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

Optimal. Leaf size=190 \[ \frac{g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A-B n\right )}{12 b^2}+\frac{g^2 i (a+b x)^3 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b}-\frac{B g^2 i n (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}-\frac{B g^2 i n (a+b x)^2 (b c-a d)^2}{24 b^2 d}+\frac{B g^2 i n x (b c-a d)^3}{12 b d^2} \]

[Out]

(B*(b*c - a*d)^3*g^2*i*n*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*n*(a + b*x)^2)/(24*b^2*d) + (g^2*i*(a + b*x)^3
*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A - B*n + B*Log[e*(
(a + b*x)/(c + d*x))^n]))/(12*b^2) - (B*(b*c - a*d)^4*g^2*i*n*Log[c + d*x])/(12*b^2*d^3)

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Rubi [A]  time = 0.315728, antiderivative size = 210, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {2528, 2525, 12, 43} \[ \frac{g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^2}+\frac{d g^2 i (a+b x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2}-\frac{B g^2 i n (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}-\frac{B g^2 i n (a+b x)^2 (b c-a d)^2}{24 b^2 d}-\frac{B g^2 i n (a+b x)^3 (b c-a d)}{12 b^2}+\frac{B g^2 i n x (b c-a d)^3}{12 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^3*g^2*i*n*x)/(12*b*d^2) - (B*(b*c - a*d)^2*g^2*i*n*(a + b*x)^2)/(24*b^2*d) - (B*(b*c - a*d)*g^2
*i*n*(a + b*x)^3)/(12*b^2) + ((b*c - a*d)*g^2*i*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^2) +
(d*g^2*i*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^2) - (B*(b*c - a*d)^4*g^2*i*n*Log[c + d*x])/
(12*b^2*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 (109 c+109 d x) (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac{109 (b c-a d) (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{109 d (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g}\right ) \, dx\\ &=\frac{(109 (b c-a d)) \int (a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}+\frac{(109 d) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}\\ &=\frac{109 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{109 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2}-\frac{(109 B d n) \int \frac{(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{4 b^2 g^2}-\frac{(109 B (b c-a d) n) \int \frac{(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{3 b^2 g}\\ &=\frac{109 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{109 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2}-\frac{\left (109 B d (b c-a d) g^2 n\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{4 b^2}-\frac{\left (109 B (b c-a d)^2 g^2 n\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b^2}\\ &=\frac{109 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{109 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2}-\frac{\left (109 B d (b c-a d) g^2 n\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}{4 b^2}-\frac{\left (109 B (b c-a d)^2 g^2 n\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac{109 B (b c-a d)^3 g^2 n x}{12 b d^2}-\frac{109 B (b c-a d)^2 g^2 n (a+b x)^2}{24 b^2 d}-\frac{109 B (b c-a d) g^2 n (a+b x)^3}{12 b^2}+\frac{109 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{109 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b^2}-\frac{109 B (b c-a d)^4 g^2 n \log (c+d x)}{12 b^2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.16844, size = 225, normalized size = 1.18 \[ \frac{g^2 i \left (6 d (a+b x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+8 (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{4 B n (b c-a d)^2 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )}{d^3}-\frac{B n (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )}{d^3}\right )}{24 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*i*(8*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*d*(a + b*x)^4*(A + B*Log[e*((a +
b*x)/(c + d*x))^n]) + (4*B*(b*c - a*d)^2*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*
x]))/d^3 - (B*(b*c - a*d)*n*(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]))/d^3))/(24*b^2)

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Maple [F]  time = 0.528, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \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)^2*(d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

int((b*g*x+a*g)^2*(d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

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Maxima [B]  time = 1.37145, size = 999, normalized size = 5.26 \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)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

1/4*B*b^2*d*g^2*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^2*d*g^2*i*x^4 + 1/3*B*b^2*c*g^2*i*x^3*l
og(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/3*B*a*b*d*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A*b
^2*c*g^2*i*x^3 + 2/3*A*a*b*d*g^2*i*x^3 + B*a*b*c*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*B*a^2*
d*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*b*c*g^2*i*x^2 + 1/2*A*a^2*d*g^2*i*x^2 - 1/24*B*b^2*d*
g^2*i*n*(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)) + 1/6*B*b^2*c*g^2*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)) + 1/3*B*a*b*d*g^2*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)) - B*a*b*
c*g^2*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^2*d*g^2*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^2*c*g^2*i*n*(a*log(b*x + a)/b - c*log(d*x + c)
/d) + B*a^2*c*g^2*i*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^2*c*g^2*i*x

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Fricas [B]  time = 0.659294, size = 1107, normalized size = 5.83 \begin{align*} \frac{6 \, A b^{4} d^{4} g^{2} i x^{4} + 2 \,{\left (4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{2} i n \log \left (b x + a\right ) - 2 \,{\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} i n \log \left (d x + c\right ) - 2 \,{\left ({\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{2} i n - 4 \,{\left (A b^{4} c d^{3} + 2 \, A a b^{3} d^{4}\right )} g^{2} i\right )} x^{3} -{\left ({\left (B b^{4} c^{2} d^{2} + 4 \, B a b^{3} c d^{3} - 5 \, B a^{2} b^{2} d^{4}\right )} g^{2} i n - 12 \,{\left (2 \, A a b^{3} c d^{3} + A a^{2} b^{2} d^{4}\right )} g^{2} i\right )} x^{2} + 2 \,{\left (12 \, A a^{2} b^{2} c d^{3} g^{2} i +{\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 2 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g^{2} i n\right )} x + 2 \,{\left (3 \, B b^{4} d^{4} g^{2} i x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i x + 4 \,{\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i x^{3} + 6 \,{\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i x^{2}\right )} \log \left (e\right ) + 2 \,{\left (3 \, B b^{4} d^{4} g^{2} i n x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i n x + 4 \,{\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i n x^{3} + 6 \,{\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i n x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )}{24 \, b^{2} 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)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

1/24*(6*A*b^4*d^4*g^2*i*x^4 + 2*(4*B*a^3*b*c*d^3 - B*a^4*d^4)*g^2*i*n*log(b*x + a) - 2*(B*b^4*c^4 - 4*B*a*b^3*
c^3*d + 6*B*a^2*b^2*c^2*d^2)*g^2*i*n*log(d*x + c) - 2*((B*b^4*c*d^3 - B*a*b^3*d^4)*g^2*i*n - 4*(A*b^4*c*d^3 +
2*A*a*b^3*d^4)*g^2*i)*x^3 - ((B*b^4*c^2*d^2 + 4*B*a*b^3*c*d^3 - 5*B*a^2*b^2*d^4)*g^2*i*n - 12*(2*A*a*b^3*c*d^3
+ A*a^2*b^2*d^4)*g^2*i)*x^2 + 2*(12*A*a^2*b^2*c*d^3*g^2*i + (B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 2*B*a^2*b^2*c*
d^3 + B*a^3*b*d^4)*g^2*i*n)*x + 2*(3*B*b^4*d^4*g^2*i*x^4 + 12*B*a^2*b^2*c*d^3*g^2*i*x + 4*(B*b^4*c*d^3 + 2*B*a
*b^3*d^4)*g^2*i*x^3 + 6*(2*B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*g^2*i*x^2)*log(e) + 2*(3*B*b^4*d^4*g^2*i*n*x^4 + 12*
B*a^2*b^2*c*d^3*g^2*i*n*x + 4*(B*b^4*c*d^3 + 2*B*a*b^3*d^4)*g^2*i*n*x^3 + 6*(2*B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*
g^2*i*n*x^2)*log((b*x + a)/(d*x + c)))/(b^2*d^3)

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

[Out]

Timed out