I*b*B*Pi*a*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)-1/2*I*b*B*Pi*a*x^2*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*B*Pi*a*x^2*csgn(I*(b*x+a)^n/((d*x+c)^n))^3-1/2*I*b*B*Pi*a*
x^2*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+1/2*I*B*Pi*a^2*x*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/2*I*B*Pi*
a^2*x*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/2*I*B*Pi*a^2*x*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/
((d*x+c)^n))^2+1/2*I*B*Pi*a^2*x*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/6*I*b^2*B*Pi
*x^3*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+1/6*I*b^2*B*Pi*x^3*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c
)^n))^2+1/6*I*b^2*B*Pi*x^3*csgn(I/((d*x+c)^n))*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+1/6*I*b^2*B*Pi*x^3*csgn(I*(b*x+
a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2+b*B*ln(e)*a*x^2+1/3/b*B*ln(d*x+c)*a^3*n+b*B*a*x^2*ln((b*x+
a)^n)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (106) = 212\).
time = 0.33, size = 300, normalized size = 2.65 \begin {gather*} \frac {1}{3} \, B b^{2} x^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + \frac {1}{3} \, A b^{2} x^{3} + B a b x^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a 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^{2} e^{\left (-1\right )} - {\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 b e^{\left (-1\right )} + \frac {1}{6} \, {\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 b^{2} e^{\left (-1\right )} + B a^{2} x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A a^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (106) = 212\).
time = 0.35, size = 248, normalized size = 2.19 \begin {gather*} \frac {2 \, {\left (A + B\right )} b^{3} d^{3} x^{3} + {\left (6 \, {\left (A + B\right )} a b^{2} d^{3} - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} + 2 \, {\left (3 \, {\left (A + B\right )} a^{2} b d^{3} + {\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + 2 \, B a^{2} b d^{3}\right )} n\right )} x + 2 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + B a^{3} d^{3} n\right )} \log \left (b x + a\right ) - 2 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B a b^{2} d^{3} n x^{2} + 3 \, B a^{2} b d^{3} n x + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} n\right )} \log \left (d x + c\right )}{6 \, b d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

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

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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)**2*(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. 235 vs. \(2 (106) = 212\).
time = 4.56, size = 235, normalized size = 2.08 \begin {gather*} \frac {B a^{3} n \log \left (b x + a\right )}{3 \, b} + \frac {1}{3} \, {\left (A b^{2} + B b^{2}\right )} x^{3} - \frac {{\left (B b^{2} c n - B a b d n - 6 \, A a b d - 6 \, B a b d\right )} x^{2}}{6 \, d} + \frac {1}{3} \, {\left (B b^{2} n x^{3} + 3 \, B a b n x^{2} + 3 \, B a^{2} n x\right )} \log \left (b x + a\right ) - \frac {1}{3} \, {\left (B b^{2} n x^{3} + 3 \, B a b n x^{2} + 3 \, B a^{2} n x\right )} \log \left (d x + c\right ) + \frac {{\left (B b^{2} c^{2} n - 3 \, B a b c d n + 2 \, B a^{2} d^{2} n + 3 \, A a^{2} d^{2} + 3 \, B a^{2} d^{2}\right )} x}{3 \, d^{2}} - \frac {{\left (B b^{2} c^{3} n - 3 \, B a b c^{2} d n + 3 \, B a^{2} c d^{2} n\right )} \log \left (-d x - c\right )}{3 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.24, size = 262, normalized size = 2.32 \begin {gather*} \ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (B\,a^2\,x+B\,a\,b\,x^2+\frac {B\,b^2\,x^3}{3}\right )+x^2\,\left (\frac {b\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{6\,d}-\frac {A\,b\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (\frac {b\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3\,d}-\frac {A\,b\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}-\frac {a\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{d}+\frac {A\,a\,b\,c}{d}\right )+\frac {A\,b^2\,x^3}{3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,B\,n\,a^2\,c\,d^2-3\,B\,n\,a\,b\,c^2\,d+B\,n\,b^2\,c^3\right )}{3\,d^3}+\frac {B\,a^3\,n\,\ln \left (a+b\,x\right )}{3\,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)^2,x)

[Out]

log((e*(a + b*x)^n)/(c + d*x)^n)*((B*b^2*x^3)/3 + B*a^2*x + B*a*b*x^2) + x^2*((b*(9*A*a*d + 3*A*b*c + B*a*d*n
- B*b*c*n))/(6*d) - (A*b*(3*a*d + 3*b*c))/(6*d)) - x*((((b*(9*A*a*d + 3*A*b*c + B*a*d*n - B*b*c*n))/(3*d) - (A
*b*(3*a*d + 3*b*c))/(3*d))*(3*a*d + 3*b*c))/(3*b*d) - (a*(3*A*a*d + 3*A*b*c + B*a*d*n - B*b*c*n))/d + (A*a*b*c
)/d) + (A*b^2*x^3)/3 - (log(c + d*x)*(B*b^2*c^3*n + 3*B*a^2*c*d^2*n - 3*B*a*b*c^2*d*n))/(3*d^3) + (B*a^3*n*log
(a + b*x))/(3*b)

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