3.110 \(\int (a g+b g x) (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\)

Optimal. Leaf size=149 \[ \frac {g i (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{6 b^2}+\frac {g i (a+b x)^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b}+\frac {B g i n (b c-a d)^3 \log (c+d x)}{6 b^2 d^2}-\frac {B g i n x (b c-a d)^2}{6 b d} \]

[Out]

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

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Rubi [B]  time = 0.37, antiderivative size = 311, normalized size of antiderivative = 2.09, number of steps used = 13, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2528, 2486, 31, 2525, 12, 72} \[ -\frac {1}{3} b B d g i n x \left (\frac {a^2}{b^2}-\frac {c^2}{d^2}\right )-\frac {a^2 B g i n (a d+b c) \log (a+b x)}{2 b^2}+\frac {a^3 B d g i n \log (a+b x)}{3 b^2}+\frac {1}{3} b d g i x^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {1}{2} g i x^2 (a d+b c) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+a A c g i x+\frac {B c^2 g i n (a d+b c) \log (c+d x)}{2 d^2}+\frac {a B c g i (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {1}{6} B g i n x^2 (b c-a d)-\frac {B g i n x (b c-a d) (a d+b c)}{2 b d}-\frac {a B c g i n (b c-a d) \log (c+d x)}{b d}-\frac {b B c^3 g i n \log (c+d x)}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int (110 c+110 d x) (a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (110 a c g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+110 (b c+a d) g x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+110 b d g x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right ) \, dx\\ &=(110 a c g) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx+(110 b d g) \int x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx+(110 (b c+a d) g) \int x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx\\ &=110 a A c g x+55 (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {110}{3} b d g x^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+(110 a B c g) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx-\frac {1}{3} (110 b B d g n) \int \frac {(b c-a d) x^3}{(a+b x) (c+d x)} \, dx-(55 B (b c+a d) g n) \int \frac {(b c-a d) x^2}{(a+b x) (c+d x)} \, dx\\ &=110 a A c g x+\frac {110 a B c g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+55 (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {110}{3} b d g x^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {(110 a B c (b c-a d) g n) \int \frac {1}{c+d x} \, dx}{b}-\frac {1}{3} (110 b B d (b c-a d) g n) \int \frac {x^3}{(a+b x) (c+d x)} \, dx-(55 B (b c-a d) (b c+a d) g n) \int \frac {x^2}{(a+b x) (c+d x)} \, dx\\ &=110 a A c g x+\frac {110 a B c g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+55 (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {110}{3} b d g x^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {110 a B c (b c-a d) g n \log (c+d x)}{b d}-\frac {1}{3} (110 b B d (b c-a d) g n) \int \left (\frac {-b c-a d}{b^2 d^2}+\frac {x}{b d}-\frac {a^3}{b^2 (b c-a d) (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx-(55 B (b c-a d) (b c+a d) g n) \int \left (\frac {1}{b d}+\frac {a^2}{b (b c-a d) (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)}\right ) \, dx\\ &=110 a A c g x-\frac {55 B (b c-a d) (b c+a d) g n x}{3 b d}-\frac {55}{3} B (b c-a d) g n x^2+\frac {110 a^3 B d g n \log (a+b x)}{3 b^2}-\frac {55 a^2 B (b c+a d) g n \log (a+b x)}{b^2}+\frac {110 a B c g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+55 (b c+a d) g x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {110}{3} b d g x^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {110 b B c^3 g n \log (c+d x)}{3 d^2}-\frac {110 a B c (b c-a d) g n \log (c+d x)}{b d}+\frac {55 B c^2 (b c+a d) g n \log (c+d x)}{d^2}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 189, normalized size = 1.27 \[ \frac {g i \left (b \left (d x \left (a^2 B d^2 n+a b d (6 A c+3 A d x+B d n x)+A b^2 d x (3 c+2 d x)+b^2 (-B) c n (c+d x)\right )+B c n \left (6 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (c+d x)+B d^2 \left (6 a^2 c+3 a b x (2 c+d x)+b^2 x^2 (3 c+2 d x)\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-a^2 B d^2 n (a d+3 b c) \log (a+b x)\right )}{6 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.97, size = 309, normalized size = 2.07 \[ \frac {2 \, A b^{3} d^{3} g i x^{3} + {\left (3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} g i n \log \left (b x + a\right ) + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} g i n \log \left (d x + c\right ) - {\left ({\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} g i n - 3 \, {\left (A b^{3} c d^{2} + A a b^{2} d^{3}\right )} g i\right )} x^{2} + {\left (6 \, A a b^{2} c d^{2} g i - {\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} g i n\right )} x + {\left (2 \, B b^{3} d^{3} g i x^{3} + 6 \, B a b^{2} c d^{2} g i x + 3 \, {\left (B b^{3} c d^{2} + B a b^{2} d^{3}\right )} g i x^{2}\right )} \log \relax (e) + {\left (2 \, B b^{3} d^{3} g i n x^{3} + 6 \, B a b^{2} c d^{2} g i n x + 3 \, {\left (B b^{3} c d^{2} + B a b^{2} d^{3}\right )} g i n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{6 \, b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.95, size = 1256, normalized size = 8.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((B*b^5*c^4*g*i*n - 4*B*a*b^4*c^3*d*g*i*n - 3*(b*x + a)*B*b^4*c^4*d*g*i*n/(d*x + c) + 6*B*a^2*b^3*c^2*d^2
*g*i*n + 12*(b*x + a)*B*a*b^3*c^3*d^2*g*i*n/(d*x + c) - 4*B*a^3*b^2*c*d^3*g*i*n - 18*(b*x + a)*B*a^2*b^2*c^2*d
^3*g*i*n/(d*x + c) + B*a^4*b*d^4*g*i*n + 12*(b*x + a)*B*a^3*b*c*d^4*g*i*n/(d*x + c) - 3*(b*x + a)*B*a^4*d^5*g*
i*n/(d*x + c))*log((b*x + a)/(d*x + c))/(b^3*d^2 - 3*(b*x + a)*b^2*d^3/(d*x + c) + 3*(b*x + a)^2*b*d^4/(d*x +
c)^2 - (b*x + a)^3*d^5/(d*x + c)^3) + ((b*x + a)*B*b^5*c^4*d*g*i*n/(d*x + c) - 4*(b*x + a)*B*a*b^4*c^3*d^2*g*i
*n/(d*x + c) - (b*x + a)^2*B*b^4*c^4*d^2*g*i*n/(d*x + c)^2 + 6*(b*x + a)*B*a^2*b^3*c^2*d^3*g*i*n/(d*x + c) + 4
*(b*x + a)^2*B*a*b^3*c^3*d^3*g*i*n/(d*x + c)^2 - 4*(b*x + a)*B*a^3*b^2*c*d^4*g*i*n/(d*x + c) - 6*(b*x + a)^2*B
*a^2*b^2*c^2*d^4*g*i*n/(d*x + c)^2 + (b*x + a)*B*a^4*b*d^5*g*i*n/(d*x + c) + 4*(b*x + a)^2*B*a^3*b*c*d^5*g*i*n
/(d*x + c)^2 - (b*x + a)^2*B*a^4*d^6*g*i*n/(d*x + c)^2 + A*b^6*c^4*g*i + B*b^6*c^4*g*i - 4*A*a*b^5*c^3*d*g*i -
4*B*a*b^5*c^3*d*g*i - 3*(b*x + a)*A*b^5*c^4*d*g*i/(d*x + c) - 3*(b*x + a)*B*b^5*c^4*d*g*i/(d*x + c) + 6*A*a^2
*b^4*c^2*d^2*g*i + 6*B*a^2*b^4*c^2*d^2*g*i + 12*(b*x + a)*A*a*b^4*c^3*d^2*g*i/(d*x + c) + 12*(b*x + a)*B*a*b^4
*c^3*d^2*g*i/(d*x + c) - 4*A*a^3*b^3*c*d^3*g*i - 4*B*a^3*b^3*c*d^3*g*i - 18*(b*x + a)*A*a^2*b^3*c^2*d^3*g*i/(d
*x + c) - 18*(b*x + a)*B*a^2*b^3*c^2*d^3*g*i/(d*x + c) + A*a^4*b^2*d^4*g*i + B*a^4*b^2*d^4*g*i + 12*(b*x + a)*
A*a^3*b^2*c*d^4*g*i/(d*x + c) + 12*(b*x + a)*B*a^3*b^2*c*d^4*g*i/(d*x + c) - 3*(b*x + a)*A*a^4*b*d^5*g*i/(d*x
+ c) - 3*(b*x + a)*B*a^4*b*d^5*g*i/(d*x + c))/(b^4*d^2 - 3*(b*x + a)*b^3*d^3/(d*x + c) + 3*(b*x + a)^2*b^2*d^4
/(d*x + c)^2 - (b*x + a)^3*b*d^5/(d*x + c)^3) + (B*b^4*c^4*g*i*n - 4*B*a*b^3*c^3*d*g*i*n + 6*B*a^2*b^2*c^2*d^2
*g*i*n - 4*B*a^3*b*c*d^3*g*i*n + B*a^4*d^4*g*i*n)*log(b - (b*x + a)*d/(d*x + c))/(b^2*d^2) - (B*b^4*c^4*g*i*n
- 4*B*a*b^3*c^3*d*g*i*n + 6*B*a^2*b^2*c^2*d^2*g*i*n - 4*B*a^3*b*c*d^3*g*i*n + B*a^4*d^4*g*i*n)*log((b*x + a)/(
d*x + c))/(b^2*d^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right ) \left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)*(d*i*x+c*i)*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

[Out]

int((b*g*x+a*g)*(d*i*x+c*i)*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

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maxima [B]  time = 1.21, size = 393, normalized size = 2.64 \[ \frac {1}{3} \, B b d g i x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{3} \, A b d g i x^{3} + \frac {1}{2} \, B b c g i x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, B a d g i x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{2} \, A b c g i x^{2} + \frac {1}{2} \, A a d g i x^{2} + \frac {1}{6} \, B b d g i n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac {1}{2} \, B b c g i n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} - \frac {1}{2} \, B a d g i n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + B a c g i n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B a c g i x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A a c g i x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.84, size = 295, normalized size = 1.98 \[ \ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,b\,d\,g\,i\,x^3}{3}+\frac {B\,g\,i\,\left (a\,d+b\,c\right )\,x^2}{2}+B\,a\,c\,g\,i\,x\right )-x\,\left (\frac {\left (\frac {g\,i\,\left (6\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3}-\frac {A\,g\,i\,\left (6\,a\,d+6\,b\,c\right )}{6}\right )\,\left (6\,a\,d+6\,b\,c\right )}{6\,b\,d}+A\,a\,c\,g\,i-\frac {g\,i\,\left (2\,A\,a^2\,d^2+2\,A\,b^2\,c^2+B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+8\,A\,a\,b\,c\,d\right )}{2\,b\,d}\right )+x^2\,\left (\frac {g\,i\,\left (6\,A\,a\,d+6\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{6}-\frac {A\,g\,i\,\left (6\,a\,d+6\,b\,c\right )}{12}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^3\,d\,g\,i\,n-3\,B\,a^2\,b\,c\,g\,i\,n\right )}{6\,b^2}+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^3\,g\,i\,n-3\,B\,a\,c^2\,d\,g\,i\,n\right )}{6\,d^2}+\frac {A\,b\,d\,g\,i\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*((a + b*x)/(c + d*x))^n)*((B*g*i*x^2*(a*d + b*c))/2 + (B*b*d*g*i*x^3)/3 + B*a*c*g*i*x) - x*((((g*i*(6*A*
a*d + 6*A*b*c + B*a*d*n - B*b*c*n))/3 - (A*g*i*(6*a*d + 6*b*c))/6)*(6*a*d + 6*b*c))/(6*b*d) + A*a*c*g*i - (g*i
*(2*A*a^2*d^2 + 2*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 8*A*a*b*c*d))/(2*b*d)) + x^2*((g*i*(6*A*a*d + 6*A*b*
c + B*a*d*n - B*b*c*n))/6 - (A*g*i*(6*a*d + 6*b*c))/12) - (log(a + b*x)*(B*a^3*d*g*i*n - 3*B*a^2*b*c*g*i*n))/(
6*b^2) + (log(c + d*x)*(B*b*c^3*g*i*n - 3*B*a*c^2*d*g*i*n))/(6*d^2) + (A*b*d*g*i*x^3)/3

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sympy [A]  time = 60.84, size = 756, normalized size = 5.07 \[ \begin {cases} a c g i x \left (A + B \log {\left (e \left (\frac {a}{c}\right )^{n} \right )}\right ) & \text {for}\: b = 0 \wedge d = 0 \\a g \left (A c i x + \frac {A d i x^{2}}{2} - \frac {B c^{2} i n \log {\left (c + d x \right )}}{2 d} + B c i n x \log {\relax (a )} - B c i n x \log {\left (c + d x \right )} + \frac {B c i n x}{2} + B c i x \log {\relax (e )} + \frac {B d i n x^{2} \log {\relax (a )}}{2} - \frac {B d i n x^{2} \log {\left (c + d x \right )}}{2} + \frac {B d i n x^{2}}{4} + \frac {B d i x^{2} \log {\relax (e )}}{2}\right ) & \text {for}\: b = 0 \\c i \left (A a g x + \frac {A b g x^{2}}{2} + \frac {B a^{2} g n \log {\left (\frac {a}{c} + \frac {b x}{c} \right )}}{2 b} + B a g n x \log {\left (\frac {a}{c} + \frac {b x}{c} \right )} - \frac {B a g n x}{2} + B a g x \log {\relax (e )} + \frac {B b g n x^{2} \log {\left (\frac {a}{c} + \frac {b x}{c} \right )}}{2} - \frac {B b g n x^{2}}{4} + \frac {B b g x^{2} \log {\relax (e )}}{2}\right ) & \text {for}\: d = 0 \\A a c g i x + \frac {A a d g i x^{2}}{2} + \frac {A b c g i x^{2}}{2} + \frac {A b d g i x^{3}}{3} - \frac {B a^{3} d g i n \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{6 b^{2}} - \frac {B a^{3} d g i n \log {\left (\frac {c}{d} + x \right )}}{6 b^{2}} + \frac {B a^{2} c g i n \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{2 b} + \frac {B a^{2} c g i n \log {\left (\frac {c}{d} + x \right )}}{2 b} + \frac {B a^{2} d g i n x}{6 b} - \frac {B a c^{2} g i n \log {\left (\frac {c}{d} + x \right )}}{2 d} + B a c g i n x \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )} + B a c g i x \log {\relax (e )} + \frac {B a d g i n x^{2} \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{2} + \frac {B a d g i n x^{2}}{6} + \frac {B a d g i x^{2} \log {\relax (e )}}{2} + \frac {B b c^{3} g i n \log {\left (\frac {c}{d} + x \right )}}{6 d^{2}} - \frac {B b c^{2} g i n x}{6 d} + \frac {B b c g i n x^{2} \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{2} - \frac {B b c g i n x^{2}}{6} + \frac {B b c g i x^{2} \log {\relax (e )}}{2} + \frac {B b d g i n x^{3} \log {\left (\frac {a}{c + d x} + \frac {b x}{c + d x} \right )}}{3} + \frac {B b d g i x^{3} \log {\relax (e )}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c*g*i*x*(A + B*log(e*(a/c)**n)), Eq(b, 0) & Eq(d, 0)), (a*g*(A*c*i*x + A*d*i*x**2/2 - B*c**2*i*n*
log(c + d*x)/(2*d) + B*c*i*n*x*log(a) - B*c*i*n*x*log(c + d*x) + B*c*i*n*x/2 + B*c*i*x*log(e) + B*d*i*n*x**2*l
og(a)/2 - B*d*i*n*x**2*log(c + d*x)/2 + B*d*i*n*x**2/4 + B*d*i*x**2*log(e)/2), Eq(b, 0)), (c*i*(A*a*g*x + A*b*
g*x**2/2 + B*a**2*g*n*log(a/c + b*x/c)/(2*b) + B*a*g*n*x*log(a/c + b*x/c) - B*a*g*n*x/2 + B*a*g*x*log(e) + B*b
*g*n*x**2*log(a/c + b*x/c)/2 - B*b*g*n*x**2/4 + B*b*g*x**2*log(e)/2), Eq(d, 0)), (A*a*c*g*i*x + A*a*d*g*i*x**2
/2 + A*b*c*g*i*x**2/2 + A*b*d*g*i*x**3/3 - B*a**3*d*g*i*n*log(a/(c + d*x) + b*x/(c + d*x))/(6*b**2) - B*a**3*d
*g*i*n*log(c/d + x)/(6*b**2) + B*a**2*c*g*i*n*log(a/(c + d*x) + b*x/(c + d*x))/(2*b) + B*a**2*c*g*i*n*log(c/d
+ x)/(2*b) + B*a**2*d*g*i*n*x/(6*b) - B*a*c**2*g*i*n*log(c/d + x)/(2*d) + B*a*c*g*i*n*x*log(a/(c + d*x) + b*x/
(c + d*x)) + B*a*c*g*i*x*log(e) + B*a*d*g*i*n*x**2*log(a/(c + d*x) + b*x/(c + d*x))/2 + B*a*d*g*i*n*x**2/6 + B
*a*d*g*i*x**2*log(e)/2 + B*b*c**3*g*i*n*log(c/d + x)/(6*d**2) - B*b*c**2*g*i*n*x/(6*d) + B*b*c*g*i*n*x**2*log(
a/(c + d*x) + b*x/(c + d*x))/2 - B*b*c*g*i*n*x**2/6 + B*b*c*g*i*x**2*log(e)/2 + B*b*d*g*i*n*x**3*log(a/(c + d*
x) + b*x/(c + d*x))/3 + B*b*d*g*i*x**3*log(e)/3, True))

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