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]

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

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Rubi [B]  time = 0.370136, 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 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]

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 31

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

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 12

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

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*}

Mathematica [A]  time = 0.268147, 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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Maple [F]  time = 0.35, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \left ( dix+ci \right ) \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.68457, size = 531, normalized size = 3.56 \begin{align*} \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 \end{align*}

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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Fricas [B]  time = 0.589842, size = 666, normalized size = 4.47 \begin{align*} \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 \left (e\right ) +{\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}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 10.146, size = 329, normalized size = 2.21 \begin{align*} \frac{1}{3} \,{\left (A b d g i + B b d g i\right )} x^{3} - \frac{1}{6} \,{\left (B b c g i n - B a d g i n - 3 \, A b c g i - 3 \, B b c g i - 3 \, A a d g i - 3 \, B a d g i\right )} x^{2} + \frac{1}{6} \,{\left (2 \, B b d g i n x^{3} + 6 \, B a c g i n x + 3 \,{\left (B b c g i n + B a d g i n\right )} x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (B b^{2} c^{2} g i n - B a^{2} d^{2} g i n - 6 \, A a b c d g i - 6 \, B a b c d g i\right )} x}{6 \, b d} + \frac{{\left (B b c^{3} g i n - 3 \, B a c^{2} d g i n\right )} \log \left (d i x + c i\right )}{6 \, d^{2}} + \frac{{\left (3 \, B a^{2} b c g i n - B a^{3} d g i n\right )} \log \left (b x + a\right )}{6 \, b^{2}} \end{align*}

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]

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