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

Optimal. Leaf size=250 \[ -\frac{g i^2 (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2}+\frac{b g i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac{B g i^2 n (b c-a d)^4 \log \left (\frac{a+b x}{c+d x}\right )}{12 b^3 d^2}+\frac{B g i^2 n (b c-a d)^4 \log (c+d x)}{12 b^3 d^2}+\frac{B g i^2 n x (b c-a d)^3}{12 b^2 d}+\frac{B g i^2 n (c+d x)^2 (b c-a d)^2}{24 b d^2}-\frac{B g i^2 n (c+d x)^3 (b c-a d)}{12 d^2} \]

[Out]

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

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Rubi [A]  time = 0.354676, antiderivative size = 210, normalized size of antiderivative = 0.84, 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} \[ -\frac{g i^2 (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2}+\frac{b g i^2 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac{B g i^2 n (b c-a d)^4 \log (a+b x)}{12 b^3 d^2}+\frac{B g i^2 n x (b c-a d)^3}{12 b^2 d}+\frac{B g i^2 n (c+d x)^2 (b c-a d)^2}{24 b d^2}-\frac{B g i^2 n (c+d x)^3 (b c-a d)}{12 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^3*g*i^2*n*x)/(12*b^2*d) + (B*(b*c - a*d)^2*g*i^2*n*(c + d*x)^2)/(24*b*d^2) - (B*(b*c - a*d)*g*i
^2*n*(c + d*x)^3)/(12*d^2) + (B*(b*c - a*d)^4*g*i^2*n*Log[a + b*x])/(12*b^3*d^2) - ((b*c - a*d)*g*i^2*(c + d*x
)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^2) + (b*g*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))/(4*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 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 (119 c+119 d x)^2 (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 (\frac{(-b c+a d) g (119 c+119 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{b g (119 c+119 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{119 d}\right ) \, dx\\ &=\frac{(b g) \int (119 c+119 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{119 d}+\frac{((-b c+a d) g) \int (119 c+119 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}\\ &=-\frac{14161 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14161 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}-\frac{(b B g n) \int \frac{200533921 (b c-a d) (c+d x)^3}{a+b x} \, dx}{56644 d^2}+\frac{(B (b c-a d) g n) \int \frac{1685159 (b c-a d) (c+d x)^2}{a+b x} \, dx}{357 d^2}\\ &=-\frac{14161 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14161 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}-\frac{(14161 b B (b c-a d) g n) \int \frac{(c+d x)^3}{a+b x} \, dx}{4 d^2}+\frac{\left (14161 B (b c-a d)^2 g n\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{3 d^2}\\ &=-\frac{14161 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14161 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}-\frac{(14161 b B (b c-a d) g n) \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^2}+\frac{\left (14161 B (b c-a d)^2 g n\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{3 d^2}\\ &=\frac{14161 B (b c-a d)^3 g n x}{12 b^2 d}+\frac{14161 B (b c-a d)^2 g n (c+d x)^2}{24 b d^2}-\frac{14161 B (b c-a d) g n (c+d x)^3}{12 d^2}+\frac{14161 B (b c-a d)^4 g n \log (a+b x)}{12 b^3 d^2}-\frac{14161 (b c-a d) g (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac{14161 b g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.199186, size = 224, normalized size = 0.9 \[ \frac{g i^2 \left (6 b (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-8 (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{4 B n (b c-a d)^2 \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )}{b^3}-\frac{B n (b c-a d) \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )}{b^3}\right )}{24 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.507, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \left ( dix+ci \right ) ^{2} \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)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

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

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Maxima [B]  time = 1.33983, size = 999, normalized size = 4. \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)*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.65275, size = 1107, normalized size = 4.43 \begin{align*} \frac{6 \, A b^{4} d^{4} g i^{2} x^{4} + 2 \,{\left (6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} g i^{2} n \log \left (b x + a\right ) + 2 \,{\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} g i^{2} n \log \left (d x + c\right ) - 2 \,{\left ({\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g i^{2} n - 4 \,{\left (2 \, A b^{4} c d^{3} + A a b^{3} d^{4}\right )} g i^{2}\right )} x^{3} -{\left ({\left (5 \, B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} - B a^{2} b^{2} d^{4}\right )} g i^{2} n - 12 \,{\left (A b^{4} c^{2} d^{2} + 2 \, A a b^{3} c d^{3}\right )} g i^{2}\right )} x^{2} + 2 \,{\left (12 \, A a b^{3} c^{2} d^{2} g i^{2} -{\left (B b^{4} c^{3} d + 2 \, B a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g i^{2} n\right )} x + 2 \,{\left (3 \, B b^{4} d^{4} g i^{2} x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} x + 4 \,{\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} x^{3} + 6 \,{\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} x^{2}\right )} \log \left (e\right ) + 2 \,{\left (3 \, B b^{4} d^{4} g i^{2} n x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} n x + 4 \,{\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} n x^{3} + 6 \,{\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} n x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )}{24 \, b^{3} 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)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out