3.1.58 \(\int (f+g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [58]

Optimal. Leaf size=235 \[ -\frac {B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n x}{4 b^3 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) n x^2}{8 b^2 d^2}-\frac {B (b c-a d) g^3 n x^3}{12 b d}-\frac {B (b f-a g)^4 n \log (a+b x)}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 g}+\frac {B (d f-c g)^4 n \log (c+d x)}{4 d^4 g} \]

[Out]

-1/4*B*(-a*d+b*c)*g*(a^2*d^2*g^2-a*b*d*g*(-c*g+4*d*f)+b^2*(c^2*g^2-4*c*d*f*g+6*d^2*f^2))*n*x/b^3/d^3-1/8*B*(-a
*d+b*c)*g^2*(-a*d*g-b*c*g+4*b*d*f)*n*x^2/b^2/d^2-1/12*B*(-a*d+b*c)*g^3*n*x^3/b/d-1/4*B*(-a*g+b*f)^4*n*ln(b*x+a
)/b^4/g+1/4*(g*x+f)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/g+1/4*B*(-c*g+d*f)^4*n*ln(d*x+c)/d^4/g

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Rubi [A]
time = 0.21, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2547, 84} \begin {gather*} -\frac {B g n x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{4 b^3 d^3}+\frac {(f+g x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 g}-\frac {B n (b f-a g)^4 \log (a+b x)}{4 b^4 g}-\frac {B g^2 n x^2 (b c-a d) (-a d g-b c g+4 b d f)}{8 b^2 d^2}-\frac {B g^3 n x^3 (b c-a d)}{12 b d}+\frac {B n (d f-c g)^4 \log (c+d x)}{4 d^4 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(B*(b*c - a*d)*g*(a^2*d^2*g^2 - a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*n*x)/(b^3*
d^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - a*d*g)*n*x^2)/(8*b^2*d^2) - (B*(b*c - a*d)*g^3*n*x^3)/(12*b*d) -
(B*(b*f - a*g)^4*n*Log[a + b*x])/(4*b^4*g) + ((f + g*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*g) + (B*(
d*f - c*g)^4*n*Log[c + d*x])/(4*d^4*g)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

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 (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 g}-\frac {(B n) \int \frac {(b c-a d) (f+g x)^4}{(a+b x) (c+d x)} \, dx}{4 g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 g}-\frac {(B (b c-a d) n) \int \frac {(f+g x)^4}{(a+b x) (c+d x)} \, dx}{4 g}\\ &=\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 g}-\frac {(B (b c-a d) n) \int \left (\frac {g^2 \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right )}{b^3 d^3}+\frac {g^3 (4 b d f-b c g-a d g) x}{b^2 d^2}+\frac {g^4 x^2}{b d}+\frac {(b f-a g)^4}{b^3 (b c-a d) (a+b x)}+\frac {(d f-c g)^4}{d^3 (-b c+a d) (c+d x)}\right ) \, dx}{4 g}\\ &=-\frac {B (b c-a d) g \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n x}{4 b^3 d^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-a d g) n x^2}{8 b^2 d^2}-\frac {B (b c-a d) g^3 n x^3}{12 b d}-\frac {B (b f-a g)^4 n \log (a+b x)}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 g}+\frac {B (d f-c g)^4 n \log (c+d x)}{4 d^4 g}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 219, normalized size = 0.93 \begin {gather*} \frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {B n \left (6 b d (b c-a d) g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x+3 b^2 d^2 (b c-a d) g^3 (4 b d f-b c g-a d g) x^2+2 b^3 d^3 (b c-a d) g^4 x^3+6 d^4 (b f-a g)^4 \log (a+b x)-6 b^4 (d f-c g)^4 \log (c+d x)\right )}{6 b^4 d^4}}{4 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (g x +f \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

int((g*x+f)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

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Maxima [A]
time = 0.29, size = 447, normalized size = 1.90 \begin {gather*} \frac {1}{4} \, B g^{3} x^{4} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{4} \, A g^{3} x^{4} + B f g^{2} x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + A f g^{2} x^{3} + \frac {3}{2} \, B f^{2} g x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {3}{2} \, A f^{2} g x^{2} - \frac {1}{24} \, B g^{3} n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{2} \, B f g^{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 )} - \frac {3}{2} \, B f^{2} g 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 f^{3} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B f^{3} x \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + A f^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (224) = 448\).
time = 0.46, size = 465, normalized size = 1.98 \begin {gather*} \frac {6 \, {\left (A + B\right )} b^{4} d^{4} g^{3} x^{4} + 2 \, {\left (12 \, {\left (A + B\right )} b^{4} d^{4} f g^{2} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{3} n\right )} x^{3} + 3 \, {\left (12 \, {\left (A + B\right )} b^{4} d^{4} f^{2} g - {\left (4 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f g^{2} - {\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} g^{3}\right )} n\right )} x^{2} + 6 \, {\left (4 \, B a b^{3} d^{4} f^{3} - 6 \, B a^{2} b^{2} d^{4} f^{2} g + 4 \, B a^{3} b d^{4} f g^{2} - B a^{4} d^{4} g^{3}\right )} n \log \left (b x + a\right ) - 6 \, {\left (4 \, B b^{4} c d^{3} f^{3} - 6 \, B b^{4} c^{2} d^{2} f^{2} g + 4 \, B b^{4} c^{3} d f g^{2} - B b^{4} c^{4} g^{3}\right )} n \log \left (d x + c\right ) + 6 \, {\left (4 \, {\left (A + B\right )} b^{4} d^{4} f^{3} - {\left (6 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f^{2} g - 4 \, {\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} f g^{2} + {\left (B b^{4} c^{3} d - B a^{3} b d^{4}\right )} g^{3}\right )} n\right )} x + 6 \, {\left (B b^{4} d^{4} g^{3} n x^{4} + 4 \, B b^{4} d^{4} f g^{2} n x^{3} + 6 \, B b^{4} d^{4} f^{2} g n x^{2} + 4 \, B b^{4} d^{4} f^{3} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((g*x+f)**3*(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. 6660 vs. \(2 (224) = 448\).
time = 5.35, size = 6660, normalized size = 28.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

1/24*(6*(4*B*b^5*c^2*d^3*f^3*n - 8*B*a*b^4*c*d^4*f^3*n - 12*(b*x + a)*B*b^4*c^2*d^4*f^3*n/(d*x + c) + 4*B*a^2*
b^3*d^5*f^3*n + 24*(b*x + a)*B*a*b^3*c*d^5*f^3*n/(d*x + c) + 12*(b*x + a)^2*B*b^3*c^2*d^5*f^3*n/(d*x + c)^2 -
12*(b*x + a)*B*a^2*b^2*d^6*f^3*n/(d*x + c) - 24*(b*x + a)^2*B*a*b^2*c*d^6*f^3*n/(d*x + c)^2 - 4*(b*x + a)^3*B*
b^2*c^2*d^6*f^3*n/(d*x + c)^3 + 12*(b*x + a)^2*B*a^2*b*d^7*f^3*n/(d*x + c)^2 + 8*(b*x + a)^3*B*a*b*c*d^7*f^3*n
/(d*x + c)^3 - 4*(b*x + a)^3*B*a^2*d^8*f^3*n/(d*x + c)^3 - 6*B*b^5*c^3*d^2*f^2*g*n + 6*B*a*b^4*c^2*d^3*f^2*g*n
+ 24*(b*x + a)*B*b^4*c^3*d^3*f^2*g*n/(d*x + c) + 6*B*a^2*b^3*c*d^4*f^2*g*n - 36*(b*x + a)*B*a*b^3*c^2*d^4*f^2
*g*n/(d*x + c) - 30*(b*x + a)^2*B*b^3*c^3*d^4*f^2*g*n/(d*x + c)^2 - 6*B*a^3*b^2*d^5*f^2*g*n + 54*(b*x + a)^2*B
*a*b^2*c^2*d^5*f^2*g*n/(d*x + c)^2 + 12*(b*x + a)^3*B*b^2*c^3*d^5*f^2*g*n/(d*x + c)^3 + 12*(b*x + a)*B*a^3*b*d
^6*f^2*g*n/(d*x + c) - 18*(b*x + a)^2*B*a^2*b*c*d^6*f^2*g*n/(d*x + c)^2 - 24*(b*x + a)^3*B*a*b*c^2*d^6*f^2*g*n
/(d*x + c)^3 - 6*(b*x + a)^2*B*a^3*d^7*f^2*g*n/(d*x + c)^2 + 12*(b*x + a)^3*B*a^2*c*d^7*f^2*g*n/(d*x + c)^3 +
4*B*b^5*c^4*d*f*g^2*n - 4*B*a*b^4*c^3*d^2*f*g^2*n - 16*(b*x + a)*B*b^4*c^4*d^2*f*g^2*n/(d*x + c) + 16*(b*x + a
)*B*a*b^3*c^3*d^3*f*g^2*n/(d*x + c) + 24*(b*x + a)^2*B*b^3*c^4*d^3*f*g^2*n/(d*x + c)^2 - 4*B*a^3*b^2*c*d^4*f*g
^2*n + 12*(b*x + a)*B*a^2*b^2*c^2*d^4*f*g^2*n/(d*x + c) - 36*(b*x + a)^2*B*a*b^2*c^3*d^4*f*g^2*n/(d*x + c)^2 -
12*(b*x + a)^3*B*b^2*c^4*d^4*f*g^2*n/(d*x + c)^3 + 4*B*a^4*b*d^5*f*g^2*n - 8*(b*x + a)*B*a^3*b*c*d^5*f*g^2*n/
(d*x + c) + 24*(b*x + a)^3*B*a*b*c^3*d^5*f*g^2*n/(d*x + c)^3 - 4*(b*x + a)*B*a^4*d^6*f*g^2*n/(d*x + c) + 12*(b
*x + a)^2*B*a^3*c*d^6*f*g^2*n/(d*x + c)^2 - 12*(b*x + a)^3*B*a^2*c^2*d^6*f*g^2*n/(d*x + c)^3 - B*b^5*c^5*g^3*n
+ B*a*b^4*c^4*d*g^3*n + 4*(b*x + a)*B*b^4*c^5*d*g^3*n/(d*x + c) - 4*(b*x + a)*B*a*b^3*c^4*d^2*g^3*n/(d*x + c)
- 6*(b*x + a)^2*B*b^3*c^5*d^2*g^3*n/(d*x + c)^2 + 6*(b*x + a)^2*B*a*b^2*c^4*d^3*g^3*n/(d*x + c)^2 + 4*(b*x +
a)^3*B*b^2*c^5*d^3*g^3*n/(d*x + c)^3 + B*a^4*b*c*d^4*g^3*n - 4*(b*x + a)*B*a^3*b*c^2*d^4*g^3*n/(d*x + c) + 6*(
b*x + a)^2*B*a^2*b*c^3*d^4*g^3*n/(d*x + c)^2 - 8*(b*x + a)^3*B*a*b*c^4*d^4*g^3*n/(d*x + c)^3 - B*a^5*d^5*g^3*n
+ 4*(b*x + a)*B*a^4*c*d^5*g^3*n/(d*x + c) - 6*(b*x + a)^2*B*a^3*c^2*d^5*g^3*n/(d*x + c)^2 + 4*(b*x + a)^3*B*a
^2*c^3*d^5*g^3*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(b^4*d^4 - 4*(b*x + a)*b^3*d^5/(d*x + c) + 6*(b*x + a)^
2*b^2*d^6/(d*x + c)^2 - 4*(b*x + a)^3*b*d^7/(d*x + c)^3 + (b*x + a)^4*d^8/(d*x + c)^4) - (36*B*b^8*c^3*d^2*f^2
*g*n - 108*B*a*b^7*c^2*d^3*f^2*g*n - 108*(b*x + a)*B*b^7*c^3*d^3*f^2*g*n/(d*x + c) + 108*B*a^2*b^6*c*d^4*f^2*g
*n + 324*(b*x + a)*B*a*b^6*c^2*d^4*f^2*g*n/(d*x + c) + 108*(b*x + a)^2*B*b^6*c^3*d^4*f^2*g*n/(d*x + c)^2 - 36*
B*a^3*b^5*d^5*f^2*g*n - 324*(b*x + a)*B*a^2*b^5*c*d^5*f^2*g*n/(d*x + c) - 324*(b*x + a)^2*B*a*b^5*c^2*d^5*f^2*
g*n/(d*x + c)^2 - 36*(b*x + a)^3*B*b^5*c^3*d^5*f^2*g*n/(d*x + c)^3 + 108*(b*x + a)*B*a^3*b^4*d^6*f^2*g*n/(d*x
+ c) + 324*(b*x + a)^2*B*a^2*b^4*c*d^6*f^2*g*n/(d*x + c)^2 + 108*(b*x + a)^3*B*a*b^4*c^2*d^6*f^2*g*n/(d*x + c)
^3 - 108*(b*x + a)^2*B*a^3*b^3*d^7*f^2*g*n/(d*x + c)^2 - 108*(b*x + a)^3*B*a^2*b^3*c*d^7*f^2*g*n/(d*x + c)^3 +
36*(b*x + a)^3*B*a^3*b^2*d^8*f^2*g*n/(d*x + c)^3 - 36*B*b^8*c^4*d*f*g^2*n + 72*B*a*b^7*c^3*d^2*f*g^2*n + 120*
(b*x + a)*B*b^7*c^4*d^2*f*g^2*n/(d*x + c) - 264*(b*x + a)*B*a*b^6*c^3*d^3*f*g^2*n/(d*x + c) - 132*(b*x + a)^2*
B*b^6*c^4*d^3*f*g^2*n/(d*x + c)^2 - 72*B*a^3*b^5*c*d^4*f*g^2*n + 72*(b*x + a)*B*a^2*b^5*c^2*d^4*f*g^2*n/(d*x +
c) + 312*(b*x + a)^2*B*a*b^5*c^3*d^4*f*g^2*n/(d*x + c)^2 + 48*(b*x + a)^3*B*b^5*c^4*d^4*f*g^2*n/(d*x + c)^3 +
36*B*a^4*b^4*d^5*f*g^2*n + 168*(b*x + a)*B*a^3*b^4*c*d^5*f*g^2*n/(d*x + c) - 144*(b*x + a)^2*B*a^2*b^4*c^2*d^
5*f*g^2*n/(d*x + c)^2 - 120*(b*x + a)^3*B*a*b^4*c^3*d^5*f*g^2*n/(d*x + c)^3 - 96*(b*x + a)*B*a^4*b^3*d^6*f*g^2
*n/(d*x + c) - 120*(b*x + a)^2*B*a^3*b^3*c*d^6*f*g^2*n/(d*x + c)^2 + 72*(b*x + a)^3*B*a^2*b^3*c^2*d^6*f*g^2*n/
(d*x + c)^3 + 84*(b*x + a)^2*B*a^4*b^2*d^7*f*g^2*n/(d*x + c)^2 + 24*(b*x + a)^3*B*a^3*b^2*c*d^7*f*g^2*n/(d*x +
c)^3 - 24*(b*x + a)^3*B*a^4*b*d^8*f*g^2*n/(d*x + c)^3 + 11*B*b^8*c^5*g^3*n - 19*B*a*b^7*c^4*d*g^3*n - 38*(b*x
+ a)*B*b^7*c^5*d*g^3*n/(d*x + c) + 2*B*a^2*b^6*c^3*d^2*g^3*n + 70*(b*x + a)*B*a*b^6*c^4*d^2*g^3*n/(d*x + c) +
45*(b*x + a)^2*B*b^6*c^5*d^2*g^3*n/(d*x + c)^2 - 2*B*a^3*b^5*c^2*d^3*g^3*n - 8*(b*x + a)*B*a^2*b^5*c^3*d^3*g^
3*n/(d*x + c) - 93*(b*x + a)^2*B*a*b^5*c^4*d^3*g^3*n/(d*x + c)^2 - 18*(b*x + a)^3*B*b^5*c^5*d^3*g^3*n/(d*x + c
)^3 + 19*B*a^4*b^4*c*d^4*g^3*n - 16*(b*x + a)*B*a^3*b^4*c^2*d^4*g^3*n/(d*x + c) + 30*(b*x + a)^2*B*a^2*b^4*c^3
*d^4*g^3*n/(d*x + c)^2 + 42*(b*x + a)^3*B*a*b^4*c^4*d^4*g^3*n/(d*x + c)^3 - 11*B*a^5*b^3*d^5*g^3*n - 34*(b*x +
a)*B*a^4*b^3*c*d^5*g^3*n/(d*x + c) + 18*(b*x + a)^2*B*a^3*b^3*c^2*d^5*g^3*n/(d*x + c)^2 - 24*(b*x + a)^3*B*a^
2*b^3*c^3*d^5*g^3*n/(d*x + c)^3 + 26*(b*x + a)*B*a^5*b^2*d^6*g^3*n/(d*x + c) + 21*(b*x + a)^2*B*a^4*b^2*c*d^6*
g^3*n/(d*x + c)^2 - 21*(b*x + a)^2*B*a^5*b*d^7*...

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Mupad [B]
time = 4.74, size = 766, normalized size = 3.26 \begin {gather*} x\,\left (\frac {4\,A\,b\,d\,f^3+12\,A\,a\,c\,f\,g^2+12\,A\,a\,d\,f^2\,g+12\,A\,b\,c\,f^2\,g+6\,B\,a\,d\,f^2\,g\,n-6\,B\,b\,c\,f^2\,g\,n}{4\,b\,d}+\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2+B\,a\,d\,g^3\,n-B\,b\,c\,g^3\,n}{4\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {4\,A\,a\,c\,g^3+12\,A\,a\,d\,f\,g^2+12\,A\,b\,c\,f\,g^2+12\,A\,b\,d\,f^2\,g+4\,B\,a\,d\,f\,g^2\,n-4\,B\,b\,c\,f\,g^2\,n}{4\,b\,d}+\frac {A\,a\,c\,g^3}{b\,d}\right )}{4\,b\,d}-\frac {a\,c\,\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2+B\,a\,d\,g^3\,n-B\,b\,c\,g^3\,n}{4\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2+B\,a\,d\,g^3\,n-B\,b\,c\,g^3\,n}{4\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {4\,A\,a\,c\,g^3+12\,A\,a\,d\,f\,g^2+12\,A\,b\,c\,f\,g^2+12\,A\,b\,d\,f^2\,g+4\,B\,a\,d\,f\,g^2\,n-4\,B\,b\,c\,f\,g^2\,n}{8\,b\,d}+\frac {A\,a\,c\,g^3}{2\,b\,d}\right )+x^3\,\left (\frac {4\,A\,a\,d\,g^3+4\,A\,b\,c\,g^3+12\,A\,b\,d\,f\,g^2+B\,a\,d\,g^3\,n-B\,b\,c\,g^3\,n}{12\,b\,d}-\frac {A\,g^3\,\left (4\,a\,d+4\,b\,c\right )}{12\,b\,d}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,f^3\,x+\frac {3\,B\,f^2\,g\,x^2}{2}+B\,f\,g^2\,x^3+\frac {B\,g^3\,x^4}{4}\right )+\frac {A\,g^3\,x^4}{4}-\frac {\ln \left (a+b\,x\right )\,\left (B\,n\,a^4\,g^3-4\,B\,n\,a^3\,b\,f\,g^2+6\,B\,n\,a^2\,b^2\,f^2\,g-4\,B\,n\,a\,b^3\,f^3\right )}{4\,b^4}+\frac {\ln \left (c+d\,x\right )\,\left (B\,n\,c^4\,g^3-4\,B\,n\,c^3\,d\,f\,g^2+6\,B\,n\,c^2\,d^2\,f^2\,g-4\,B\,n\,c\,d^3\,f^3\right )}{4\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)

[Out]

x*((4*A*b*d*f^3 + 12*A*a*c*f*g^2 + 12*A*a*d*f^2*g + 12*A*b*c*f^2*g + 6*B*a*d*f^2*g*n - 6*B*b*c*f^2*g*n)/(4*b*d
) + ((4*a*d + 4*b*c)*((((4*A*a*d*g^3 + 4*A*b*c*g^3 + 12*A*b*d*f*g^2 + B*a*d*g^3*n - B*b*c*g^3*n)/(4*b*d) - (A*
g^3*(4*a*d + 4*b*c))/(4*b*d))*(4*a*d + 4*b*c))/(4*b*d) - (4*A*a*c*g^3 + 12*A*a*d*f*g^2 + 12*A*b*c*f*g^2 + 12*A
*b*d*f^2*g + 4*B*a*d*f*g^2*n - 4*B*b*c*f*g^2*n)/(4*b*d) + (A*a*c*g^3)/(b*d)))/(4*b*d) - (a*c*((4*A*a*d*g^3 + 4
*A*b*c*g^3 + 12*A*b*d*f*g^2 + B*a*d*g^3*n - B*b*c*g^3*n)/(4*b*d) - (A*g^3*(4*a*d + 4*b*c))/(4*b*d)))/(b*d)) -
x^2*((((4*A*a*d*g^3 + 4*A*b*c*g^3 + 12*A*b*d*f*g^2 + B*a*d*g^3*n - B*b*c*g^3*n)/(4*b*d) - (A*g^3*(4*a*d + 4*b*
c))/(4*b*d))*(4*a*d + 4*b*c))/(8*b*d) - (4*A*a*c*g^3 + 12*A*a*d*f*g^2 + 12*A*b*c*f*g^2 + 12*A*b*d*f^2*g + 4*B*
a*d*f*g^2*n - 4*B*b*c*f*g^2*n)/(8*b*d) + (A*a*c*g^3)/(2*b*d)) + x^3*((4*A*a*d*g^3 + 4*A*b*c*g^3 + 12*A*b*d*f*g
^2 + B*a*d*g^3*n - B*b*c*g^3*n)/(12*b*d) - (A*g^3*(4*a*d + 4*b*c))/(12*b*d)) + log(e*((a + b*x)/(c + d*x))^n)*
((B*g^3*x^4)/4 + B*f^3*x + (3*B*f^2*g*x^2)/2 + B*f*g^2*x^3) + (A*g^3*x^4)/4 - (log(a + b*x)*(B*a^4*g^3*n - 4*B
*a*b^3*f^3*n - 4*B*a^3*b*f*g^2*n + 6*B*a^2*b^2*f^2*g*n))/(4*b^4) + (log(c + d*x)*(B*c^4*g^3*n - 4*B*c*d^3*f^3*
n - 4*B*c^3*d*f*g^2*n + 6*B*c^2*d^2*f^2*g*n))/(4*d^4)

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