3.2.20 \(\int (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [120]

Optimal. Leaf size=124 \[ -\frac {B (b c-a d)^2 i^2 n x}{3 b^2}-\frac {B (b c-a d) i^2 n (c+d x)^2}{6 b d}-\frac {B (b c-a d)^3 i^2 n \log (a+b x)}{3 b^3 d}+\frac {i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2547, 21, 45} \begin {gather*} \frac {i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d}-\frac {B i^2 n (b c-a d)^3 \log (a+b x)}{3 b^3 d}-\frac {B i^2 n x (b c-a d)^2}{3 b^2}-\frac {B i^2 n (c+d x)^2 (b c-a d)}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(B*(b*c - a*d)^2*i^2*n*x)/b^2 - (B*(b*c - a*d)*i^2*n*(c + d*x)^2)/(6*b*d) - (B*(b*c - a*d)^3*i^2*n*Log[a
+ b*x])/(3*b^3*d) + (i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d)

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]

Rubi steps

\begin {align*} \int (120 c+120 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac {4800 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}-\frac {(B n) \int \frac {1728000 (b c-a d) (c+d x)^2}{a+b x} \, dx}{360 d}\\ &=\frac {4800 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}-\frac {(4800 B (b c-a d) n) \int \frac {(c+d x)^2}{a+b x} \, dx}{d}\\ &=\frac {4800 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}-\frac {(4800 B (b c-a d) n) \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}{d}\\ &=-\frac {4800 B (b c-a d)^2 n x}{b^2}-\frac {2400 B (b c-a d) n (c+d x)^2}{b d}-\frac {4800 B (b c-a d)^3 n \log (a+b x)}{b^3 d}+\frac {4800 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 101, normalized size = 0.81 \begin {gather*} \frac {i^2 \left (-\frac {B (b c-a d) n \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )}{2 b^3}+(c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(i^2*(-1/2*(B*(b*c - a*d)*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 + (c +
d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(3*d)

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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (105) = 210\).
time = 0.27, size = 289, normalized size = 2.33 \begin {gather*} -\frac {1}{3} \, B d^{2} x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {1}{3} \, A d^{2} x^{3} - B c d x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - A c d x^{2} - \frac {1}{6} \, B d^{2} 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 )} + B c d 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 )} - B c^{2} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} - B c^{2} x \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - A c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (105) = 210\).
time = 0.42, size = 232, normalized size = 1.87 \begin {gather*} -\frac {2 \, {\left (A + B\right )} b^{3} d^{3} x^{3} - 2 \, B b^{3} c^{3} n \log \left (\frac {d x + c}{d}\right ) + {\left (6 \, {\left (A + B\right )} b^{3} c d^{2} - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} + 2 \, {\left (3 \, B a b^{2} c^{2} d - 3 \, B a^{2} b c d^{2} + B a^{3} d^{3}\right )} n \log \left (\frac {b x + a}{b}\right ) + 2 \, {\left (3 \, {\left (A + B\right )} b^{3} c^{2} d - {\left (2 \, B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} n\right )} x + 2 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B b^{3} c d^{2} n x^{2} + 3 \, B b^{3} c^{2} d n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{6 \, b^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((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 [B] Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (105) = 210\).
time = 3.11, size = 860, normalized size = 6.94 \begin {gather*} -\frac {1}{6} \, {\left (\frac {2 \, {\left (B b^{4} c^{4} n - 4 \, B a b^{3} c^{3} d n + 6 \, B a^{2} b^{2} c^{2} d^{2} n - 4 \, B a^{3} b c d^{3} n + B a^{4} d^{4} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{3} d - \frac {3 \, {\left (b x + a\right )} b^{2} d^{2}}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} b d^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{3} d^{4}}{{\left (d x + c\right )}^{3}}} - \frac {3 \, B b^{6} c^{4} n - 12 \, B a b^{5} c^{3} d n - \frac {5 \, {\left (b x + a\right )} B b^{5} c^{4} d n}{d x + c} + 18 \, B a^{2} b^{4} c^{2} d^{2} n + \frac {20 \, {\left (b x + a\right )} B a b^{4} c^{3} d^{2} n}{d x + c} + \frac {2 \, {\left (b x + a\right )}^{2} B b^{4} c^{4} d^{2} n}{{\left (d x + c\right )}^{2}} - 12 \, B a^{3} b^{3} c d^{3} n - \frac {30 \, {\left (b x + a\right )} B a^{2} b^{3} c^{2} d^{3} n}{d x + c} - \frac {8 \, {\left (b x + a\right )}^{2} B a b^{3} c^{3} d^{3} n}{{\left (d x + c\right )}^{2}} + 3 \, B a^{4} b^{2} d^{4} n + \frac {20 \, {\left (b x + a\right )} B a^{3} b^{2} c d^{4} n}{d x + c} + \frac {12 \, {\left (b x + a\right )}^{2} B a^{2} b^{2} c^{2} d^{4} n}{{\left (d x + c\right )}^{2}} - \frac {5 \, {\left (b x + a\right )} B a^{4} b d^{5} n}{d x + c} - \frac {8 \, {\left (b x + a\right )}^{2} B a^{3} b c d^{5} n}{{\left (d x + c\right )}^{2}} + \frac {2 \, {\left (b x + a\right )}^{2} B a^{4} d^{6} n}{{\left (d x + c\right )}^{2}} - 2 \, A b^{6} c^{4} - 2 \, B b^{6} c^{4} + 8 \, A a b^{5} c^{3} d + 8 \, B a b^{5} c^{3} d - 12 \, A a^{2} b^{4} c^{2} d^{2} - 12 \, B a^{2} b^{4} c^{2} d^{2} + 8 \, A a^{3} b^{3} c d^{3} + 8 \, B a^{3} b^{3} c d^{3} - 2 \, A a^{4} b^{2} d^{4} - 2 \, B a^{4} b^{2} d^{4}}{b^{5} d - \frac {3 \, {\left (b x + a\right )} b^{4} d^{2}}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} b^{3} d^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{3} b^{2} d^{4}}{{\left (d x + c\right )}^{3}}} + \frac {2 \, {\left (B b^{4} c^{4} n - 4 \, B a b^{3} c^{3} d n + 6 \, B a^{2} b^{2} c^{2} d^{2} n - 4 \, B a^{3} b c d^{3} n + B a^{4} d^{4} n\right )} \log \left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}{b^{3} d} - \frac {2 \, {\left (B b^{4} c^{4} n - 4 \, B a b^{3} c^{3} d n + 6 \, B a^{2} b^{2} c^{2} d^{2} n - 4 \, B a^{3} b c d^{3} n + B a^{4} d^{4} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{3} d}\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((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

-1/6*(2*(B*b^4*c^4*n - 4*B*a*b^3*c^3*d*n + 6*B*a^2*b^2*c^2*d^2*n - 4*B*a^3*b*c*d^3*n + B*a^4*d^4*n)*log((b*x +
a)/(d*x + c))/(b^3*d - 3*(b*x + a)*b^2*d^2/(d*x + c) + 3*(b*x + a)^2*b*d^3/(d*x + c)^2 - (b*x + a)^3*d^4/(d*x
+ c)^3) - (3*B*b^6*c^4*n - 12*B*a*b^5*c^3*d*n - 5*(b*x + a)*B*b^5*c^4*d*n/(d*x + c) + 18*B*a^2*b^4*c^2*d^2*n
+ 20*(b*x + a)*B*a*b^4*c^3*d^2*n/(d*x + c) + 2*(b*x + a)^2*B*b^4*c^4*d^2*n/(d*x + c)^2 - 12*B*a^3*b^3*c*d^3*n
- 30*(b*x + a)*B*a^2*b^3*c^2*d^3*n/(d*x + c) - 8*(b*x + a)^2*B*a*b^3*c^3*d^3*n/(d*x + c)^2 + 3*B*a^4*b^2*d^4*n
+ 20*(b*x + a)*B*a^3*b^2*c*d^4*n/(d*x + c) + 12*(b*x + a)^2*B*a^2*b^2*c^2*d^4*n/(d*x + c)^2 - 5*(b*x + a)*B*a
^4*b*d^5*n/(d*x + c) - 8*(b*x + a)^2*B*a^3*b*c*d^5*n/(d*x + c)^2 + 2*(b*x + a)^2*B*a^4*d^6*n/(d*x + c)^2 - 2*A
*b^6*c^4 - 2*B*b^6*c^4 + 8*A*a*b^5*c^3*d + 8*B*a*b^5*c^3*d - 12*A*a^2*b^4*c^2*d^2 - 12*B*a^2*b^4*c^2*d^2 + 8*A
*a^3*b^3*c*d^3 + 8*B*a^3*b^3*c*d^3 - 2*A*a^4*b^2*d^4 - 2*B*a^4*b^2*d^4)/(b^5*d - 3*(b*x + a)*b^4*d^2/(d*x + c)
+ 3*(b*x + a)^2*b^3*d^3/(d*x + c)^2 - (b*x + a)^3*b^2*d^4/(d*x + c)^3) + 2*(B*b^4*c^4*n - 4*B*a*b^3*c^3*d*n +
6*B*a^2*b^2*c^2*d^2*n - 4*B*a^3*b*c*d^3*n + B*a^4*d^4*n)*log(b - (b*x + a)*d/(d*x + c))/(b^3*d) - 2*(B*b^4*c^
4*n - 4*B*a*b^3*c^3*d*n + 6*B*a^2*b^2*c^2*d^2*n - 4*B*a^3*b*c*d^3*n + B*a^4*d^4*n)*log((b*x + a)/(d*x + c))/(b
^3*d))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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Mupad [B]
time = 4.62, size = 303, normalized size = 2.44 \begin {gather*} \ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,c^2\,i^2\,x+B\,c\,d\,i^2\,x^2+\frac {B\,d^2\,i^2\,x^3}{3}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {d\,i^2\,\left (3\,A\,a\,d+9\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{3\,b}-\frac {A\,d\,i^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,b}\right )}{3\,b\,d}-\frac {c\,i^2\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{b}+\frac {A\,a\,c\,d\,i^2}{b}\right )+x^2\,\left (\frac {d\,i^2\,\left (3\,A\,a\,d+9\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{6\,b}-\frac {A\,d\,i^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,n\,a^3\,d^2\,i^2-3\,B\,n\,a^2\,b\,c\,d\,i^2+3\,B\,n\,a\,b^2\,c^2\,i^2\right )}{3\,b^3}+\frac {A\,d^2\,i^2\,x^3}{3}-\frac {B\,c^3\,i^2\,n\,\ln \left (c+d\,x\right )}{3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*((a + b*x)/(c + d*x))^n)*((B*d^2*i^2*x^3)/3 + B*c^2*i^2*x + B*c*d*i^2*x^2) - x*(((3*a*d + 3*b*c)*((d*i^2
*(3*A*a*d + 9*A*b*c + B*a*d*n - B*b*c*n))/(3*b) - (A*d*i^2*(3*a*d + 3*b*c))/(3*b)))/(3*b*d) - (c*i^2*(3*A*a*d
+ 3*A*b*c + B*a*d*n - B*b*c*n))/b + (A*a*c*d*i^2)/b) + x^2*((d*i^2*(3*A*a*d + 9*A*b*c + B*a*d*n - B*b*c*n))/(6
*b) - (A*d*i^2*(3*a*d + 3*b*c))/(6*b)) + (log(a + b*x)*(B*a^3*d^2*i^2*n + 3*B*a*b^2*c^2*i^2*n - 3*B*a^2*b*c*d*
i^2*n))/(3*b^3) + (A*d^2*i^2*x^3)/3 - (B*c^3*i^2*n*log(c + d*x))/(3*d)

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