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

Optimal. Leaf size=227 \[ -\frac{B g x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{4 b^3 d^3}+\frac{(f+g x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g}-\frac{B g^2 x^2 (b c-a d) (-a d g-b c g+4 b d f)}{8 b^2 d^2}-\frac{B (b f-a g)^4 \log (a+b x)}{4 b^4 g}-\frac{B g^3 x^3 (b c-a d)}{12 b d}+\frac{B (d f-c g)^4 \log (c+d x)}{4 d^4 g} \]

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

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Rubi [A]  time = 0.341136, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2525, 12, 72} \[ -\frac{B g x (b c-a d) \left (a^2 d^2 g^2-a b d g (4 d f-c g)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )}{4 b^3 d^3}+\frac{(f+g x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g}-\frac{B g^2 x^2 (b c-a d) (-a d g-b c g+4 b d f)}{8 b^2 d^2}-\frac{B (b f-a g)^4 \log (a+b x)}{4 b^4 g}-\frac{B g^3 x^3 (b c-a d)}{12 b d}+\frac{B (d f-c g)^4 \log (c+d x)}{4 d^4 g} \]

Antiderivative was successfully verified.

[In]

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.261989, size = 215, normalized size = 0.95 \[ \frac{(f+g x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{B \left (6 b d g^2 x (b c-a d) \left (a^2 d^2 g^2+a b d g (c g-4 d f)+b^2 \left (c^2 g^2-4 c d f g+6 d^2 f^2\right )\right )+3 b^2 d^2 g^3 x^2 (b c-a d) (-a d g-b c g+4 b d f)+2 b^3 d^3 g^4 x^3 (b c-a d)+6 d^4 (b f-a g)^4 \log (a+b x)-6 b^4 (d f-c g)^4 \log (c+d x)\right )}{6 b^4 d^4}}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.2, size = 8605, normalized size = 37.9 \begin{align*} \text{output too large to display} \end{align*}

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))),x)

[Out]

result too large to display

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Maxima [A]  time = 1.20154, size = 560, normalized size = 2.47 \begin{align*} \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 e x}{d x + c} + \frac{a e}{d x + c}\right ) + \frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} B f^{3} + \frac{3}{2} \,{\left (x^{2} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \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^{2} g + \frac{1}{2} \,{\left (2 \, x^{3} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\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}{24} \,{\left (6 \, x^{4} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\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 \end{align*}

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))),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*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c
*log(d*x + c)/d)*B*f^3 + 3/2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 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^2*g + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 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*f*g^2 + 1/24*(6
*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 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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Fricas [B]  time = 1.83661, size = 895, normalized size = 3.94 \begin{align*} \frac{6 \, A b^{4} d^{4} g^{3} x^{4} + 2 \,{\left (12 \, 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 (12 \, 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 (4 \, 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 ) + 6 \,{\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 e x + a e}{d x + c}\right )}{24 \, b^{4} d^{4}} \end{align*}

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))),x, algorithm="fricas")

[Out]

1/24*(6*A*b^4*d^4*g^3*x^4 + 2*(12*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*(12*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*(4*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) + 6*(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*e*x + a*e)/(d*x + c)))/(b^4*d^4
)

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Sympy [B]  time = 20.5577, size = 1049, normalized size = 4.62 \begin{align*} \text{result too large to display} \end{align*}

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

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

[Out]

Timed out