3.10 \(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)
Optimal. Leaf size=423 \[ -\frac{3 b^2 g^3 i^2 (c+d x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^4}+\frac{b^3 g^3 i^2 (c+d x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 d^4}-\frac{g^3 i^2 (c+d x)^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 d^4}+\frac{3 b g^3 i^2 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^4}+\frac{B g^3 i^2 x (b c-a d)^5}{60 b^2 d^3}-\frac{b^2 B g^3 i^2 (c+d x)^5 (b c-a d)}{30 d^4}+\frac{B g^3 i^2 (b c-a d)^6 \log \left (\frac{a+b x}{c+d x}\right )}{60 b^3 d^4}+\frac{B g^3 i^2 (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}+\frac{B g^3 i^2 (c+d x)^2 (b c-a d)^4}{120 b d^4}-\frac{19 B g^3 i^2 (c+d x)^3 (b c-a d)^3}{180 d^4}+\frac{13 b B g^3 i^2 (c+d x)^4 (b c-a d)^2}{120 d^4} \]
[Out]
(B*(b*c - a*d)^5*g^3*i^2*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*(c + d*x)^2)/(120*b*d^4) - (19*B*(b*c - a*
d)^3*g^3*i^2*(c + d*x)^3)/(180*d^4) + (13*b*B*(b*c - a*d)^2*g^3*i^2*(c + d*x)^4)/(120*d^4) - (b^2*B*(b*c - a*d
)*g^3*i^2*(c + d*x)^5)/(30*d^4) + (B*(b*c - a*d)^6*g^3*i^2*Log[(a + b*x)/(c + d*x)])/(60*b^3*d^4) - ((b*c - a*
d)^3*g^3*i^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d^4) + (3*b*(b*c - a*d)^2*g^3*i^2*(c + d*x)^
4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^4) - (3*b^2*(b*c - a*d)*g^3*i^2*(c + d*x)^5*(A + B*Log[(e*(a + b*
x))/(c + d*x)]))/(5*d^4) + (b^3*g^3*i^2*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^4) + (B*(b*c -
a*d)^6*g^3*i^2*Log[c + d*x])/(60*b^3*d^4)
________________________________________________________________________________________
Rubi [A] time = 0.655071, antiderivative size = 330, normalized size of antiderivative =
0.78, number of steps used = 14, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules
used = {2528, 2525, 12, 43} \[ \frac{d^2 g^3 i^2 (a+b x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 b^3}+\frac{g^3 i^2 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^3}+\frac{2 d g^3 i^2 (a+b x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^3}-\frac{B g^3 i^2 x (b c-a d)^5}{60 b^2 d^3}+\frac{B g^3 i^2 (a+b x)^2 (b c-a d)^4}{120 b^3 d^2}+\frac{B g^3 i^2 (b c-a d)^6 \log (c+d x)}{60 b^3 d^4}-\frac{B g^3 i^2 (a+b x)^3 (b c-a d)^3}{180 b^3 d}-\frac{7 B g^3 i^2 (a+b x)^4 (b c-a d)^2}{120 b^3}-\frac{B d g^3 i^2 (a+b x)^5 (b c-a d)}{30 b^3} \]
Antiderivative was successfully verified.
[In]
Int[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
-(B*(b*c - a*d)^5*g^3*i^2*x)/(60*b^2*d^3) + (B*(b*c - a*d)^4*g^3*i^2*(a + b*x)^2)/(120*b^3*d^2) - (B*(b*c - a*
d)^3*g^3*i^2*(a + b*x)^3)/(180*b^3*d) - (7*B*(b*c - a*d)^2*g^3*i^2*(a + b*x)^4)/(120*b^3) - (B*d*(b*c - a*d)*g
^3*i^2*(a + b*x)^5)/(30*b^3) + ((b*c - a*d)^2*g^3*i^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^3
) + (2*d*(b*c - a*d)*g^3*i^2*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b^3) + (d^2*g^3*i^2*(a + b*x
)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*b^3) + (B*(b*c - a*d)^6*g^3*i^2*Log[c + d*x])/(60*b^3*d^4)
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 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 43
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])
Rubi steps
\begin{align*} \int (10 c+10 d x)^2 (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac{100 (b c-a d)^2 (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2}+\frac{200 d (b c-a d) (a g+b g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g}+\frac{100 d^2 (a g+b g x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}\right ) \, dx\\ &=\frac{\left (100 (b c-a d)^2\right ) \int (a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2}+\frac{\left (100 d^2\right ) \int (a g+b g x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g^2}+\frac{(200 d (b c-a d)) \int (a g+b g x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g}\\ &=\frac{25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac{\left (50 B d^2\right ) \int \frac{(b c-a d) g^6 (a+b x)^5}{c+d x} \, dx}{3 b^3 g^3}-\frac{(40 B d (b c-a d)) \int \frac{(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{b^3 g^2}-\frac{\left (25 B (b c-a d)^2\right ) \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{b^3 g}\\ &=\frac{25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac{\left (50 B d^2 (b c-a d) g^3\right ) \int \frac{(a+b x)^5}{c+d x} \, dx}{3 b^3}-\frac{\left (40 B d (b c-a d)^2 g^3\right ) \int \frac{(a+b x)^4}{c+d x} \, dx}{b^3}-\frac{\left (25 B (b c-a d)^3 g^3\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{b^3}\\ &=\frac{25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac{\left (50 B d^2 (b c-a d) g^3\right ) \int \left (\frac{b (b c-a d)^4}{d^5}-\frac{b (b c-a d)^3 (a+b x)}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2}{d^3}-\frac{b (b c-a d) (a+b x)^3}{d^2}+\frac{b (a+b x)^4}{d}+\frac{(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{3 b^3}-\frac{\left (40 B d (b c-a d)^2 g^3\right ) \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}{b^3}-\frac{\left (25 B (b c-a d)^3 g^3\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{b^3}\\ &=-\frac{5 B (b c-a d)^5 g^3 x}{3 b^2 d^3}+\frac{5 B (b c-a d)^4 g^3 (a+b x)^2}{6 b^3 d^2}-\frac{5 B (b c-a d)^3 g^3 (a+b x)^3}{9 b^3 d}-\frac{35 B (b c-a d)^2 g^3 (a+b x)^4}{6 b^3}-\frac{10 B d (b c-a d) g^3 (a+b x)^5}{3 b^3}+\frac{25 (b c-a d)^2 g^3 (a+b x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{40 d (b c-a d) g^3 (a+b x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3}+\frac{50 d^2 g^3 (a+b x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3}+\frac{5 B (b c-a d)^6 g^3 \log (c+d x)}{3 b^3 d^4}\\ \end{align*}
Mathematica [A] time = 0.351404, size = 429, normalized size = 1.01 \[ \frac{g^3 i^2 \left (60 d^6 (a+b x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+144 d^5 (a+b x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+90 d^4 (a+b x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-15 B (b c-a d)^3 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+12 B (b c-a d)^2 \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )-B (b c-a d) \left (20 d^3 (a+b x)^3 (b c-a d)^2+15 d^4 (a+b x)^4 (a d-b c)+30 d^2 (a+b x)^2 (a d-b c)^3+60 b d x (b c-a d)^4-60 (b c-a d)^5 \log (c+d x)+12 d^5 (a+b x)^5\right )\right )}{360 b^3 d^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]),x]
[Out]
(g^3*i^2*(90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^5*(b*c - a*d)*(a + b*x
)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 60*d^6*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 15*B*(b*c
- a*d)^3*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[
c + d*x]) + 12*B*(b*c - a*d)^2*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(
a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) - B*(b*c - a*d)*(60*b*d*(b*c - a*d)^4*x + 30*d
^2*(-(b*c) + a*d)^3*(a + b*x)^2 + 20*d^3*(b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b*x)^4 + 12*d^
5*(a + b*x)^5 - 60*(b*c - a*d)^5*Log[c + d*x])))/(360*b^3*d^4)
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Maple [B] time = 0.242, size = 9298, normalized size = 22. \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
result too large to display
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Maxima [B] time = 1.80372, size = 2415, normalized size = 5.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")
[Out]
1/6*A*b^3*d^2*g^3*i^2*x^6 + 2/5*A*b^3*c*d*g^3*i^2*x^5 + 3/5*A*a*b^2*d^2*g^3*i^2*x^5 + 1/4*A*b^3*c^2*g^3*i^2*x^
4 + 3/2*A*a*b^2*c*d*g^3*i^2*x^4 + 3/4*A*a^2*b*d^2*g^3*i^2*x^4 + A*a*b^2*c^2*g^3*i^2*x^3 + 2*A*a^2*b*c*d*g^3*i^
2*x^3 + 1/3*A*a^3*d^2*g^3*i^2*x^3 + 3/2*A*a^2*b*c^2*g^3*i^2*x^2 + A*a^3*c*d*g^3*i^2*x^2 + (x*log(b*e*x/(d*x +
c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*B*a^3*c^2*g^3*i^2 + 3/2*(x^2*log(b*e*x/(d*x + c) +
a*e/(d*x + c)) - 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^2*b*c^2*g^3*i^2 + 1/2*
(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 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))*B*a*b^2*c^2*g^3*i^2 + 1/24*(6*x^4*log(b*e*x/(d*x + c) + a*e/(
d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d -
a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^3*c^2*g^3*i^2 + (x^2*log(b*e*x/(d*x + c) + a*e/(d*x +
c)) - 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^3*c*d*g^3*i^2 + (2*x^3*log(b*e*x
/(d*x + c) + a*e/(d*x + c)) + 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))*B*a^2*b*c*d*g^3*i^2 + 1/4*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4
*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 +
6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*a*b^2*c*d*g^3*i^2 + 1/30*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) +
12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*
d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*b^3*c*d*g^3*i^2 + 1/6*(2*x^3
*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 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))*B*a^3*d^2*g^3*i^2 + 1/8*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3
)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*a^2*b*d^2*g^3*i^2 + 1/20*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x
+ c)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 -
a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*a*b^2*d^2*g^3*i^2 +
1/360*(60*x^6*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 60*a^6*log(b*x + a)/b^6 + 60*c^6*log(d*x + c)/d^6 - (12*(
b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5
*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5))*B*b^3*d^2*g^3*i^2 + A*a^3*c^2*g^3*i^2*x
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Fricas [A] time = 1.76867, size = 1509, normalized size = 3.57 \begin{align*} \frac{60 \, A b^{6} d^{6} g^{3} i^{2} x^{6} + 12 \,{\left ({\left (12 \, A - B\right )} b^{6} c d^{5} +{\left (18 \, A + B\right )} a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 3 \,{\left ({\left (30 \, A - 7 \, B\right )} b^{6} c^{2} d^{4} + 6 \,{\left (30 \, A - B\right )} a b^{5} c d^{5} +{\left (90 \, A + 13 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} - 2 \,{\left (B b^{6} c^{3} d^{3} - 3 \,{\left (60 \, A - 13 \, B\right )} a b^{5} c^{2} d^{4} - 3 \,{\left (120 \, A + 7 \, B\right )} a^{2} b^{4} c d^{5} -{\left (60 \, A + 19 \, B\right )} a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 3 \,{\left (B b^{6} c^{4} d^{2} - 6 \, B a b^{5} c^{3} d^{3} + 30 \,{\left (6 \, A - B\right )} a^{2} b^{4} c^{2} d^{4} + 2 \,{\left (60 \, A + 17 \, B\right )} a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{3} i^{2} x^{2} - 6 \,{\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 15 \, B a^{2} b^{4} c^{3} d^{3} - 5 \,{\left (12 \, A + B\right )} a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{3} i^{2} x + 6 \,{\left (15 \, B a^{4} b^{2} c^{2} d^{4} - 6 \, B a^{5} b c d^{5} + B a^{6} d^{6}\right )} g^{3} i^{2} \log \left (b x + a\right ) + 6 \,{\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2} - 20 \, B a^{3} b^{3} c^{3} d^{3}\right )} g^{3} i^{2} \log \left (d x + c\right ) + 6 \,{\left (10 \, B b^{6} d^{6} g^{3} i^{2} x^{6} + 60 \, B a^{3} b^{3} c^{2} d^{4} g^{3} i^{2} x + 12 \,{\left (2 \, B b^{6} c d^{5} + 3 \, B a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 15 \,{\left (B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + 3 \, B a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} + 20 \,{\left (3 \, B a b^{5} c^{2} d^{4} + 6 \, B a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 30 \,{\left (3 \, B a^{2} b^{4} c^{2} d^{4} + 2 \, B a^{3} b^{3} c d^{5}\right )} g^{3} i^{2} x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{360 \, b^{3} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")
[Out]
1/360*(60*A*b^6*d^6*g^3*i^2*x^6 + 12*((12*A - B)*b^6*c*d^5 + (18*A + B)*a*b^5*d^6)*g^3*i^2*x^5 + 3*((30*A - 7*
B)*b^6*c^2*d^4 + 6*(30*A - B)*a*b^5*c*d^5 + (90*A + 13*B)*a^2*b^4*d^6)*g^3*i^2*x^4 - 2*(B*b^6*c^3*d^3 - 3*(60*
A - 13*B)*a*b^5*c^2*d^4 - 3*(120*A + 7*B)*a^2*b^4*c*d^5 - (60*A + 19*B)*a^3*b^3*d^6)*g^3*i^2*x^3 + 3*(B*b^6*c^
4*d^2 - 6*B*a*b^5*c^3*d^3 + 30*(6*A - B)*a^2*b^4*c^2*d^4 + 2*(60*A + 17*B)*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^3*
i^2*x^2 - 6*(B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 15*B*a^2*b^4*c^3*d^3 - 5*(12*A + B)*a^3*b^3*c^2*d^4 - 6*B*a^4*b
^2*c*d^5 + B*a^5*b*d^6)*g^3*i^2*x + 6*(15*B*a^4*b^2*c^2*d^4 - 6*B*a^5*b*c*d^5 + B*a^6*d^6)*g^3*i^2*log(b*x + a
) + 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2 - 20*B*a^3*b^3*c^3*d^3)*g^3*i^2*log(d*x + c) + 6*(10
*B*b^6*d^6*g^3*i^2*x^6 + 60*B*a^3*b^3*c^2*d^4*g^3*i^2*x + 12*(2*B*b^6*c*d^5 + 3*B*a*b^5*d^6)*g^3*i^2*x^5 + 15*
(B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + 3*B*a^2*b^4*d^6)*g^3*i^2*x^4 + 20*(3*B*a*b^5*c^2*d^4 + 6*B*a^2*b^4*c*d^5 +
B*a^3*b^3*d^6)*g^3*i^2*x^3 + 30*(3*B*a^2*b^4*c^2*d^4 + 2*B*a^3*b^3*c*d^5)*g^3*i^2*x^2)*log((b*e*x + a*e)/(d*x
+ c)))/(b^3*d^4)
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Sympy [B] time = 14.5041, size = 1761, normalized size = 4.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)**3*(d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c))),x)
[Out]
A*b**3*d**2*g**3*i**2*x**6/6 + B*a**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)*log(x + (B*a**6*c*d**5*
g**3*i**2 - 6*B*a**5*b*c**2*d**4*g**3*i**2 + B*a**5*d**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)/b +
35*B*a**4*b**2*c**3*d**3*g**3*i**2 - B*a**4*c*d**3*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2) - 15*B*a**
3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2*b**4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2)/(B*a**6*d**6*g**3*i**2
- 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**3*b**3*c**3*d**3*g**3*i**2 - 15*B
*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g**3*i**2))/(60*b**3) - B*c**3*g**3
*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3)*log(x + (B*a**6*c*d**5*g**3*i**2 - 6*B*a
**5*b*c**2*d**4*g**3*i**2 + 35*B*a**4*b**2*c**3*d**3*g**3*i**2 - 15*B*a**3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2
*b**4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2 - B*a*b**2*c**3*g**3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 +
6*a*b**2*c**2*d - b**3*c**3) + B*b**3*c**4*g**3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3
*c**3)/d)/(B*a**6*d**6*g**3*i**2 - 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**
3*b**3*c**3*d**3*g**3*i**2 - 15*B*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g*
*3*i**2))/(60*d**4) + x**5*(3*A*a*b**2*d**2*g**3*i**2/5 + 2*A*b**3*c*d*g**3*i**2/5 + B*a*b**2*d**2*g**3*i**2/3
0 - B*b**3*c*d*g**3*i**2/30) + x**4*(3*A*a**2*b*d**2*g**3*i**2/4 + 3*A*a*b**2*c*d*g**3*i**2/2 + A*b**3*c**2*g*
*3*i**2/4 + 13*B*a**2*b*d**2*g**3*i**2/120 - B*a*b**2*c*d*g**3*i**2/20 - 7*B*b**3*c**2*g**3*i**2/120) + (B*a**
3*c**2*g**3*i**2*x + B*a**3*c*d*g**3*i**2*x**2 + B*a**3*d**2*g**3*i**2*x**3/3 + 3*B*a**2*b*c**2*g**3*i**2*x**2
/2 + 2*B*a**2*b*c*d*g**3*i**2*x**3 + 3*B*a**2*b*d**2*g**3*i**2*x**4/4 + B*a*b**2*c**2*g**3*i**2*x**3 + 3*B*a*b
**2*c*d*g**3*i**2*x**4/2 + 3*B*a*b**2*d**2*g**3*i**2*x**5/5 + B*b**3*c**2*g**3*i**2*x**4/4 + 2*B*b**3*c*d*g**3
*i**2*x**5/5 + B*b**3*d**2*g**3*i**2*x**6/6)*log(e*(a + b*x)/(c + d*x)) + x**3*(60*A*a**3*d**3*g**3*i**2 + 360
*A*a**2*b*c*d**2*g**3*i**2 + 180*A*a*b**2*c**2*d*g**3*i**2 + 19*B*a**3*d**3*g**3*i**2 + 21*B*a**2*b*c*d**2*g**
3*i**2 - 39*B*a*b**2*c**2*d*g**3*i**2 - B*b**3*c**3*g**3*i**2)/(180*d) + x**2*(120*A*a**3*b*c*d**3*g**3*i**2 +
180*A*a**2*b**2*c**2*d**2*g**3*i**2 + B*a**4*d**4*g**3*i**2 + 34*B*a**3*b*c*d**3*g**3*i**2 - 30*B*a**2*b**2*c
**2*d**2*g**3*i**2 - 6*B*a*b**3*c**3*d*g**3*i**2 + B*b**4*c**4*g**3*i**2)/(120*b*d**2) - x*(-60*A*a**3*b**2*c*
*2*d**3*g**3*i**2 + B*a**5*d**5*g**3*i**2 - 6*B*a**4*b*c*d**4*g**3*i**2 - 5*B*a**3*b**2*c**2*d**3*g**3*i**2 +
15*B*a**2*b**3*c**3*d**2*g**3*i**2 - 6*B*a*b**4*c**4*d*g**3*i**2 + B*b**5*c**5*g**3*i**2)/(60*b**2*d**3)
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Giac [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)^3*(d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")
[Out]
Timed out