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

Optimal. Leaf size=229 \[ -\frac {B g x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{2 b^3 d^3}+\frac {(f+g x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 g}-\frac {B (b f-a g)^4 \log (a+b x)}{2 b^4 g}-\frac {B g^2 x^2 (b c-a d) (-a d g-b c g+4 b d f)}{4 b^2 d^2}-\frac {B g^3 x^3 (b c-a d)}{6 b d}+\frac {B (d f-c g)^4 \log (c+d x)}{2 d^4 g} \]

[Out]

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

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Rubi [A]  time = 0.32, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2525, 12, 72} \[ -\frac {B g x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{2 b^3 d^3}+\frac {(f+g x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 g}-\frac {B g^2 x^2 (b c-a d) (-a d g-b c g+4 b d f)}{4 b^2 d^2}-\frac {B (b f-a g)^4 \log (a+b x)}{2 b^4 g}-\frac {B g^3 x^3 (b c-a d)}{6 b d}+\frac {B (d f-c g)^4 \log (c+d x)}{2 d^4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 72

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.26, size = 217, normalized size = 0.95 \[ \frac {(f+g x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )-\frac {B \left (6 b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+2 b^3 d^3 g^4 x^3 (b c-a d)+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f)+6 d^4 (b f-a g)^4 \log (a+b x)-6 b^4 (d f-c g)^4 \log (c+d x)\right )}{3 b^4 d^4}}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.31, size = 468, normalized size = 2.04 \[ \frac {3 \, A b^{4} d^{4} g^{3} x^{4} + 2 \, {\left (6 \, A b^{4} d^{4} f g^{2} - {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{3}\right )} x^{3} + 3 \, {\left (6 \, A b^{4} d^{4} f^{2} g - 4 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f g^{2} + {\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} g^{3}\right )} x^{2} + 6 \, {\left (2 \, A b^{4} d^{4} f^{3} - 6 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} f^{2} g + 4 \, {\left (B b^{4} c^{2} d^{2} - B a^{2} b^{2} d^{4}\right )} f g^{2} - {\left (B b^{4} c^{3} d - B a^{3} b d^{4}\right )} g^{3}\right )} x + 6 \, {\left (4 \, B a b^{3} d^{4} f^{3} - 6 \, B a^{2} b^{2} d^{4} f^{2} g + 4 \, B a^{3} b d^{4} f g^{2} - B a^{4} d^{4} g^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (4 \, B b^{4} c d^{3} f^{3} - 6 \, B b^{4} c^{2} d^{2} f^{2} g + 4 \, B b^{4} c^{3} d f g^{2} - B b^{4} c^{4} g^{3}\right )} \log \left (d x + c\right ) + 3 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B b^{4} d^{4} f g^{2} x^{3} + 6 \, B b^{4} d^{4} f^{2} g x^{2} + 4 \, B b^{4} d^{4} f^{3} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{12 \, b^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 167.45, size = 447, normalized size = 1.95 \[ \frac {1}{4} \, {\left (A g^{3} + B g^{3}\right )} x^{4} + \frac {{\left (6 \, A b d f g^{2} + 6 \, B b d f g^{2} - B b c g^{3} + B a d g^{3}\right )} x^{3}}{6 \, b d} + \frac {1}{4} \, {\left (B g^{3} x^{4} + 4 \, B f g^{2} x^{3} + 6 \, B f^{2} g x^{2} + 4 \, B f^{3} x\right )} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {{\left (6 \, A b^{2} d^{2} f^{2} g + 6 \, B b^{2} d^{2} f^{2} g - 4 \, B b^{2} c d f g^{2} + 4 \, B a b d^{2} f g^{2} + B b^{2} c^{2} g^{3} - B a^{2} d^{2} g^{3}\right )} x^{2}}{4 \, b^{2} d^{2}} + \frac {{\left (4 \, B a b^{3} f^{3} - 6 \, B a^{2} b^{2} f^{2} g + 4 \, B a^{3} b f g^{2} - B a^{4} g^{3}\right )} \log \left (b x + a\right )}{2 \, b^{4}} - \frac {{\left (4 \, B c d^{3} f^{3} - 6 \, B c^{2} d^{2} f^{2} g + 4 \, B c^{3} d f g^{2} - B c^{4} g^{3}\right )} \log \left (-d x - c\right )}{2 \, d^{4}} + \frac {{\left (2 \, A b^{3} d^{3} f^{3} + 2 \, B b^{3} d^{3} f^{3} - 6 \, B b^{3} c d^{2} f^{2} g + 6 \, B a b^{2} d^{3} f^{2} g + 4 \, B b^{3} c^{2} d f g^{2} - 4 \, B a^{2} b d^{3} f g^{2} - B b^{3} c^{3} g^{3} + B a^{3} d^{3} g^{3}\right )} x}{2 \, b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 1783, normalized size = 7.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(B*ln((b*x+a)^2/(d*x+c)^2*e)+A),x)

[Out]

-3/2/d^2*A*c^2*f^2*g-3/d^2*B*c^2*f^2*g+1/d^3*A*c^3*f*g^2+3/d^3*B*c^3*f*g^2-1/2/d^3*B*c^3*g^3*x+1/4/d^2*B*c^2*g
^3*x^2+B*g^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*f*x^3+3/2*B*g*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^
2*e)*f^2*x^2-2*B/b*ln(1/(d*x+c))*a*f^3+2/d*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^2*b*f^3+2/d^3*B*g^3
*a/b*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^3+2*d*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*f^3-11/12/d
^4*B*g^3*c^4-1/4/d^4*A*c^4*g^3+1/d*A*c*f^3+A*x^3*f*g^2+3/2*A*x^2*f^2*g+1/4*B*g^3*ln((1/(d*x+c)*a*d-1/(d*x+c)*b
*c+b)^2/d^2*e)*x^4+B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*x*f^3-2/d^2*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/
(d*x+c)*b*c+b)*a^2*c^3*g^3+6/d^3*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^4*b*f*g^2-12/d^2*B/(a*d-b*c)*
ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^3*f*g^2-6/d^2*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^3*b*f^2*g-
6/d^2*B/b*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^2*f*g^2+6/d*B*g/b*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c*f^2+12
/d*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^2*f^2*g-6*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)
*a^2*c*f^2*g+1/4*A*x^4*g^3+A*x*f^3+6/d*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c^2*f*g^2+1/2/d*B*g
^3*a^3/b^3*c+1/4/d^2*B*g^3*a^2/b^2*c^2+1/6/d^3*B*g^3*a/b*c^3+1/d^3*B*g^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/
d^2*e)*f*c^3-3/d^2*B*g*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^2*f^2-3/2/d^2*B*g*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b
)^2/d^2*e)*f^2*c^2+4/d^3*B*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^3*f*g^2-1/2/d^4*B*ln(1/(d*x+c))*c^4*g^3-1/4/d^4
*B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c^4*g^3+1/d*B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c*f^3
+2/d*B*ln(1/(d*x+c))*c*f^3-3/2/d^4*B*g^3*c^4*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)-1/2*B*g^3*a^4/b^4*ln(1/(d*x+c)*
a*d-1/(d*x+c)*b*c+b)+1/2*B*g^3*a^4/b^4*ln(1/(d*x+c))+1/6*B*g^3*a/b*x^3-1/4*B*g^3*a^2/b^2*x^2+1/2*B*g^3*a^3/b^3
*x-1/6/d*B*g^3*c*x^3+2*B*g^2*a^3/b^3*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*f-2*B*g^2*a^3/b^3*ln(1/(d*x+c))*f+3*B*g
/b*a*f^2*x-2*B*g^2*a^2/b^2*f*x+2/d^2*B*c^2*f*g^2*x-1/d*B*g^2*c*f*x^2-3/d*B*g*c*f^2*x+2/d^3*B*ln(1/(d*x+c))*c^3
*f*g^2-3*B*g/b^2*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*f^2+3*B*g/b^2*ln(1/(d*x+c))*a^2*f^2+B*g^2*a/b*f*x^2-3/d
^2*B*ln(1/(d*x+c))*c^2*f^2*g-4*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c*f^3+4/d^3*B/(a*d-b*c)*ln(1/(d
*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^4*g^3-2/d^4*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^5*b*g^3+3/d*B*g/b*a
*f^2*c-1/d^2*B/b*a*c^2*f*g^2-2/d*B*g^2*a^2/b^2*f*c

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maxima [B]  time = 1.06, size = 623, normalized size = 2.72 \[ \frac {1}{4} \, A g^{3} x^{4} + A f g^{2} x^{3} + \frac {3}{2} \, A f^{2} g x^{2} + {\left (x \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B f^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B f^{2} g + {\left (x^{3} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B f g^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B g^{3} + A f^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.03, size = 743, normalized size = 3.24 \[ \ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (B\,f^3\,x+\frac {3\,B\,f^2\,g\,x^2}{2}+B\,f\,g^2\,x^3+\frac {B\,g^3\,x^4}{4}\right )+x\,\left (\frac {2\,A\,b\,d\,f^3+6\,A\,a\,c\,f\,g^2+6\,A\,a\,d\,f^2\,g+6\,A\,b\,c\,f^2\,g+6\,B\,a\,d\,f^2\,g-6\,B\,b\,c\,f^2\,g}{2\,b\,d}+\frac {\left (2\,a\,d+2\,b\,c\right )\,\left (\frac {\left (\frac {2\,A\,a\,d\,g^3+2\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+6\,A\,b\,d\,f\,g^2}{2\,b\,d}-\frac {A\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}\right )\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}-\frac {2\,A\,a\,c\,g^3+6\,A\,a\,d\,f\,g^2+6\,A\,b\,c\,f\,g^2+6\,A\,b\,d\,f^2\,g+4\,B\,a\,d\,f\,g^2-4\,B\,b\,c\,f\,g^2}{2\,b\,d}+\frac {A\,a\,c\,g^3}{b\,d}\right )}{2\,b\,d}-\frac {a\,c\,\left (\frac {2\,A\,a\,d\,g^3+2\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+6\,A\,b\,d\,f\,g^2}{2\,b\,d}-\frac {A\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (\frac {2\,A\,a\,d\,g^3+2\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+6\,A\,b\,d\,f\,g^2}{2\,b\,d}-\frac {A\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}\right )\,\left (2\,a\,d+2\,b\,c\right )}{4\,b\,d}-\frac {2\,A\,a\,c\,g^3+6\,A\,a\,d\,f\,g^2+6\,A\,b\,c\,f\,g^2+6\,A\,b\,d\,f^2\,g+4\,B\,a\,d\,f\,g^2-4\,B\,b\,c\,f\,g^2}{4\,b\,d}+\frac {A\,a\,c\,g^3}{2\,b\,d}\right )+x^3\,\left (\frac {2\,A\,a\,d\,g^3+2\,A\,b\,c\,g^3+B\,a\,d\,g^3-B\,b\,c\,g^3+6\,A\,b\,d\,f\,g^2}{6\,b\,d}-\frac {A\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{6\,b\,d}\right )+\frac {A\,g^3\,x^4}{4}-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,g^3-4\,B\,a^3\,b\,f\,g^2+6\,B\,a^2\,b^2\,f^2\,g-4\,B\,a\,b^3\,f^3\right )}{2\,b^4}+\frac {\ln \left (c+d\,x\right )\,\left (B\,c^4\,g^3-4\,B\,c^3\,d\,f\,g^2+6\,B\,c^2\,d^2\,f^2\,g-4\,B\,c\,d^3\,f^3\right )}{2\,d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)^3*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2)),x)

[Out]

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

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sympy [B]  time = 12.64, size = 998, normalized size = 4.36 \[ \frac {A g^{3} x^{4}}{4} - \frac {B a \left (a g - 2 b f\right ) \left (a^{2} g^{2} - 2 a b f g + 2 b^{2} f^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{3} - 4 B a^{3} b c d^{3} f g^{2} + 6 B a^{2} b^{2} c d^{3} f^{2} g + \frac {B a^{2} d^{4} \left (a g - 2 b f\right ) \left (a^{2} g^{2} - 2 a b f g + 2 b^{2} f^{2}\right )}{b} + B a b^{3} c^{4} g^{3} - 4 B a b^{3} c^{3} d f g^{2} + 6 B a b^{3} c^{2} d^{2} f^{2} g - 8 B a b^{3} c d^{3} f^{3} - B a c d^{3} \left (a g - 2 b f\right ) \left (a^{2} g^{2} - 2 a b f g + 2 b^{2} f^{2}\right )}{B a^{4} d^{4} g^{3} - 4 B a^{3} b d^{4} f g^{2} + 6 B a^{2} b^{2} d^{4} f^{2} g - 4 B a b^{3} d^{4} f^{3} + B b^{4} c^{4} g^{3} - 4 B b^{4} c^{3} d f g^{2} + 6 B b^{4} c^{2} d^{2} f^{2} g - 4 B b^{4} c d^{3} f^{3}} \right )}}{2 b^{4}} + \frac {B c \left (c g - 2 d f\right ) \left (c^{2} g^{2} - 2 c d f g + 2 d^{2} f^{2}\right ) \log {\left (x + \frac {B a^{4} c d^{3} g^{3} - 4 B a^{3} b c d^{3} f g^{2} + 6 B a^{2} b^{2} c d^{3} f^{2} g + B a b^{3} c^{4} g^{3} - 4 B a b^{3} c^{3} d f g^{2} + 6 B a b^{3} c^{2} d^{2} f^{2} g - 8 B a b^{3} c d^{3} f^{3} - B a b^{3} c \left (c g - 2 d f\right ) \left (c^{2} g^{2} - 2 c d f g + 2 d^{2} f^{2}\right ) + \frac {B b^{4} c^{2} \left (c g - 2 d f\right ) \left (c^{2} g^{2} - 2 c d f g + 2 d^{2} f^{2}\right )}{d}}{B a^{4} d^{4} g^{3} - 4 B a^{3} b d^{4} f g^{2} + 6 B a^{2} b^{2} d^{4} f^{2} g - 4 B a b^{3} d^{4} f^{3} + B b^{4} c^{4} g^{3} - 4 B b^{4} c^{3} d f g^{2} + 6 B b^{4} c^{2} d^{2} f^{2} g - 4 B b^{4} c d^{3} f^{3}} \right )}}{2 d^{4}} + x^{3} \left (A f g^{2} + \frac {B a g^{3}}{6 b} - \frac {B c g^{3}}{6 d}\right ) + x^{2} \left (\frac {3 A f^{2} g}{2} - \frac {B a^{2} g^{3}}{4 b^{2}} + \frac {B a f g^{2}}{b} + \frac {B c^{2} g^{3}}{4 d^{2}} - \frac {B c f g^{2}}{d}\right ) + x \left (A f^{3} + \frac {B a^{3} g^{3}}{2 b^{3}} - \frac {2 B a^{2} f g^{2}}{b^{2}} + \frac {3 B a f^{2} g}{b} - \frac {B c^{3} g^{3}}{2 d^{3}} + \frac {2 B c^{2} f g^{2}}{d^{2}} - \frac {3 B c f^{2} g}{d}\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 ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(A+B*ln(e*(b*x+a)**2/(d*x+c)**2)),x)

[Out]

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

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