3.8 \(\int (a+b x)^3 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\)

Optimal. Leaf size=172 \[ \frac{q r x (b c-a d)^3}{4 d^3}-\frac{q r (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac{q r (b c-a d)^4 \log (c+d x)}{4 b d^4}+\frac{(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac{q r (a+b x)^3 (b c-a d)}{12 b d}-\frac{p r (a+b x)^4}{16 b}-\frac{q r (a+b x)^4}{16 b} \]

[Out]

((b*c - a*d)^3*q*r*x)/(4*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^2)/(8*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^3)/(12*
b*d) - (p*r*(a + b*x)^4)/(16*b) - (q*r*(a + b*x)^4)/(16*b) - ((b*c - a*d)^4*q*r*Log[c + d*x])/(4*b*d^4) + ((a
+ b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b)

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Rubi [A]  time = 0.0722046, antiderivative size = 172, normalized size of antiderivative = 1., 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 = {2495, 32, 43} \[ \frac{q r x (b c-a d)^3}{4 d^3}-\frac{q r (a+b x)^2 (b c-a d)^2}{8 b d^2}-\frac{q r (b c-a d)^4 \log (c+d x)}{4 b d^4}+\frac{(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac{q r (a+b x)^3 (b c-a d)}{12 b d}-\frac{p r (a+b x)^4}{16 b}-\frac{q r (a+b x)^4}{16 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

((b*c - a*d)^3*q*r*x)/(4*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^2)/(8*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^3)/(12*
b*d) - (p*r*(a + b*x)^4)/(16*b) - (q*r*(a + b*x)^4)/(16*b) - ((b*c - a*d)^4*q*r*Log[c + d*x])/(4*b*d^4) + ((a
+ b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b)

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac{1}{4} (p r) \int (a+b x)^3 \, dx-\frac{(d q r) \int \frac{(a+b x)^4}{c+d x} \, dx}{4 b}\\ &=-\frac{p r (a+b x)^4}{16 b}+\frac{(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac{(d q r) \int \left (-\frac{b (b c-a d)^3}{d^4}+\frac{b (b c-a d)^2 (a+b x)}{d^3}-\frac{b (b c-a d) (a+b x)^2}{d^2}+\frac{b (a+b x)^3}{d}+\frac{(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{4 b}\\ &=\frac{(b c-a d)^3 q r x}{4 d^3}-\frac{(b c-a d)^2 q r (a+b x)^2}{8 b d^2}+\frac{(b c-a d) q r (a+b x)^3}{12 b d}-\frac{p r (a+b x)^4}{16 b}-\frac{q r (a+b x)^4}{16 b}-\frac{(b c-a d)^4 q r \log (c+d x)}{4 b d^4}+\frac{(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.211291, size = 154, normalized size = 0.9 \[ \frac{\frac{r \left (-18 b^2 (p+2 q) (c+d x)^2 (b c-a d)^2+4 b^3 (3 p+4 q) (c+d x)^3 (b c-a d)+12 b d x (p+4 q) (b c-a d)^3-12 q (b c-a d)^4 \log (c+d x)-3 b^4 (p+q) (c+d x)^4\right )}{12 d^4}+(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

((r*(12*b*d*(b*c - a*d)^3*(p + 4*q)*x - 18*b^2*(b*c - a*d)^2*(p + 2*q)*(c + d*x)^2 + 4*b^3*(b*c - a*d)*(3*p +
4*q)*(c + d*x)^3 - 3*b^4*(p + q)*(c + d*x)^4 - 12*(b*c - a*d)^4*q*Log[c + d*x]))/(12*d^4) + (a + b*x)^4*Log[e*
(f*(a + b*x)^p*(c + d*x)^q)^r])/(4*b)

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Maple [F]  time = 0.422, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{3}\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

[Out]

int((b*x+a)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

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Maxima [A]  time = 1.33621, size = 385, normalized size = 2.24 \begin{align*} \frac{1}{4} \,{\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac{{\left (\frac{12 \, a^{4} f p \log \left (b x + a\right )}{b} - \frac{3 \, b^{3} d^{3} f{\left (p + q\right )} x^{4} + 4 \,{\left (a b^{2} d^{3} f{\left (3 \, p + 4 \, q\right )} - b^{3} c d^{2} f q\right )} x^{3} + 6 \,{\left (3 \, a^{2} b d^{3} f{\left (p + 2 \, q\right )} + b^{3} c^{2} d f q - 4 \, a b^{2} c d^{2} f q\right )} x^{2} + 12 \,{\left (a^{3} d^{3} f{\left (p + 4 \, q\right )} - b^{3} c^{3} f q + 4 \, a b^{2} c^{2} d f q - 6 \, a^{2} b c d^{2} f q\right )} x}{d^{3}} - \frac{12 \,{\left (b^{3} c^{4} f q - 4 \, a b^{2} c^{3} d f q + 6 \, a^{2} b c^{2} d^{2} f q - 4 \, a^{3} c d^{3} f q\right )} \log \left (d x + c\right )}{d^{4}}\right )} r}{48 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima")

[Out]

1/4*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2 + 4*a^3*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1/48*(12*a^4*f*p*lo
g(b*x + a)/b - (3*b^3*d^3*f*(p + q)*x^4 + 4*(a*b^2*d^3*f*(3*p + 4*q) - b^3*c*d^2*f*q)*x^3 + 6*(3*a^2*b*d^3*f*(
p + 2*q) + b^3*c^2*d*f*q - 4*a*b^2*c*d^2*f*q)*x^2 + 12*(a^3*d^3*f*(p + 4*q) - b^3*c^3*f*q + 4*a*b^2*c^2*d*f*q
- 6*a^2*b*c*d^2*f*q)*x)/d^3 - 12*(b^3*c^4*f*q - 4*a*b^2*c^3*d*f*q + 6*a^2*b*c^2*d^2*f*q - 4*a^3*c*d^3*f*q)*log
(d*x + c)/d^4)*r/f

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Fricas [B]  time = 0.956594, size = 980, normalized size = 5.7 \begin{align*} -\frac{3 \,{\left (b^{4} d^{4} p + b^{4} d^{4} q\right )} r x^{4} + 4 \,{\left (3 \, a b^{3} d^{4} p -{\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} q\right )} r x^{3} + 6 \,{\left (3 \, a^{2} b^{2} d^{4} p +{\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} q\right )} r x^{2} + 12 \,{\left (a^{3} b d^{4} p -{\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} q\right )} r x - 12 \,{\left (b^{4} d^{4} p r x^{4} + 4 \, a b^{3} d^{4} p r x^{3} + 6 \, a^{2} b^{2} d^{4} p r x^{2} + 4 \, a^{3} b d^{4} p r x + a^{4} d^{4} p r\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} d^{4} q r x^{4} + 4 \, a b^{3} d^{4} q r x^{3} + 6 \, a^{2} b^{2} d^{4} q r x^{2} + 4 \, a^{3} b d^{4} q r x -{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3}\right )} q r\right )} \log \left (d x + c\right ) - 12 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x\right )} \log \left (e\right ) - 12 \,{\left (b^{4} d^{4} r x^{4} + 4 \, a b^{3} d^{4} r x^{3} + 6 \, a^{2} b^{2} d^{4} r x^{2} + 4 \, a^{3} b d^{4} r x\right )} \log \left (f\right )}{48 \, b d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas")

[Out]

-1/48*(3*(b^4*d^4*p + b^4*d^4*q)*r*x^4 + 4*(3*a*b^3*d^4*p - (b^4*c*d^3 - 4*a*b^3*d^4)*q)*r*x^3 + 6*(3*a^2*b^2*
d^4*p + (b^4*c^2*d^2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*q)*r*x^2 + 12*(a^3*b*d^4*p - (b^4*c^3*d - 4*a*b^3*c^2*d^
2 + 6*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*q)*r*x - 12*(b^4*d^4*p*r*x^4 + 4*a*b^3*d^4*p*r*x^3 + 6*a^2*b^2*d^4*p*r*x^2
+ 4*a^3*b*d^4*p*r*x + a^4*d^4*p*r)*log(b*x + a) - 12*(b^4*d^4*q*r*x^4 + 4*a*b^3*d^4*q*r*x^3 + 6*a^2*b^2*d^4*q*
r*x^2 + 4*a^3*b*d^4*q*r*x - (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3)*q*r)*log(d*x + c) -
12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x)*log(e) - 12*(b^4*d^4*r*x^4 + 4*a*b^3*d^
4*r*x^3 + 6*a^2*b^2*d^4*r*x^2 + 4*a^3*b*d^4*r*x)*log(f))/(b*d^4)

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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*x+a)**3*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)

[Out]

Timed out

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Giac [B]  time = 1.40639, size = 774, normalized size = 4.5 \begin{align*} -\frac{1}{16} \,{\left (b^{3} p r + b^{3} q r - 4 \, b^{3} r \log \left (f\right ) - 4 \, b^{3}\right )} x^{4} - \frac{{\left (3 \, a b^{2} d p r - b^{3} c q r + 4 \, a b^{2} d q r - 12 \, a b^{2} d r \log \left (f\right ) - 12 \, a b^{2} d\right )} x^{3}}{12 \, d} + \frac{1}{4} \,{\left (b^{3} p r x^{4} + 4 \, a b^{2} p r x^{3} + 6 \, a^{2} b p r x^{2} + 4 \, a^{3} p r x\right )} \log \left (b x + a\right ) + \frac{1}{4} \,{\left (b^{3} q r x^{4} + 4 \, a b^{2} q r x^{3} + 6 \, a^{2} b q r x^{2} + 4 \, a^{3} q r x\right )} \log \left (d x + c\right ) - \frac{{\left (3 \, a^{2} b d^{2} p r + b^{3} c^{2} q r - 4 \, a b^{2} c d q r + 6 \, a^{2} b d^{2} q r - 12 \, a^{2} b d^{2} r \log \left (f\right ) - 12 \, a^{2} b d^{2}\right )} x^{2}}{8 \, d^{2}} - \frac{{\left (a^{3} d^{3} p r - b^{3} c^{3} q r + 4 \, a b^{2} c^{2} d q r - 6 \, a^{2} b c d^{2} q r + 4 \, a^{3} d^{3} q r - 4 \, a^{3} d^{3} r \log \left (f\right ) - 4 \, a^{3} d^{3}\right )} x}{4 \, d^{3}} + \frac{{\left (a^{4} d^{4} p r - b^{4} c^{4} q r + 4 \, a b^{3} c^{3} d q r - 6 \, a^{2} b^{2} c^{2} d^{2} q r + 4 \, a^{3} b c d^{3} q r\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{8 \, b d^{4}} + \frac{{\left (a^{4} b c d^{4} p r - a^{5} d^{5} p r + b^{5} c^{5} q r - 5 \, a b^{4} c^{4} d q r + 10 \, a^{2} b^{3} c^{3} d^{2} q r - 10 \, a^{3} b^{2} c^{2} d^{3} q r + 4 \, a^{4} b c d^{4} q r\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | b c - a d \right |}}{2 \, b d x + b c + a d +{\left | b c - a d \right |}} \right |}\right )}{8 \, b d^{4}{\left | b c - a d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")

[Out]

-1/16*(b^3*p*r + b^3*q*r - 4*b^3*r*log(f) - 4*b^3)*x^4 - 1/12*(3*a*b^2*d*p*r - b^3*c*q*r + 4*a*b^2*d*q*r - 12*
a*b^2*d*r*log(f) - 12*a*b^2*d)*x^3/d + 1/4*(b^3*p*r*x^4 + 4*a*b^2*p*r*x^3 + 6*a^2*b*p*r*x^2 + 4*a^3*p*r*x)*log
(b*x + a) + 1/4*(b^3*q*r*x^4 + 4*a*b^2*q*r*x^3 + 6*a^2*b*q*r*x^2 + 4*a^3*q*r*x)*log(d*x + c) - 1/8*(3*a^2*b*d^
2*p*r + b^3*c^2*q*r - 4*a*b^2*c*d*q*r + 6*a^2*b*d^2*q*r - 12*a^2*b*d^2*r*log(f) - 12*a^2*b*d^2)*x^2/d^2 - 1/4*
(a^3*d^3*p*r - b^3*c^3*q*r + 4*a*b^2*c^2*d*q*r - 6*a^2*b*c*d^2*q*r + 4*a^3*d^3*q*r - 4*a^3*d^3*r*log(f) - 4*a^
3*d^3)*x/d^3 + 1/8*(a^4*d^4*p*r - b^4*c^4*q*r + 4*a*b^3*c^3*d*q*r - 6*a^2*b^2*c^2*d^2*q*r + 4*a^3*b*c*d^3*q*r)
*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b*d^4) + 1/8*(a^4*b*c*d^4*p*r - a^5*d^5*p*r + b^5*c^5*q*r - 5*a*b^4*
c^4*d*q*r + 10*a^2*b^3*c^3*d^2*q*r - 10*a^3*b^2*c^2*d^3*q*r + 4*a^4*b*c*d^4*q*r)*log(abs((2*b*d*x + b*c + a*d
- abs(b*c - a*d))/(2*b*d*x + b*c + a*d + abs(b*c - a*d))))/(b*d^4*abs(b*c - a*d))