3.2.10 \(\int (a g+b g x) (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [110]

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

[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 [A]
time = 0.08, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2559, 2547, 21, 45} \begin {gather*} \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} \end {gather*}

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]

-1/6*(B*(b*c - a*d)^2*g*i*n*x)/(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)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 2547

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x
_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)/(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] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]

Rule 2559

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

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 = 0.19, size = 189, normalized size = 1.27 \begin {gather*} \frac {g i \left (-a^2 B d^2 (3 b c+a d) n \log (a+b x)+b \left (d x \left (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)\right )+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 )+B c \left (b^2 c^2-3 a b c d+6 a^2 d^2\right ) n \log (c+d x)\right )\right )}{6 b^2 d^2} \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right ) \left (d i x +c i \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )\, dx\]

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (141) = 282\).
time = 0.28, size = 388, normalized size = 2.60 \begin {gather*} \frac {1}{3} i \, B b d g x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{3} i \, A b d g x^{3} + \frac {1}{2} i \, B b c g x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{2} i \, B a d g x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{2} i \, A b c g x^{2} + \frac {1}{2} i \, A a d g x^{2} + \frac {1}{6} i \, B b d g 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} i \, B b c g 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} i \, B a d g 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 )} + i \, B a c g n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + i \, B a c g x \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + i \, A a c g x \end {gather*}

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

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Fricas [A]
time = 0.41, size = 281, normalized size = 1.89 \begin {gather*} -\frac {2 \, {\left (-i \, A - i \, B\right )} b^{3} d^{3} g x^{3} - {\left (3 i \, B a^{2} b c d^{2} - i \, B a^{3} d^{3}\right )} g n \log \left (\frac {b x + a}{b}\right ) - {\left (i \, B b^{3} c^{3} - 3 i \, B a b^{2} c^{2} d\right )} g n \log \left (\frac {d x + c}{d}\right ) - {\left ({\left (-i \, B b^{3} c d^{2} + i \, B a b^{2} d^{3}\right )} g n - 3 \, {\left ({\left (-i \, A - i \, B\right )} b^{3} c d^{2} + {\left (-i \, A - i \, B\right )} a b^{2} d^{3}\right )} g\right )} x^{2} + {\left (6 \, {\left (-i \, A - i \, B\right )} a b^{2} c d^{2} g - {\left (-i \, B b^{3} c^{2} d + i \, B a^{2} b d^{3}\right )} g n\right )} x - {\left (2 i \, B b^{3} d^{3} g n x^{3} + 6 i \, B a b^{2} c d^{2} g n x - 3 \, {\left (-i \, B b^{3} c d^{2} - i \, B a b^{2} d^{3}\right )} g n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{6 \, b^{2} d^{2}} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1277 vs. \(2 (141) = 282\).
time = 2.69, size = 1277, normalized size = 8.57 \begin {gather*} -\frac {1}{6} \, {\left (\frac {{\left (i \, B b^{5} c^{4} g n - 4 i \, B a b^{4} c^{3} d g n - \frac {3 \, {\left (i \, b x + i \, a\right )} B b^{4} c^{4} d g n}{d x + c} + 6 i \, B a^{2} b^{3} c^{2} d^{2} g n - \frac {12 \, {\left (-i \, b x - i \, a\right )} B a b^{3} c^{3} d^{2} g n}{d x + c} - 4 i \, B a^{3} b^{2} c d^{3} g n - \frac {18 \, {\left (i \, b x + i \, a\right )} B a^{2} b^{2} c^{2} d^{3} g n}{d x + c} + i \, B a^{4} b d^{4} g n - \frac {12 \, {\left (-i \, b x - i \, a\right )} B a^{3} b c d^{4} g n}{d x + c} - \frac {3 \, {\left (i \, b x + i \, a\right )} B a^{4} d^{5} g n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{3} d^{2} - \frac {3 \, {\left (b x + a\right )} b^{2} d^{3}}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} b d^{4}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{3} d^{5}}{{\left (d x + c\right )}^{3}}} + \frac {\frac {{\left (i \, b x + i \, a\right )} B b^{5} c^{4} d g n}{d x + c} - \frac {4 \, {\left (i \, b x + i \, a\right )} B a b^{4} c^{3} d^{2} g n}{d x + c} - \frac {i \, {\left (b x + a\right )}^{2} B b^{4} c^{4} d^{2} g n}{{\left (d x + c\right )}^{2}} - \frac {6 \, {\left (-i \, b x - i \, a\right )} B a^{2} b^{3} c^{2} d^{3} g n}{d x + c} + \frac {4 i \, {\left (b x + a\right )}^{2} B a b^{3} c^{3} d^{3} g n}{{\left (d x + c\right )}^{2}} - \frac {4 \, {\left (i \, b x + i \, a\right )} B a^{3} b^{2} c d^{4} g n}{d x + c} - \frac {6 i \, {\left (b x + a\right )}^{2} B a^{2} b^{2} c^{2} d^{4} g n}{{\left (d x + c\right )}^{2}} + \frac {{\left (i \, b x + i \, a\right )} B a^{4} b d^{5} g n}{d x + c} + \frac {4 i \, {\left (b x + a\right )}^{2} B a^{3} b c d^{5} g n}{{\left (d x + c\right )}^{2}} - \frac {i \, {\left (b x + a\right )}^{2} B a^{4} d^{6} g n}{{\left (d x + c\right )}^{2}} + i \, A b^{6} c^{4} g + i \, B b^{6} c^{4} g - 4 i \, A a b^{5} c^{3} d g - 4 i \, B a b^{5} c^{3} d g - \frac {3 \, {\left (i \, b x + i \, a\right )} A b^{5} c^{4} d g}{d x + c} - \frac {3 \, {\left (i \, b x + i \, a\right )} B b^{5} c^{4} d g}{d x + c} + 6 i \, A a^{2} b^{4} c^{2} d^{2} g + 6 i \, B a^{2} b^{4} c^{2} d^{2} g - \frac {12 \, {\left (-i \, b x - i \, a\right )} A a b^{4} c^{3} d^{2} g}{d x + c} - \frac {12 \, {\left (-i \, b x - i \, a\right )} B a b^{4} c^{3} d^{2} g}{d x + c} - 4 i \, A a^{3} b^{3} c d^{3} g - 4 i \, B a^{3} b^{3} c d^{3} g - \frac {18 \, {\left (i \, b x + i \, a\right )} A a^{2} b^{3} c^{2} d^{3} g}{d x + c} - \frac {18 \, {\left (i \, b x + i \, a\right )} B a^{2} b^{3} c^{2} d^{3} g}{d x + c} + i \, A a^{4} b^{2} d^{4} g + i \, B a^{4} b^{2} d^{4} g - \frac {12 \, {\left (-i \, b x - i \, a\right )} A a^{3} b^{2} c d^{4} g}{d x + c} - \frac {12 \, {\left (-i \, b x - i \, a\right )} B a^{3} b^{2} c d^{4} g}{d x + c} - \frac {3 \, {\left (i \, b x + i \, a\right )} A a^{4} b d^{5} g}{d x + c} - \frac {3 \, {\left (i \, b x + i \, a\right )} B a^{4} b d^{5} g}{d x + c}}{b^{4} d^{2} - \frac {3 \, {\left (b x + a\right )} b^{3} d^{3}}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} b^{2} d^{4}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{3} b d^{5}}{{\left (d x + c\right )}^{3}}} - \frac {{\left (-i \, B b^{4} c^{4} g n + 4 i \, B a b^{3} c^{3} d g n - 6 i \, B a^{2} b^{2} c^{2} d^{2} g n + 4 i \, B a^{3} b c d^{3} g n - i \, B a^{4} d^{4} g n\right )} \log \left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}{b^{2} d^{2}} - \frac {{\left (i \, B b^{4} c^{4} g n - 4 i \, B a b^{3} c^{3} d g n + 6 i \, B a^{2} b^{2} c^{2} d^{2} g n - 4 i \, B a^{3} b c d^{3} g n + i \, B a^{4} d^{4} g n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{2} d^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \end {gather*}

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*((I*B*b^5*c^4*g*n - 4*I*B*a*b^4*c^3*d*g*n - 3*(I*b*x + I*a)*B*b^4*c^4*d*g*n/(d*x + c) + 6*I*B*a^2*b^3*c^2
*d^2*g*n - 12*(-I*b*x - I*a)*B*a*b^3*c^3*d^2*g*n/(d*x + c) - 4*I*B*a^3*b^2*c*d^3*g*n - 18*(I*b*x + I*a)*B*a^2*
b^2*c^2*d^3*g*n/(d*x + c) + I*B*a^4*b*d^4*g*n - 12*(-I*b*x - I*a)*B*a^3*b*c*d^4*g*n/(d*x + c) - 3*(I*b*x + I*a
)*B*a^4*d^5*g*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) + ((I*b*x + I*a)*B*b^5*c^4*d*g*n/(d*x + c) - 4*(I*b*x + I*a)*B
*a*b^4*c^3*d^2*g*n/(d*x + c) - I*(b*x + a)^2*B*b^4*c^4*d^2*g*n/(d*x + c)^2 - 6*(-I*b*x - I*a)*B*a^2*b^3*c^2*d^
3*g*n/(d*x + c) + 4*I*(b*x + a)^2*B*a*b^3*c^3*d^3*g*n/(d*x + c)^2 - 4*(I*b*x + I*a)*B*a^3*b^2*c*d^4*g*n/(d*x +
c) - 6*I*(b*x + a)^2*B*a^2*b^2*c^2*d^4*g*n/(d*x + c)^2 + (I*b*x + I*a)*B*a^4*b*d^5*g*n/(d*x + c) + 4*I*(b*x +
a)^2*B*a^3*b*c*d^5*g*n/(d*x + c)^2 - I*(b*x + a)^2*B*a^4*d^6*g*n/(d*x + c)^2 + I*A*b^6*c^4*g + I*B*b^6*c^4*g
- 4*I*A*a*b^5*c^3*d*g - 4*I*B*a*b^5*c^3*d*g - 3*(I*b*x + I*a)*A*b^5*c^4*d*g/(d*x + c) - 3*(I*b*x + I*a)*B*b^5*
c^4*d*g/(d*x + c) + 6*I*A*a^2*b^4*c^2*d^2*g + 6*I*B*a^2*b^4*c^2*d^2*g - 12*(-I*b*x - I*a)*A*a*b^4*c^3*d^2*g/(d
*x + c) - 12*(-I*b*x - I*a)*B*a*b^4*c^3*d^2*g/(d*x + c) - 4*I*A*a^3*b^3*c*d^3*g - 4*I*B*a^3*b^3*c*d^3*g - 18*(
I*b*x + I*a)*A*a^2*b^3*c^2*d^3*g/(d*x + c) - 18*(I*b*x + I*a)*B*a^2*b^3*c^2*d^3*g/(d*x + c) + I*A*a^4*b^2*d^4*
g + I*B*a^4*b^2*d^4*g - 12*(-I*b*x - I*a)*A*a^3*b^2*c*d^4*g/(d*x + c) - 12*(-I*b*x - I*a)*B*a^3*b^2*c*d^4*g/(d
*x + c) - 3*(I*b*x + I*a)*A*a^4*b*d^5*g/(d*x + c) - 3*(I*b*x + I*a)*B*a^4*b*d^5*g/(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) - (-I*B*b^4*c^4*g
*n + 4*I*B*a*b^3*c^3*d*g*n - 6*I*B*a^2*b^2*c^2*d^2*g*n + 4*I*B*a^3*b*c*d^3*g*n - I*B*a^4*d^4*g*n)*log(b - (b*x
+ a)*d/(d*x + c))/(b^2*d^2) - (I*B*b^4*c^4*g*n - 4*I*B*a*b^3*c^3*d*g*n + 6*I*B*a^2*b^2*c^2*d^2*g*n - 4*I*B*a^
3*b*c*d^3*g*n + I*B*a^4*d^4*g*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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Mupad [B]
time = 4.84, size = 295, normalized size = 1.98 \begin {gather*} \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} \end {gather*}

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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