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

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

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 201, 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 = {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 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])

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]

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*}

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Mathematica [A]  time = 0.31, size = 185, normalized size = 0.92 \[ \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-\frac {r \left (-15 b^4 (4 p+5 q) (c+d x)^4 (b c-a d)+40 b^3 (3 p+5 q) (c+d x)^3 (b c-a d)^2-60 b^2 (2 p+5 q) (c+d x)^2 (b c-a d)^3+60 b d x (p+5 q) (b c-a d)^4-60 q (b c-a d)^5 \log (c+d x)+12 b^5 (p+q) (c+d x)^5\right )}{60 d^5}}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/60*(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]))/d^5 + (a + b*x)^5*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(5*b)

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fricas [B]  time = 0.43, size = 624, normalized size = 3.10 \[ -\frac {12 \, {\left (b^{5} d^{5} p + b^{5} d^{5} q\right )} r x^{5} + 15 \, {\left (4 \, a b^{4} d^{5} p - {\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} q\right )} r x^{4} + 20 \, {\left (6 \, a^{2} b^{3} d^{5} p + {\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} q\right )} r x^{3} + 30 \, {\left (4 \, a^{3} b^{2} d^{5} p - {\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} q\right )} r x^{2} + 60 \, {\left (a^{4} b d^{5} p + {\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} q\right )} r x - 60 \, {\left (b^{5} d^{5} p r x^{5} + 5 \, a b^{4} d^{5} p r x^{4} + 10 \, a^{2} b^{3} d^{5} p r x^{3} + 10 \, a^{3} b^{2} d^{5} p r x^{2} + 5 \, a^{4} b d^{5} p r x + a^{5} d^{5} p r\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} d^{5} q r x^{5} + 5 \, a b^{4} d^{5} q r x^{4} + 10 \, a^{2} b^{3} d^{5} q r x^{3} + 10 \, a^{3} b^{2} d^{5} q r x^{2} + 5 \, a^{4} b d^{5} q r x + {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4}\right )} q r\right )} \log \left (d x + c\right ) - 60 \, {\left (b^{5} d^{5} x^{5} + 5 \, a b^{4} d^{5} x^{4} + 10 \, a^{2} b^{3} d^{5} x^{3} + 10 \, a^{3} b^{2} d^{5} x^{2} + 5 \, a^{4} b d^{5} x\right )} \log \relax (e) - 60 \, {\left (b^{5} d^{5} r x^{5} + 5 \, a b^{4} d^{5} r x^{4} + 10 \, a^{2} b^{3} d^{5} r x^{3} + 10 \, a^{3} b^{2} d^{5} r x^{2} + 5 \, a^{4} b d^{5} r x\right )} \log \relax (f)}{300 \, b d^{5}} \]

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)

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giac [B]  time = 34.44, size = 560, normalized size = 2.79 \[ \frac {a^{5} p r \log \left (b x + a\right )}{5 \, b} - \frac {1}{25} \, {\left (b^{4} p r + b^{4} q r - 5 \, b^{4} r \log \relax (f) - 5 \, b^{4}\right )} x^{5} - \frac {{\left (4 \, a b^{3} d p r - b^{4} c q r + 5 \, a b^{3} d q r - 20 \, a b^{3} d r \log \relax (f) - 20 \, a b^{3} d\right )} x^{4}}{20 \, d} - \frac {{\left (6 \, a^{2} b^{2} d^{2} p r + b^{4} c^{2} q r - 5 \, a b^{3} c d q r + 10 \, a^{2} b^{2} d^{2} q r - 30 \, a^{2} b^{2} d^{2} r \log \relax (f) - 30 \, a^{2} b^{2} d^{2}\right )} x^{3}}{15 \, d^{2}} + \frac {1}{5} \, {\left (b^{4} p r x^{5} + 5 \, a b^{3} p r x^{4} + 10 \, a^{2} b^{2} p r x^{3} + 10 \, a^{3} b p r x^{2} + 5 \, a^{4} p r x\right )} \log \left (b x + a\right ) + \frac {1}{5} \, {\left (b^{4} q r x^{5} + 5 \, a b^{3} q r x^{4} + 10 \, a^{2} b^{2} q r x^{3} + 10 \, a^{3} b q r x^{2} + 5 \, a^{4} q r x\right )} \log \left (d x + c\right ) - \frac {{\left (4 \, a^{3} b d^{3} p r - b^{4} c^{3} q r + 5 \, a b^{3} c^{2} d q r - 10 \, a^{2} b^{2} c d^{2} q r + 10 \, a^{3} b d^{3} q r - 20 \, a^{3} b d^{3} r \log \relax (f) - 20 \, a^{3} b d^{3}\right )} x^{2}}{10 \, d^{3}} - \frac {{\left (a^{4} d^{4} p r + b^{4} c^{4} q r - 5 \, a b^{3} c^{3} d q r + 10 \, a^{2} b^{2} c^{2} d^{2} q r - 10 \, a^{3} b c d^{3} q r + 5 \, a^{4} d^{4} q r - 5 \, a^{4} d^{4} r \log \relax (f) - 5 \, a^{4} d^{4}\right )} x}{5 \, d^{4}} + \frac {{\left (b^{4} c^{5} q r - 5 \, a b^{3} c^{4} d q r + 10 \, a^{2} b^{2} c^{3} d^{2} q r - 10 \, a^{3} b c^{2} d^{3} q r + 5 \, a^{4} c d^{4} q r\right )} \log \left (-d x - c\right )}{5 \, d^{5}} \]

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]

1/5*a^5*p*r*log(b*x + a)/b - 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 + 1/5*(b^4*c^5*q*r - 5*a*b^3*c^4*d*q*r + 10*a^2*b^2*c^3*d^2*q
*r - 10*a^3*b*c^2*d^3*q*r + 5*a^4*c*d^4*q*r)*log(-d*x - c)/d^5

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maple [F]  time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{4} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )\, dx \]

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)

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maxima [B]  time = 0.74, size = 395, normalized size = 1.97 \[ \frac {1}{5} \, {\left (b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {{\left (\frac {60 \, a^{5} f p \log \left (b x + a\right )}{b} - \frac {12 \, b^{4} d^{4} f {\left (p + q\right )} x^{5} + 15 \, {\left (a b^{3} d^{4} f {\left (4 \, p + 5 \, q\right )} - b^{4} c d^{3} f q\right )} x^{4} + 20 \, {\left (2 \, a^{2} b^{2} d^{4} f {\left (3 \, p + 5 \, q\right )} + b^{4} c^{2} d^{2} f q - 5 \, a b^{3} c d^{3} f q\right )} x^{3} + 30 \, {\left (2 \, a^{3} b d^{4} f {\left (2 \, p + 5 \, q\right )} - b^{4} c^{3} d f q + 5 \, a b^{3} c^{2} d^{2} f q - 10 \, a^{2} b^{2} c d^{3} f q\right )} x^{2} + 60 \, {\left (a^{4} d^{4} f {\left (p + 5 \, q\right )} + b^{4} c^{4} f q - 5 \, a b^{3} c^{3} d f q + 10 \, a^{2} b^{2} c^{2} d^{2} f q - 10 \, a^{3} b c d^{3} f q\right )} x}{d^{4}} + \frac {60 \, {\left (b^{4} c^{5} f q - 5 \, a b^{3} c^{4} d f q + 10 \, a^{2} b^{2} c^{3} d^{2} f q - 10 \, a^{3} b c^{2} d^{3} f q + 5 \, a^{4} c d^{4} f q\right )} \log \left (d x + c\right )}{d^{5}}\right )} r}{300 \, f} \]

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

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mupad [B]  time = 0.70, size = 886, normalized size = 4.41 \[ \ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (a^4\,x+2\,a^3\,b\,x^2+2\,a^2\,b^2\,x^3+a\,b^3\,x^4+\frac {b^4\,x^5}{5}\right )-x^4\,\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{20\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{100\,d}\right )+x^3\,\left (\frac {\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{5\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{15\,b\,d}-\frac {a\,b^2\,r\,\left (2\,a\,d\,p+b\,c\,p+3\,a\,d\,q\right )}{3\,d}+\frac {a\,b^3\,c\,r\,\left (p+q\right )}{15\,d}\right )-x\,\left (\frac {a^3\,r\,\left (a\,d\,p+2\,b\,c\,p+3\,a\,d\,q\right )}{d}-\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{5\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,r\,\left (2\,a\,d\,p+b\,c\,p+3\,a\,d\,q\right )}{d}+\frac {a\,b^3\,c\,r\,\left (p+q\right )}{5\,d}\right )}{5\,b\,d}-\frac {a\,c\,\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{5\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,d}\right )}{b\,d}+\frac {2\,a^2\,b\,r\,\left (a\,d\,p+b\,c\,p+2\,a\,d\,q\right )}{d}\right )}{5\,b\,d}+\frac {a\,c\,\left (\frac {\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{5\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,r\,\left (2\,a\,d\,p+b\,c\,p+3\,a\,d\,q\right )}{d}+\frac {a\,b^3\,c\,r\,\left (p+q\right )}{5\,d}\right )}{b\,d}\right )-x^2\,\left (\frac {\left (5\,a\,d+5\,b\,c\right )\,\left (\frac {\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{5\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,d}\right )\,\left (5\,a\,d+5\,b\,c\right )}{5\,b\,d}-\frac {a\,b^2\,r\,\left (2\,a\,d\,p+b\,c\,p+3\,a\,d\,q\right )}{d}+\frac {a\,b^3\,c\,r\,\left (p+q\right )}{5\,d}\right )}{10\,b\,d}-\frac {a\,c\,\left (\frac {b^3\,r\,\left (5\,a\,d\,p+b\,c\,p+6\,a\,d\,q\right )}{5\,d}-\frac {b^3\,r\,\left (p+q\right )\,\left (5\,a\,d+5\,b\,c\right )}{25\,d}\right )}{2\,b\,d}+\frac {a^2\,b\,r\,\left (a\,d\,p+b\,c\,p+2\,a\,d\,q\right )}{d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (q\,r\,a^4\,c\,d^4-2\,q\,r\,a^3\,b\,c^2\,d^3+2\,q\,r\,a^2\,b^2\,c^3\,d^2-q\,r\,a\,b^3\,c^4\,d+\frac {q\,r\,b^4\,c^5}{5}\right )}{d^5}-\frac {b^4\,r\,x^5\,\left (p+q\right )}{25}+\frac {a^5\,p\,r\,\ln \left (a+b\,x\right )}{5\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(a^4*x + (b^4*x^5)/5 + 2*a^3*b*x^2 + a*b^3*x^4 + 2*a^2*b^2*x^3) - x^4*((b
^3*r*(5*a*d*p + b*c*p + 6*a*d*q))/(20*d) - (b^3*r*(p + q)*(5*a*d + 5*b*c))/(100*d)) + x^3*((((b^3*r*(5*a*d*p +
b*c*p + 6*a*d*q))/(5*d) - (b^3*r*(p + q)*(5*a*d + 5*b*c))/(25*d))*(5*a*d + 5*b*c))/(15*b*d) - (a*b^2*r*(2*a*d
*p + b*c*p + 3*a*d*q))/(3*d) + (a*b^3*c*r*(p + q))/(15*d)) - x*((a^3*r*(a*d*p + 2*b*c*p + 3*a*d*q))/d - ((5*a*
d + 5*b*c)*(((5*a*d + 5*b*c)*((((b^3*r*(5*a*d*p + b*c*p + 6*a*d*q))/(5*d) - (b^3*r*(p + q)*(5*a*d + 5*b*c))/(2
5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*r*(2*a*d*p + b*c*p + 3*a*d*q))/d + (a*b^3*c*r*(p + q))/(5*d)))/(5*b*d)
- (a*c*((b^3*r*(5*a*d*p + b*c*p + 6*a*d*q))/(5*d) - (b^3*r*(p + q)*(5*a*d + 5*b*c))/(25*d)))/(b*d) + (2*a^2*b
*r*(a*d*p + b*c*p + 2*a*d*q))/d))/(5*b*d) + (a*c*((((b^3*r*(5*a*d*p + b*c*p + 6*a*d*q))/(5*d) - (b^3*r*(p + q)
*(5*a*d + 5*b*c))/(25*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*r*(2*a*d*p + b*c*p + 3*a*d*q))/d + (a*b^3*c*r*(p +
q))/(5*d)))/(b*d)) - x^2*(((5*a*d + 5*b*c)*((((b^3*r*(5*a*d*p + b*c*p + 6*a*d*q))/(5*d) - (b^3*r*(p + q)*(5*a
*d + 5*b*c))/(25*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*r*(2*a*d*p + b*c*p + 3*a*d*q))/d + (a*b^3*c*r*(p + q))/
(5*d)))/(10*b*d) - (a*c*((b^3*r*(5*a*d*p + b*c*p + 6*a*d*q))/(5*d) - (b^3*r*(p + q)*(5*a*d + 5*b*c))/(25*d)))/
(2*b*d) + (a^2*b*r*(a*d*p + b*c*p + 2*a*d*q))/d) + (log(c + d*x)*((b^4*c^5*q*r)/5 + a^4*c*d^4*q*r + 2*a^2*b^2*
c^3*d^2*q*r - a*b^3*c^4*d*q*r - 2*a^3*b*c^2*d^3*q*r))/d^5 - (b^4*r*x^5*(p + q))/25 + (a^5*p*r*log(a + b*x))/(5
*b)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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