3.7 \(\int (a+b x)^4 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\)

Optimal. Leaf size=201 \[ -\frac{q r x (b c-a d)^4}{5 d^4}+\frac{q r (a+b x)^2 (b c-a d)^3}{10 b d^3}-\frac{q r (a+b x)^3 (b c-a d)^2}{15 b d^2}+\frac{q r (b c-a d)^5 \log (c+d x)}{5 b d^5}+\frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac{q r (a+b x)^4 (b c-a d)}{20 b d}-\frac{p r (a+b x)^5}{25 b}-\frac{q r (a+b x)^5}{25 b} \]

[Out]

-((b*c - a*d)^4*q*r*x)/(5*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)^2)/(10*b*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^3)/
(15*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^4)/(20*b*d) - (p*r*(a + b*x)^5)/(25*b) - (q*r*(a + b*x)^5)/(25*b) + ((
b*c - a*d)^5*q*r*Log[c + d*x])/(5*b*d^5) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0949281, antiderivative size = 201, 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} \[ -\frac{q r x (b c-a d)^4}{5 d^4}+\frac{q r (a+b x)^2 (b c-a d)^3}{10 b d^3}-\frac{q r (a+b x)^3 (b c-a d)^2}{15 b d^2}+\frac{q r (b c-a d)^5 \log (c+d x)}{5 b d^5}+\frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac{q r (a+b x)^4 (b c-a d)}{20 b d}-\frac{p r (a+b x)^5}{25 b}-\frac{q r (a+b x)^5}{25 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

-((b*c - a*d)^4*q*r*x)/(5*d^4) + ((b*c - a*d)^3*q*r*(a + b*x)^2)/(10*b*d^3) - ((b*c - a*d)^2*q*r*(a + b*x)^3)/
(15*b*d^2) + ((b*c - a*d)*q*r*(a + b*x)^4)/(20*b*d) - (p*r*(a + b*x)^5)/(25*b) - (q*r*(a + b*x)^5)/(25*b) + ((
b*c - a*d)^5*q*r*Log[c + d*x])/(5*b*d^5) + ((a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*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)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{1}{5} (p r) \int (a+b x)^4 \, dx-\frac{(d q r) \int \frac{(a+b x)^5}{c+d x} \, dx}{5 b}\\ &=-\frac{p r (a+b x)^5}{25 b}+\frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac{(d q r) \int \left (\frac{b (b c-a d)^4}{d^5}-\frac{b (b c-a d)^3 (a+b x)}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2}{d^3}-\frac{b (b c-a d) (a+b x)^3}{d^2}+\frac{b (a+b x)^4}{d}+\frac{(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{5 b}\\ &=-\frac{(b c-a d)^4 q r x}{5 d^4}+\frac{(b c-a d)^3 q r (a+b x)^2}{10 b d^3}-\frac{(b c-a d)^2 q r (a+b x)^3}{15 b d^2}+\frac{(b c-a d) q r (a+b x)^4}{20 b d}-\frac{p r (a+b x)^5}{25 b}-\frac{q r (a+b x)^5}{25 b}+\frac{(b c-a d)^5 q r \log (c+d x)}{5 b d^5}+\frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.308289, size = 185, normalized size = 0.92 \[ \frac{(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac{r \left (-60 b^2 (2 p+5 q) (c+d x)^2 (b c-a d)^3+40 b^3 (3 p+5 q) (c+d x)^3 (b c-a d)^2-15 b^4 (4 p+5 q) (c+d x)^4 (b c-a d)+60 b d x (p+5 q) (b c-a d)^4-60 q (b c-a d)^5 \log (c+d x)+12 b^5 (p+q) (c+d x)^5\right )}{60 d^5}}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]

[Out]

(-(r*(60*b*d*(b*c - a*d)^4*(p + 5*q)*x - 60*b^2*(b*c - a*d)^3*(2*p + 5*q)*(c + d*x)^2 + 40*b^3*(b*c - a*d)^2*(
3*p + 5*q)*(c + d*x)^3 - 15*b^4*(b*c - a*d)*(4*p + 5*q)*(c + d*x)^4 + 12*b^5*(p + q)*(c + d*x)^5 - 60*(b*c - a
*d)^5*q*Log[c + d*x]))/(60*d^5) + (a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b)

________________________________________________________________________________________

Maple [F]  time = 0.734, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{4}\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)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

[Out]

int((b*x+a)^4*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)

________________________________________________________________________________________

Maxima [B]  time = 1.30716, size = 533, normalized size = 2.65 \begin{align*} \frac{1}{5} \,{\left (b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} 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{60 \, a^{5} f p \log \left (b x + a\right )}{b} - \frac{12 \, b^{4} d^{4} f{\left (p + q\right )} x^{5} + 15 \,{\left (a b^{3} d^{4} f{\left (4 \, p + 5 \, q\right )} - b^{4} c d^{3} f q\right )} x^{4} + 20 \,{\left (2 \, a^{2} b^{2} d^{4} f{\left (3 \, p + 5 \, q\right )} + b^{4} c^{2} d^{2} f q - 5 \, a b^{3} c d^{3} f q\right )} x^{3} + 30 \,{\left (2 \, a^{3} b d^{4} f{\left (2 \, p + 5 \, q\right )} - b^{4} c^{3} d f q + 5 \, a b^{3} c^{2} d^{2} f q - 10 \, a^{2} b^{2} c d^{3} f q\right )} x^{2} + 60 \,{\left (a^{4} d^{4} f{\left (p + 5 \, q\right )} + b^{4} c^{4} f q - 5 \, a b^{3} c^{3} d f q + 10 \, a^{2} b^{2} c^{2} d^{2} f q - 10 \, a^{3} b c d^{3} f q\right )} x}{d^{4}} + \frac{60 \,{\left (b^{4} c^{5} f q - 5 \, a b^{3} c^{4} d f q + 10 \, a^{2} b^{2} c^{3} d^{2} f q - 10 \, a^{3} b c^{2} d^{3} f q + 5 \, a^{4} c d^{4} f q\right )} \log \left (d x + c\right )}{d^{5}}\right )} r}{300 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima")

[Out]

1/5*(b^4*x^5 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3 + 10*a^3*b*x^2 + 5*a^4*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 1
/300*(60*a^5*f*p*log(b*x + a)/b - (12*b^4*d^4*f*(p + q)*x^5 + 15*(a*b^3*d^4*f*(4*p + 5*q) - b^4*c*d^3*f*q)*x^4
+ 20*(2*a^2*b^2*d^4*f*(3*p + 5*q) + b^4*c^2*d^2*f*q - 5*a*b^3*c*d^3*f*q)*x^3 + 30*(2*a^3*b*d^4*f*(2*p + 5*q)
- b^4*c^3*d*f*q + 5*a*b^3*c^2*d^2*f*q - 10*a^2*b^2*c*d^3*f*q)*x^2 + 60*(a^4*d^4*f*(p + 5*q) + b^4*c^4*f*q - 5*
a*b^3*c^3*d*f*q + 10*a^2*b^2*c^2*d^2*f*q - 10*a^3*b*c*d^3*f*q)*x)/d^4 + 60*(b^4*c^5*f*q - 5*a*b^3*c^4*d*f*q +
10*a^2*b^2*c^3*d^2*f*q - 10*a^3*b*c^2*d^3*f*q + 5*a^4*c*d^4*f*q)*log(d*x + c)/d^5)*r/f

________________________________________________________________________________________

Fricas [B]  time = 0.864394, size = 1315, normalized size = 6.54 \begin{align*} -\frac{12 \,{\left (b^{5} d^{5} p + b^{5} d^{5} q\right )} r x^{5} + 15 \,{\left (4 \, a b^{4} d^{5} p -{\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} q\right )} r x^{4} + 20 \,{\left (6 \, a^{2} b^{3} d^{5} p +{\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} q\right )} r x^{3} + 30 \,{\left (4 \, a^{3} b^{2} d^{5} p -{\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} q\right )} r x^{2} + 60 \,{\left (a^{4} b d^{5} p +{\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} q\right )} r x - 60 \,{\left (b^{5} d^{5} p r x^{5} + 5 \, a b^{4} d^{5} p r x^{4} + 10 \, a^{2} b^{3} d^{5} p r x^{3} + 10 \, a^{3} b^{2} d^{5} p r x^{2} + 5 \, a^{4} b d^{5} p r x + a^{5} d^{5} p r\right )} \log \left (b x + a\right ) - 60 \,{\left (b^{5} d^{5} q r x^{5} + 5 \, a b^{4} d^{5} q r x^{4} + 10 \, a^{2} b^{3} d^{5} q r x^{3} + 10 \, a^{3} b^{2} d^{5} q r x^{2} + 5 \, a^{4} b d^{5} q r x +{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4}\right )} q r\right )} \log \left (d x + c\right ) - 60 \,{\left (b^{5} d^{5} x^{5} + 5 \, a b^{4} d^{5} x^{4} + 10 \, a^{2} b^{3} d^{5} x^{3} + 10 \, a^{3} b^{2} d^{5} x^{2} + 5 \, a^{4} b d^{5} x\right )} \log \left (e\right ) - 60 \,{\left (b^{5} d^{5} r x^{5} + 5 \, a b^{4} d^{5} r x^{4} + 10 \, a^{2} b^{3} d^{5} r x^{3} + 10 \, a^{3} b^{2} d^{5} r x^{2} + 5 \, a^{4} b d^{5} r x\right )} \log \left (f\right )}{300 \, b d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas")

[Out]

-1/300*(12*(b^5*d^5*p + b^5*d^5*q)*r*x^5 + 15*(4*a*b^4*d^5*p - (b^5*c*d^4 - 5*a*b^4*d^5)*q)*r*x^4 + 20*(6*a^2*
b^3*d^5*p + (b^5*c^2*d^3 - 5*a*b^4*c*d^4 + 10*a^2*b^3*d^5)*q)*r*x^3 + 30*(4*a^3*b^2*d^5*p - (b^5*c^3*d^2 - 5*a
*b^4*c^2*d^3 + 10*a^2*b^3*c*d^4 - 10*a^3*b^2*d^5)*q)*r*x^2 + 60*(a^4*b*d^5*p + (b^5*c^4*d - 5*a*b^4*c^3*d^2 +
10*a^2*b^3*c^2*d^3 - 10*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*q)*r*x - 60*(b^5*d^5*p*r*x^5 + 5*a*b^4*d^5*p*r*x^4 + 10*a
^2*b^3*d^5*p*r*x^3 + 10*a^3*b^2*d^5*p*r*x^2 + 5*a^4*b*d^5*p*r*x + a^5*d^5*p*r)*log(b*x + a) - 60*(b^5*d^5*q*r*
x^5 + 5*a*b^4*d^5*q*r*x^4 + 10*a^2*b^3*d^5*q*r*x^3 + 10*a^3*b^2*d^5*q*r*x^2 + 5*a^4*b*d^5*q*r*x + (b^5*c^5 - 5
*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4)*q*r)*log(d*x + c) - 60*(b^5*d^5*x^5 +
5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x)*log(e) - 60*(b^5*d^5*r*x^5 + 5*a*b^
4*d^5*r*x^4 + 10*a^2*b^3*d^5*r*x^3 + 10*a^3*b^2*d^5*r*x^2 + 5*a^4*b*d^5*r*x)*log(f))/(b*d^5)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.26035, size = 1002, normalized size = 4.99 \begin{align*} -\frac{1}{25} \,{\left (b^{4} p r + b^{4} q r - 5 \, b^{4} r \log \left (f\right ) - 5 \, b^{4}\right )} x^{5} - \frac{{\left (4 \, a b^{3} d p r - b^{4} c q r + 5 \, a b^{3} d q r - 20 \, a b^{3} d r \log \left (f\right ) - 20 \, a b^{3} d\right )} x^{4}}{20 \, d} - \frac{{\left (6 \, a^{2} b^{2} d^{2} p r + b^{4} c^{2} q r - 5 \, a b^{3} c d q r + 10 \, a^{2} b^{2} d^{2} q r - 30 \, a^{2} b^{2} d^{2} r \log \left (f\right ) - 30 \, a^{2} b^{2} d^{2}\right )} x^{3}}{15 \, d^{2}} + \frac{1}{5} \,{\left (b^{4} p r x^{5} + 5 \, a b^{3} p r x^{4} + 10 \, a^{2} b^{2} p r x^{3} + 10 \, a^{3} b p r x^{2} + 5 \, a^{4} p r x\right )} \log \left (b x + a\right ) + \frac{1}{5} \,{\left (b^{4} q r x^{5} + 5 \, a b^{3} q r x^{4} + 10 \, a^{2} b^{2} q r x^{3} + 10 \, a^{3} b q r x^{2} + 5 \, a^{4} q r x\right )} \log \left (d x + c\right ) - \frac{{\left (4 \, a^{3} b d^{3} p r - b^{4} c^{3} q r + 5 \, a b^{3} c^{2} d q r - 10 \, a^{2} b^{2} c d^{2} q r + 10 \, a^{3} b d^{3} q r - 20 \, a^{3} b d^{3} r \log \left (f\right ) - 20 \, a^{3} b d^{3}\right )} x^{2}}{10 \, d^{3}} - \frac{{\left (a^{4} d^{4} p r + b^{4} c^{4} q r - 5 \, a b^{3} c^{3} d q r + 10 \, a^{2} b^{2} c^{2} d^{2} q r - 10 \, a^{3} b c d^{3} q r + 5 \, a^{4} d^{4} q r - 5 \, a^{4} d^{4} r \log \left (f\right ) - 5 \, a^{4} d^{4}\right )} x}{5 \, d^{4}} + \frac{{\left (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 + 5 \, a^{4} b c d^{4} q r\right )} \log \left ({\left | b d x^{2} + b c x + a d x + a c \right |}\right )}{10 \, b d^{5}} + \frac{{\left (a^{5} b c d^{5} p r - a^{6} d^{6} p r - b^{6} c^{6} q r + 6 \, a b^{5} c^{5} d q r - 15 \, a^{2} b^{4} c^{4} d^{2} q r + 20 \, a^{3} b^{3} c^{3} d^{3} q r - 15 \, a^{4} b^{2} c^{2} d^{4} q r + 5 \, a^{5} b c d^{5} 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 )}{10 \, b d^{5}{\left | b c - a d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")

[Out]

-1/25*(b^4*p*r + b^4*q*r - 5*b^4*r*log(f) - 5*b^4)*x^5 - 1/20*(4*a*b^3*d*p*r - b^4*c*q*r + 5*a*b^3*d*q*r - 20*
a*b^3*d*r*log(f) - 20*a*b^3*d)*x^4/d - 1/15*(6*a^2*b^2*d^2*p*r + b^4*c^2*q*r - 5*a*b^3*c*d*q*r + 10*a^2*b^2*d^
2*q*r - 30*a^2*b^2*d^2*r*log(f) - 30*a^2*b^2*d^2)*x^3/d^2 + 1/5*(b^4*p*r*x^5 + 5*a*b^3*p*r*x^4 + 10*a^2*b^2*p*
r*x^3 + 10*a^3*b*p*r*x^2 + 5*a^4*p*r*x)*log(b*x + a) + 1/5*(b^4*q*r*x^5 + 5*a*b^3*q*r*x^4 + 10*a^2*b^2*q*r*x^3
+ 10*a^3*b*q*r*x^2 + 5*a^4*q*r*x)*log(d*x + c) - 1/10*(4*a^3*b*d^3*p*r - b^4*c^3*q*r + 5*a*b^3*c^2*d*q*r - 10
*a^2*b^2*c*d^2*q*r + 10*a^3*b*d^3*q*r - 20*a^3*b*d^3*r*log(f) - 20*a^3*b*d^3)*x^2/d^3 - 1/5*(a^4*d^4*p*r + b^4
*c^4*q*r - 5*a*b^3*c^3*d*q*r + 10*a^2*b^2*c^2*d^2*q*r - 10*a^3*b*c*d^3*q*r + 5*a^4*d^4*q*r - 5*a^4*d^4*r*log(f
) - 5*a^4*d^4)*x/d^4 + 1/10*(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 + 5*a^4*b*c*d^4*q*r)*log(abs(b*d*x^2 + b*c*x + a*d*x + a*c))/(b*d^5) + 1/10*(a^5*b*c*d^5*p*r -
a^6*d^6*p*r - b^6*c^6*q*r + 6*a*b^5*c^5*d*q*r - 15*a^2*b^4*c^4*d^2*q*r + 20*a^3*b^3*c^3*d^3*q*r - 15*a^4*b^2*c
^2*d^4*q*r + 5*a^5*b*c*d^5*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^5*abs(b*c - a*d))