3.21 \(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x})) \, dx\)

Optimal. Leaf size=371 \[ \frac{b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 d^3}+\frac{g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^3}-\frac{2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^3}-\frac{B g^2 i^3 x (b c-a d)^5}{60 b^3 d^2}-\frac{B g^2 i^3 (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac{B g^2 i^3 (b c-a d)^6 \log \left (\frac{a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac{B g^2 i^3 (b c-a d)^6 \log (c+d x)}{60 b^4 d^3}-\frac{B g^2 i^3 (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac{7 B g^2 i^3 (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac{b B g^2 i^3 (c+d x)^5 (b c-a d)}{30 d^3} \]

[Out]

-(B*(b*c - a*d)^5*g^2*i^3*x)/(60*b^3*d^2) - (B*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*c - a*
d)^3*g^2*i^3*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*(c + d*x)^4)/(120*d^3) - (b*B*(b*c - a*d)*g
^2*i^3*(c + d*x)^5)/(30*d^3) - (B*(b*c - a*d)^6*g^2*i^3*Log[(a + b*x)/(c + d*x)])/(60*b^4*d^3) + ((b*c - a*d)^
2*g^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A
+ B*Log[(e*(a + b*x))/(c + d*x)]))/(5*d^3) + (b^2*g^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6
*d^3) - (B*(b*c - a*d)^6*g^2*i^3*Log[c + d*x])/(60*b^4*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.674729, antiderivative size = 330, normalized size of antiderivative = 0.89, 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} \[ \frac{b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{6 d^3}+\frac{g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 d^3}-\frac{2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 d^3}-\frac{B g^2 i^3 x (b c-a d)^5}{60 b^3 d^2}-\frac{B g^2 i^3 (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac{B g^2 i^3 (b c-a d)^6 \log (a+b x)}{60 b^4 d^3}-\frac{B g^2 i^3 (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac{7 B g^2 i^3 (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac{b B g^2 i^3 (c+d x)^5 (b c-a d)}{30 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)^5*g^2*i^3*x)/(60*b^3*d^2) - (B*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*c - a*
d)^3*g^2*i^3*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*(c + d*x)^4)/(120*d^3) - (b*B*(b*c - a*d)*g
^2*i^3*(c + d*x)^5)/(30*d^3) - (B*(b*c - a*d)^6*g^2*i^3*Log[a + b*x])/(60*b^4*d^3) + ((b*c - a*d)^2*g^2*i^3*(c
+ d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[(e*(
a + b*x))/(c + d*x)]))/(5*d^3) + (b^2*g^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^3)

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 (21 c+21 d x)^3 (a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (\frac{(-b c+a d)^2 g^2 (21 c+21 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{d^2}-\frac{2 b (b c-a d) g^2 (21 c+21 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{21 d^2}+\frac{b^2 g^2 (21 c+21 d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{441 d^2}\right ) \, dx\\ &=\frac{\left (b^2 g^2\right ) \int (21 c+21 d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{441 d^2}-\frac{\left (2 b (b c-a d) g^2\right ) \int (21 c+21 d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{21 d^2}+\frac{\left ((b c-a d)^2 g^2\right ) \int (21 c+21 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx}{d^2}\\ &=\frac{9261 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac{18522 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}+\frac{3087 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac{\left (b^2 B g^2\right ) \int \frac{85766121 (b c-a d) (c+d x)^5}{a+b x} \, dx}{55566 d^3}+\frac{\left (2 b B (b c-a d) g^2\right ) \int \frac{4084101 (b c-a d) (c+d x)^4}{a+b x} \, dx}{2205 d^3}-\frac{\left (B (b c-a d)^2 g^2\right ) \int \frac{194481 (b c-a d) (c+d x)^3}{a+b x} \, dx}{84 d^3}\\ &=\frac{9261 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac{18522 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}+\frac{3087 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac{\left (3087 b^2 B (b c-a d) g^2\right ) \int \frac{(c+d x)^5}{a+b x} \, dx}{2 d^3}+\frac{\left (18522 b B (b c-a d)^2 g^2\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^3}-\frac{\left (9261 B (b c-a d)^3 g^2\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{4 d^3}\\ &=\frac{9261 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac{18522 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}+\frac{3087 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}-\frac{\left (3087 b^2 B (b c-a d) g^2\right ) \int \left (\frac{d (b c-a d)^4}{b^5}+\frac{(b c-a d)^5}{b^5 (a+b x)}+\frac{d (b c-a d)^3 (c+d x)}{b^4}+\frac{d (b c-a d)^2 (c+d x)^2}{b^3}+\frac{d (b c-a d) (c+d x)^3}{b^2}+\frac{d (c+d x)^4}{b}\right ) \, dx}{2 d^3}+\frac{\left (18522 b B (b c-a d)^2 g^2\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^3}-\frac{\left (9261 B (b c-a d)^3 g^2\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{4 d^3}\\ &=-\frac{3087 B (b c-a d)^5 g^2 x}{20 b^3 d^2}-\frac{3087 B (b c-a d)^4 g^2 (c+d x)^2}{40 b^2 d^3}-\frac{1029 B (b c-a d)^3 g^2 (c+d x)^3}{20 b d^3}+\frac{21609 B (b c-a d)^2 g^2 (c+d x)^4}{40 d^3}-\frac{3087 b B (b c-a d) g^2 (c+d x)^5}{10 d^3}-\frac{3087 B (b c-a d)^6 g^2 \log (a+b x)}{20 b^4 d^3}+\frac{9261 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 d^3}-\frac{18522 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 d^3}+\frac{3087 b^2 g^2 (c+d x)^6 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.320609, size = 429, normalized size = 1.16 \[ \frac{g^2 i^3 \left (60 b^6 (c+d x)^6 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-144 b^5 (c+d x)^5 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+90 b^4 (c+d x)^4 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-15 B (b c-a d)^3 \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )+12 B (b c-a d)^2 \left (6 b^2 (c+d x)^2 (b c-a d)^2+4 b^3 (c+d x)^3 (b c-a d)+12 b d x (b c-a d)^3+12 (b c-a d)^4 \log (a+b x)+3 b^4 (c+d x)^4\right )-B (b c-a d) \left (30 b^2 (c+d x)^2 (b c-a d)^3+20 b^3 (c+d x)^3 (b c-a d)^2+15 b^4 (c+d x)^4 (b c-a d)+60 b d x (b c-a d)^4+60 (b c-a d)^5 \log (a+b x)+12 b^5 (c+d x)^5\right )\right )}{360 b^4 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*i^3*(-15*B*(b*c - a*d)^3*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(
b*c - a*d)^3*Log[a + b*x]) + 12*B*(b*c - a*d)^2*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2 + 4*
b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]) - B*(b*c - a*d)*(60*b*d*(b*c
- a*d)^4*x + 30*b^2*(b*c - a*d)^3*(c + d*x)^2 + 20*b^3*(b*c - a*d)^2*(c + d*x)^3 + 15*b^4*(b*c - a*d)*(c + d*x
)^4 + 12*b^5*(c + d*x)^5 + 60*(b*c - a*d)^5*Log[a + b*x]) + 90*b^4*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]) - 144*b^5*(b*c - a*d)*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 60*b^6*(c + d*x)^
6*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/(360*b^4*d^3)

________________________________________________________________________________________

Maple [B]  time = 0.211, size = 7597, normalized size = 20.5 \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)^2*(d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 1.49185, size = 2415, normalized size = 6.51 \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)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="maxima")

[Out]

1/6*A*b^2*d^3*g^2*i^3*x^6 + 3/5*A*b^2*c*d^2*g^2*i^3*x^5 + 2/5*A*a*b*d^3*g^2*i^3*x^5 + 3/4*A*b^2*c^2*d*g^2*i^3*
x^4 + 3/2*A*a*b*c*d^2*g^2*i^3*x^4 + 1/4*A*a^2*d^3*g^2*i^3*x^4 + 1/3*A*b^2*c^3*g^2*i^3*x^3 + 2*A*a*b*c^2*d*g^2*
i^3*x^3 + A*a^2*c*d^2*g^2*i^3*x^3 + A*a*b*c^3*g^2*i^3*x^2 + 3/2*A*a^2*c^2*d*g^2*i^3*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^2*c^3*g^2*i^3 + (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*b*c^3*g^2*i^3 + 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*b^2*c^3*g^2*i^3 + 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*c^2*d*g^2*i^3 + (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*c^2*d*g^2*i^3 + 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*b^2*c^2*d*g^2*i^3 + 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
^2*c*d^2*g^2*i^3 + 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*c*d^2*g^2*i^3 + 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*b^2*c*d^2*g^2*i^3 + 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*a^2*d^3*g^2*i^3 + 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*a*b*d^3*g^2*i^3 +
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^2*d^3*g^2*i^3 + A*a^2*c^3*g^2*i^3*x

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Fricas [B]  time = 1.50269, size = 1509, normalized size = 4.07 \begin{align*} \frac{60 \, A b^{6} d^{6} g^{2} i^{3} x^{6} + 12 \,{\left ({\left (18 \, A - B\right )} b^{6} c d^{5} +{\left (12 \, A + B\right )} a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 3 \,{\left ({\left (90 \, A - 13 \, B\right )} b^{6} c^{2} d^{4} + 6 \,{\left (30 \, A + B\right )} a b^{5} c d^{5} +{\left (30 \, A + 7 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 2 \,{\left ({\left (60 \, A - 19 \, B\right )} b^{6} c^{3} d^{3} + 3 \,{\left (120 \, A - 7 \, B\right )} a b^{5} c^{2} d^{4} + 3 \,{\left (60 \, A + 13 \, B\right )} a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{2} i^{3} x^{3} - 3 \,{\left (B b^{6} c^{4} d^{2} - 2 \,{\left (60 \, A - 17 \, B\right )} a b^{5} c^{3} d^{3} - 30 \,{\left (6 \, A + B\right )} a^{2} b^{4} c^{2} d^{4} - 6 \, B a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{2} i^{3} x^{2} + 6 \,{\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 5 \,{\left (12 \, A - B\right )} a^{2} b^{4} c^{3} d^{3} + 15 \, B 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^{2} i^{3} x + 6 \,{\left (20 \, B a^{3} b^{3} c^{3} d^{3} - 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^{2} i^{3} \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}\right )} g^{2} i^{3} \log \left (d x + c\right ) + 6 \,{\left (10 \, B b^{6} d^{6} g^{2} i^{3} x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} x + 12 \,{\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 15 \,{\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 20 \,{\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} x^{3} + 30 \,{\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} x^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{360 \, b^{4} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="fricas")

[Out]

1/360*(60*A*b^6*d^6*g^2*i^3*x^6 + 12*((18*A - B)*b^6*c*d^5 + (12*A + B)*a*b^5*d^6)*g^2*i^3*x^5 + 3*((90*A - 13
*B)*b^6*c^2*d^4 + 6*(30*A + B)*a*b^5*c*d^5 + (30*A + 7*B)*a^2*b^4*d^6)*g^2*i^3*x^4 + 2*((60*A - 19*B)*b^6*c^3*
d^3 + 3*(120*A - 7*B)*a*b^5*c^2*d^4 + 3*(60*A + 13*B)*a^2*b^4*c*d^5 + B*a^3*b^3*d^6)*g^2*i^3*x^3 - 3*(B*b^6*c^
4*d^2 - 2*(60*A - 17*B)*a*b^5*c^3*d^3 - 30*(6*A + B)*a^2*b^4*c^2*d^4 - 6*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^2*
i^3*x^2 + 6*(B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 5*(12*A - B)*a^2*b^4*c^3*d^3 + 15*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^2*i^3*x + 6*(20*B*a^3*b^3*c^3*d^3 - 15*B*a^4*b^2*c^2*d^4 + 6*B*a^5*b*c*d^5 - B*a^6*d
^6)*g^2*i^3*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)*g^2*i^3*log(d*x + c) + 6*(10
*B*b^6*d^6*g^2*i^3*x^6 + 60*B*a^2*b^4*c^3*d^3*g^2*i^3*x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*x^5 + 15*
(3*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + B*a^2*b^4*d^6)*g^2*i^3*x^4 + 20*(B*b^6*c^3*d^3 + 6*B*a*b^5*c^2*d^4 + 3*B*
a^2*b^4*c*d^5)*g^2*i^3*x^3 + 30*(2*B*a*b^5*c^3*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*x^2)*log((b*e*x + a*e)/(d*x
+ c)))/(b^4*d^3)

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Sympy [B]  time = 13.9354, size = 1761, normalized size = 4.75 \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)**2*(d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c))),x)

[Out]

A*b**2*d**3*g**2*i**3*x**6/6 - B*a**3*g**2*i**3*(a**3*d**3 - 6*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 20*b**3*c**3
)*log(x + (B*a**6*c*d**5*g**2*i**3 - 6*B*a**5*b*c**2*d**4*g**2*i**3 + 15*B*a**4*b**2*c**3*d**3*g**2*i**3 + B*a
**4*d**3*g**2*i**3*(a**3*d**3 - 6*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 20*b**3*c**3)/b - 35*B*a**3*b**3*c**4*d**
2*g**2*i**3 - B*a**3*c*d**2*g**2*i**3*(a**3*d**3 - 6*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 20*b**3*c**3) + 6*B*a*
*2*b**4*c**5*d*g**2*i**3 - B*a*b**5*c**6*g**2*i**3)/(B*a**6*d**6*g**2*i**3 - 6*B*a**5*b*c*d**5*g**2*i**3 + 15*
B*a**4*b**2*c**2*d**4*g**2*i**3 - 20*B*a**3*b**3*c**3*d**3*g**2*i**3 - 15*B*a**2*b**4*c**4*d**2*g**2*i**3 + 6*
B*a*b**5*c**5*d*g**2*i**3 - B*b**6*c**6*g**2*i**3))/(60*b**4) - B*c**4*g**2*i**3*(15*a**2*d**2 - 6*a*b*c*d + b
**2*c**2)*log(x + (B*a**6*c*d**5*g**2*i**3 - 6*B*a**5*b*c**2*d**4*g**2*i**3 + 15*B*a**4*b**2*c**3*d**3*g**2*i*
*3 - 35*B*a**3*b**3*c**4*d**2*g**2*i**3 + 6*B*a**2*b**4*c**5*d*g**2*i**3 - B*a*b**5*c**6*g**2*i**3 + B*a*b**3*
c**4*g**2*i**3*(15*a**2*d**2 - 6*a*b*c*d + b**2*c**2) - B*b**4*c**5*g**2*i**3*(15*a**2*d**2 - 6*a*b*c*d + b**2
*c**2)/d)/(B*a**6*d**6*g**2*i**3 - 6*B*a**5*b*c*d**5*g**2*i**3 + 15*B*a**4*b**2*c**2*d**4*g**2*i**3 - 20*B*a**
3*b**3*c**3*d**3*g**2*i**3 - 15*B*a**2*b**4*c**4*d**2*g**2*i**3 + 6*B*a*b**5*c**5*d*g**2*i**3 - B*b**6*c**6*g*
*2*i**3))/(60*d**3) + x**5*(2*A*a*b*d**3*g**2*i**3/5 + 3*A*b**2*c*d**2*g**2*i**3/5 + B*a*b*d**3*g**2*i**3/30 -
B*b**2*c*d**2*g**2*i**3/30) + x**4*(A*a**2*d**3*g**2*i**3/4 + 3*A*a*b*c*d**2*g**2*i**3/2 + 3*A*b**2*c**2*d*g*
*2*i**3/4 + 7*B*a**2*d**3*g**2*i**3/120 + B*a*b*c*d**2*g**2*i**3/20 - 13*B*b**2*c**2*d*g**2*i**3/120) + (B*a**
2*c**3*g**2*i**3*x + 3*B*a**2*c**2*d*g**2*i**3*x**2/2 + B*a**2*c*d**2*g**2*i**3*x**3 + B*a**2*d**3*g**2*i**3*x
**4/4 + B*a*b*c**3*g**2*i**3*x**2 + 2*B*a*b*c**2*d*g**2*i**3*x**3 + 3*B*a*b*c*d**2*g**2*i**3*x**4/2 + 2*B*a*b*
d**3*g**2*i**3*x**5/5 + B*b**2*c**3*g**2*i**3*x**3/3 + 3*B*b**2*c**2*d*g**2*i**3*x**4/4 + 3*B*b**2*c*d**2*g**2
*i**3*x**5/5 + B*b**2*d**3*g**2*i**3*x**6/6)*log(e*(a + b*x)/(c + d*x)) + x**3*(180*A*a**2*b*c*d**2*g**2*i**3
+ 360*A*a*b**2*c**2*d*g**2*i**3 + 60*A*b**3*c**3*g**2*i**3 + B*a**3*d**3*g**2*i**3 + 39*B*a**2*b*c*d**2*g**2*i
**3 - 21*B*a*b**2*c**2*d*g**2*i**3 - 19*B*b**3*c**3*g**2*i**3)/(180*b) - x**2*(-180*A*a**2*b**2*c**2*d**2*g**2
*i**3 - 120*A*a*b**3*c**3*d*g**2*i**3 + B*a**4*d**4*g**2*i**3 - 6*B*a**3*b*c*d**3*g**2*i**3 - 30*B*a**2*b**2*c
**2*d**2*g**2*i**3 + 34*B*a*b**3*c**3*d*g**2*i**3 + B*b**4*c**4*g**2*i**3)/(120*b**2*d) + x*(60*A*a**2*b**3*c*
*3*d**2*g**2*i**3 + B*a**5*d**5*g**2*i**3 - 6*B*a**4*b*c*d**4*g**2*i**3 + 15*B*a**3*b**2*c**2*d**3*g**2*i**3 -
5*B*a**2*b**3*c**3*d**2*g**2*i**3 - 6*B*a*b**4*c**4*d*g**2*i**3 + B*b**5*c**5*g**2*i**3)/(60*b**3*d**2)

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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)^2*(d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c))),x, algorithm="giac")

[Out]

Timed out