3.440 \(\int (f x)^m (d+e x^r)^3 (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=233 \[ \frac{3 d^2 e x^{r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+r+1}+\frac{d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{3 d e^2 x^{2 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+2 r+1}+\frac{e^3 x^{3 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+3 r+1}-\frac{3 b d^2 e n x^{r+1} (f x)^m}{(m+r+1)^2}-\frac{b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac{3 b d e^2 n x^{2 r+1} (f x)^m}{(m+2 r+1)^2}-\frac{b e^3 n x^{3 r+1} (f x)^m}{(m+3 r+1)^2} \]

[Out]

(-3*b*d^2*e*n*x^(1 + r)*(f*x)^m)/(1 + m + r)^2 - (3*b*d*e^2*n*x^(1 + 2*r)*(f*x)^m)/(1 + m + 2*r)^2 - (b*e^3*n*
x^(1 + 3*r)*(f*x)^m)/(1 + m + 3*r)^2 - (b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2) + (3*d^2*e*x^(1 + r)*(f*x)^m*(a +
b*Log[c*x^n]))/(1 + m + r) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (e^3*x^(1 + 3*r
)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 3*r) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))

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Rubi [A]  time = 1.98725, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {270, 20, 30, 2350, 14} \[ \frac{3 d^2 e x^{r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+r+1}+\frac{d^3 (f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}+\frac{3 d e^2 x^{2 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+2 r+1}+\frac{e^3 x^{3 r+1} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{m+3 r+1}-\frac{3 b d^2 e n x^{r+1} (f x)^m}{(m+r+1)^2}-\frac{b d^3 n (f x)^{m+1}}{f (m+1)^2}-\frac{3 b d e^2 n x^{2 r+1} (f x)^m}{(m+2 r+1)^2}-\frac{b e^3 n x^{3 r+1} (f x)^m}{(m+3 r+1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d^2*e*n*x^(1 + r)*(f*x)^m)/(1 + m + r)^2 - (3*b*d*e^2*n*x^(1 + 2*r)*(f*x)^m)/(1 + m + 2*r)^2 - (b*e^3*n*
x^(1 + 3*r)*(f*x)^m)/(1 + m + 3*r)^2 - (b*d^3*n*(f*x)^(1 + m))/(f*(1 + m)^2) + (3*d^2*e*x^(1 + r)*(f*x)^m*(a +
b*Log[c*x^n]))/(1 + m + r) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (e^3*x^(1 + 3*r
)*(f*x)^m*(a + b*Log[c*x^n]))/(1 + m + 3*r) + (d^3*(f*x)^(1 + m)*(a + b*Log[c*x^n]))/(f*(1 + m))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*x)^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x,
x], x], x] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int (f x)^m \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac{3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac{e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-(b n) \int (f x)^m \left (\frac{d^3}{1+m}+\frac{3 d^2 e x^r}{1+m+r}+\frac{3 d e^2 x^{2 r}}{1+m+2 r}+\frac{e^3 x^{3 r}}{1+m+3 r}\right ) \, dx\\ &=\frac{3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac{3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac{e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-(b n) \int \left (\frac{d^3 (f x)^m}{1+m}+\frac{3 d^2 e x^r (f x)^m}{1+m+r}+\frac{3 d e^2 x^{2 r} (f x)^m}{1+m+2 r}+\frac{e^3 x^{3 r} (f x)^m}{1+m+3 r}\right ) \, dx\\ &=-\frac{b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac{3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac{3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac{e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-\frac{\left (3 b d^2 e n\right ) \int x^r (f x)^m \, dx}{1+m+r}-\frac{\left (3 b d e^2 n\right ) \int x^{2 r} (f x)^m \, dx}{1+m+2 r}-\frac{\left (b e^3 n\right ) \int x^{3 r} (f x)^m \, dx}{1+m+3 r}\\ &=-\frac{b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac{3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac{3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac{e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}-\frac{\left (3 b d^2 e n x^{-m} (f x)^m\right ) \int x^{m+r} \, dx}{1+m+r}-\frac{\left (3 b d e^2 n x^{-m} (f x)^m\right ) \int x^{m+2 r} \, dx}{1+m+2 r}-\frac{\left (b e^3 n x^{-m} (f x)^m\right ) \int x^{m+3 r} \, dx}{1+m+3 r}\\ &=-\frac{3 b d^2 e n x^{1+r} (f x)^m}{(1+m+r)^2}-\frac{3 b d e^2 n x^{1+2 r} (f x)^m}{(1+m+2 r)^2}-\frac{b e^3 n x^{1+3 r} (f x)^m}{(1+m+3 r)^2}-\frac{b d^3 n (f x)^{1+m}}{f (1+m)^2}+\frac{3 d^2 e x^{1+r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+r}+\frac{3 d e^2 x^{1+2 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+2 r}+\frac{e^3 x^{1+3 r} (f x)^m \left (a+b \log \left (c x^n\right )\right )}{1+m+3 r}+\frac{d^3 (f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.476407, size = 178, normalized size = 0.76 \[ x (f x)^m \left (\frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{m+r+1}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{m+1}+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{m+2 r+1}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{m+3 r+1}-\frac{3 b d^2 e n x^r}{(m+r+1)^2}-\frac{b d^3 n}{(m+1)^2}-\frac{3 b d e^2 n x^{2 r}}{(m+2 r+1)^2}-\frac{b e^3 n x^{3 r}}{(m+3 r+1)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(f*x)^m*(-((b*d^3*n)/(1 + m)^2) - (3*b*d^2*e*n*x^r)/(1 + m + r)^2 - (3*b*d*e^2*n*x^(2*r))/(1 + m + 2*r)^2 -
(b*e^3*n*x^(3*r))/(1 + m + 3*r)^2 + (d^3*(a + b*Log[c*x^n]))/(1 + m) + (3*d^2*e*x^r*(a + b*Log[c*x^n]))/(1 + m
+ r) + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(1 + m + 2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(1 + m + 3*r))

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Maple [C]  time = 1.512, size = 22706, normalized size = 97.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.00417, size = 10967, normalized size = 47.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((b*e^3*m^7 + 7*b*e^3*m^6 + 21*b*e^3*m^5 + 35*b*e^3*m^4 + 35*b*e^3*m^3 + 21*b*e^3*m^2 + 12*(b*e^3*m^2 + 2*b*e
^3*m + b*e^3)*r^5 + 7*b*e^3*m + 40*(b*e^3*m^3 + 3*b*e^3*m^2 + 3*b*e^3*m + b*e^3)*r^4 + b*e^3 + 51*(b*e^3*m^4 +
4*b*e^3*m^3 + 6*b*e^3*m^2 + 4*b*e^3*m + b*e^3)*r^3 + 31*(b*e^3*m^5 + 5*b*e^3*m^4 + 10*b*e^3*m^3 + 10*b*e^3*m^
2 + 5*b*e^3*m + b*e^3)*r^2 + 9*(b*e^3*m^6 + 6*b*e^3*m^5 + 15*b*e^3*m^4 + 20*b*e^3*m^3 + 15*b*e^3*m^2 + 6*b*e^3
*m + b*e^3)*r)*x*log(c) + (12*(b*e^3*m^2 + 2*b*e^3*m + b*e^3)*n*r^5 + 40*(b*e^3*m^3 + 3*b*e^3*m^2 + 3*b*e^3*m
+ b*e^3)*n*r^4 + 51*(b*e^3*m^4 + 4*b*e^3*m^3 + 6*b*e^3*m^2 + 4*b*e^3*m + b*e^3)*n*r^3 + 31*(b*e^3*m^5 + 5*b*e^
3*m^4 + 10*b*e^3*m^3 + 10*b*e^3*m^2 + 5*b*e^3*m + b*e^3)*n*r^2 + 9*(b*e^3*m^6 + 6*b*e^3*m^5 + 15*b*e^3*m^4 + 2
0*b*e^3*m^3 + 15*b*e^3*m^2 + 6*b*e^3*m + b*e^3)*n*r + (b*e^3*m^7 + 7*b*e^3*m^6 + 21*b*e^3*m^5 + 35*b*e^3*m^4 +
35*b*e^3*m^3 + 21*b*e^3*m^2 + 7*b*e^3*m + b*e^3)*n)*x*log(x) + (a*e^3*m^7 + 7*a*e^3*m^6 + 21*a*e^3*m^5 + 35*a
*e^3*m^4 + 35*a*e^3*m^3 + 21*a*e^3*m^2 + 12*(a*e^3*m^2 + 2*a*e^3*m + a*e^3)*r^5 + 7*a*e^3*m + 4*(10*a*e^3*m^3
+ 30*a*e^3*m^2 + 30*a*e^3*m + 10*a*e^3 - (b*e^3*m^2 + 2*b*e^3*m + b*e^3)*n)*r^4 + a*e^3 + 3*(17*a*e^3*m^4 + 68
*a*e^3*m^3 + 102*a*e^3*m^2 + 68*a*e^3*m + 17*a*e^3 - 4*(b*e^3*m^3 + 3*b*e^3*m^2 + 3*b*e^3*m + b*e^3)*n)*r^3 +
(31*a*e^3*m^5 + 155*a*e^3*m^4 + 310*a*e^3*m^3 + 310*a*e^3*m^2 + 155*a*e^3*m + 31*a*e^3 - 13*(b*e^3*m^4 + 4*b*e
^3*m^3 + 6*b*e^3*m^2 + 4*b*e^3*m + b*e^3)*n)*r^2 - (b*e^3*m^6 + 6*b*e^3*m^5 + 15*b*e^3*m^4 + 20*b*e^3*m^3 + 15
*b*e^3*m^2 + 6*b*e^3*m + b*e^3)*n + 3*(3*a*e^3*m^6 + 18*a*e^3*m^5 + 45*a*e^3*m^4 + 60*a*e^3*m^3 + 45*a*e^3*m^2
+ 18*a*e^3*m + 3*a*e^3 - 2*(b*e^3*m^5 + 5*b*e^3*m^4 + 10*b*e^3*m^3 + 10*b*e^3*m^2 + 5*b*e^3*m + b*e^3)*n)*r)*
x)*x^(3*r)*e^(m*log(f) + m*log(x)) + 3*((b*d*e^2*m^7 + 7*b*d*e^2*m^6 + 21*b*d*e^2*m^5 + 35*b*d*e^2*m^4 + 35*b*
d*e^2*m^3 + 21*b*d*e^2*m^2 + 18*(b*d*e^2*m^2 + 2*b*d*e^2*m + b*d*e^2)*r^5 + 7*b*d*e^2*m + 57*(b*d*e^2*m^3 + 3*
b*d*e^2*m^2 + 3*b*d*e^2*m + b*d*e^2)*r^4 + b*d*e^2 + 68*(b*d*e^2*m^4 + 4*b*d*e^2*m^3 + 6*b*d*e^2*m^2 + 4*b*d*e
^2*m + b*d*e^2)*r^3 + 38*(b*d*e^2*m^5 + 5*b*d*e^2*m^4 + 10*b*d*e^2*m^3 + 10*b*d*e^2*m^2 + 5*b*d*e^2*m + b*d*e^
2)*r^2 + 10*(b*d*e^2*m^6 + 6*b*d*e^2*m^5 + 15*b*d*e^2*m^4 + 20*b*d*e^2*m^3 + 15*b*d*e^2*m^2 + 6*b*d*e^2*m + b*
d*e^2)*r)*x*log(c) + (18*(b*d*e^2*m^2 + 2*b*d*e^2*m + b*d*e^2)*n*r^5 + 57*(b*d*e^2*m^3 + 3*b*d*e^2*m^2 + 3*b*d
*e^2*m + b*d*e^2)*n*r^4 + 68*(b*d*e^2*m^4 + 4*b*d*e^2*m^3 + 6*b*d*e^2*m^2 + 4*b*d*e^2*m + b*d*e^2)*n*r^3 + 38*
(b*d*e^2*m^5 + 5*b*d*e^2*m^4 + 10*b*d*e^2*m^3 + 10*b*d*e^2*m^2 + 5*b*d*e^2*m + b*d*e^2)*n*r^2 + 10*(b*d*e^2*m^
6 + 6*b*d*e^2*m^5 + 15*b*d*e^2*m^4 + 20*b*d*e^2*m^3 + 15*b*d*e^2*m^2 + 6*b*d*e^2*m + b*d*e^2)*n*r + (b*d*e^2*m
^7 + 7*b*d*e^2*m^6 + 21*b*d*e^2*m^5 + 35*b*d*e^2*m^4 + 35*b*d*e^2*m^3 + 21*b*d*e^2*m^2 + 7*b*d*e^2*m + b*d*e^2
)*n)*x*log(x) + (a*d*e^2*m^7 + 7*a*d*e^2*m^6 + 21*a*d*e^2*m^5 + 35*a*d*e^2*m^4 + 35*a*d*e^2*m^3 + 21*a*d*e^2*m
^2 + 18*(a*d*e^2*m^2 + 2*a*d*e^2*m + a*d*e^2)*r^5 + 7*a*d*e^2*m + 3*(19*a*d*e^2*m^3 + 57*a*d*e^2*m^2 + 57*a*d*
e^2*m + 19*a*d*e^2 - 3*(b*d*e^2*m^2 + 2*b*d*e^2*m + b*d*e^2)*n)*r^4 + a*d*e^2 + 4*(17*a*d*e^2*m^4 + 68*a*d*e^2
*m^3 + 102*a*d*e^2*m^2 + 68*a*d*e^2*m + 17*a*d*e^2 - 6*(b*d*e^2*m^3 + 3*b*d*e^2*m^2 + 3*b*d*e^2*m + b*d*e^2)*n
)*r^3 + 2*(19*a*d*e^2*m^5 + 95*a*d*e^2*m^4 + 190*a*d*e^2*m^3 + 190*a*d*e^2*m^2 + 95*a*d*e^2*m + 19*a*d*e^2 - 1
1*(b*d*e^2*m^4 + 4*b*d*e^2*m^3 + 6*b*d*e^2*m^2 + 4*b*d*e^2*m + b*d*e^2)*n)*r^2 - (b*d*e^2*m^6 + 6*b*d*e^2*m^5
+ 15*b*d*e^2*m^4 + 20*b*d*e^2*m^3 + 15*b*d*e^2*m^2 + 6*b*d*e^2*m + b*d*e^2)*n + 2*(5*a*d*e^2*m^6 + 30*a*d*e^2*
m^5 + 75*a*d*e^2*m^4 + 100*a*d*e^2*m^3 + 75*a*d*e^2*m^2 + 30*a*d*e^2*m + 5*a*d*e^2 - 4*(b*d*e^2*m^5 + 5*b*d*e^
2*m^4 + 10*b*d*e^2*m^3 + 10*b*d*e^2*m^2 + 5*b*d*e^2*m + b*d*e^2)*n)*r)*x)*x^(2*r)*e^(m*log(f) + m*log(x)) + 3*
((b*d^2*e*m^7 + 7*b*d^2*e*m^6 + 21*b*d^2*e*m^5 + 35*b*d^2*e*m^4 + 35*b*d^2*e*m^3 + 21*b*d^2*e*m^2 + 36*(b*d^2*
e*m^2 + 2*b*d^2*e*m + b*d^2*e)*r^5 + 7*b*d^2*e*m + 96*(b*d^2*e*m^3 + 3*b*d^2*e*m^2 + 3*b*d^2*e*m + b*d^2*e)*r^
4 + b*d^2*e + 97*(b*d^2*e*m^4 + 4*b*d^2*e*m^3 + 6*b*d^2*e*m^2 + 4*b*d^2*e*m + b*d^2*e)*r^3 + 47*(b*d^2*e*m^5 +
5*b*d^2*e*m^4 + 10*b*d^2*e*m^3 + 10*b*d^2*e*m^2 + 5*b*d^2*e*m + b*d^2*e)*r^2 + 11*(b*d^2*e*m^6 + 6*b*d^2*e*m^
5 + 15*b*d^2*e*m^4 + 20*b*d^2*e*m^3 + 15*b*d^2*e*m^2 + 6*b*d^2*e*m + b*d^2*e)*r)*x*log(c) + (36*(b*d^2*e*m^2 +
2*b*d^2*e*m + b*d^2*e)*n*r^5 + 96*(b*d^2*e*m^3 + 3*b*d^2*e*m^2 + 3*b*d^2*e*m + b*d^2*e)*n*r^4 + 97*(b*d^2*e*m
^4 + 4*b*d^2*e*m^3 + 6*b*d^2*e*m^2 + 4*b*d^2*e*m + b*d^2*e)*n*r^3 + 47*(b*d^2*e*m^5 + 5*b*d^2*e*m^4 + 10*b*d^2
*e*m^3 + 10*b*d^2*e*m^2 + 5*b*d^2*e*m + b*d^2*e)*n*r^2 + 11*(b*d^2*e*m^6 + 6*b*d^2*e*m^5 + 15*b*d^2*e*m^4 + 20
*b*d^2*e*m^3 + 15*b*d^2*e*m^2 + 6*b*d^2*e*m + b*d^2*e)*n*r + (b*d^2*e*m^7 + 7*b*d^2*e*m^6 + 21*b*d^2*e*m^5 + 3
5*b*d^2*e*m^4 + 35*b*d^2*e*m^3 + 21*b*d^2*e*m^2 + 7*b*d^2*e*m + b*d^2*e)*n)*x*log(x) + (a*d^2*e*m^7 + 7*a*d^2*
e*m^6 + 21*a*d^2*e*m^5 + 35*a*d^2*e*m^4 + 35*a*d^2*e*m^3 + 21*a*d^2*e*m^2 + 36*(a*d^2*e*m^2 + 2*a*d^2*e*m + a*
d^2*e)*r^5 + 7*a*d^2*e*m + 12*(8*a*d^2*e*m^3 + 24*a*d^2*e*m^2 + 24*a*d^2*e*m + 8*a*d^2*e - 3*(b*d^2*e*m^2 + 2*
b*d^2*e*m + b*d^2*e)*n)*r^4 + a*d^2*e + (97*a*d^2*e*m^4 + 388*a*d^2*e*m^3 + 582*a*d^2*e*m^2 + 388*a*d^2*e*m +
97*a*d^2*e - 60*(b*d^2*e*m^3 + 3*b*d^2*e*m^2 + 3*b*d^2*e*m + b*d^2*e)*n)*r^3 + (47*a*d^2*e*m^5 + 235*a*d^2*e*m
^4 + 470*a*d^2*e*m^3 + 470*a*d^2*e*m^2 + 235*a*d^2*e*m + 47*a*d^2*e - 37*(b*d^2*e*m^4 + 4*b*d^2*e*m^3 + 6*b*d^
2*e*m^2 + 4*b*d^2*e*m + b*d^2*e)*n)*r^2 - (b*d^2*e*m^6 + 6*b*d^2*e*m^5 + 15*b*d^2*e*m^4 + 20*b*d^2*e*m^3 + 15*
b*d^2*e*m^2 + 6*b*d^2*e*m + b*d^2*e)*n + (11*a*d^2*e*m^6 + 66*a*d^2*e*m^5 + 165*a*d^2*e*m^4 + 220*a*d^2*e*m^3
+ 165*a*d^2*e*m^2 + 66*a*d^2*e*m + 11*a*d^2*e - 10*(b*d^2*e*m^5 + 5*b*d^2*e*m^4 + 10*b*d^2*e*m^3 + 10*b*d^2*e*
m^2 + 5*b*d^2*e*m + b*d^2*e)*n)*r)*x)*x^r*e^(m*log(f) + m*log(x)) + ((b*d^3*m^7 + 7*b*d^3*m^6 + 21*b*d^3*m^5 +
35*b*d^3*m^4 + 35*b*d^3*m^3 + 36*(b*d^3*m + b*d^3)*r^6 + 21*b*d^3*m^2 + 132*(b*d^3*m^2 + 2*b*d^3*m + b*d^3)*r
^5 + 7*b*d^3*m + 193*(b*d^3*m^3 + 3*b*d^3*m^2 + 3*b*d^3*m + b*d^3)*r^4 + b*d^3 + 144*(b*d^3*m^4 + 4*b*d^3*m^3
+ 6*b*d^3*m^2 + 4*b*d^3*m + b*d^3)*r^3 + 58*(b*d^3*m^5 + 5*b*d^3*m^4 + 10*b*d^3*m^3 + 10*b*d^3*m^2 + 5*b*d^3*m
+ b*d^3)*r^2 + 12*(b*d^3*m^6 + 6*b*d^3*m^5 + 15*b*d^3*m^4 + 20*b*d^3*m^3 + 15*b*d^3*m^2 + 6*b*d^3*m + b*d^3)*
r)*x*log(c) + (36*(b*d^3*m + b*d^3)*n*r^6 + 132*(b*d^3*m^2 + 2*b*d^3*m + b*d^3)*n*r^5 + 193*(b*d^3*m^3 + 3*b*d
^3*m^2 + 3*b*d^3*m + b*d^3)*n*r^4 + 144*(b*d^3*m^4 + 4*b*d^3*m^3 + 6*b*d^3*m^2 + 4*b*d^3*m + b*d^3)*n*r^3 + 58
*(b*d^3*m^5 + 5*b*d^3*m^4 + 10*b*d^3*m^3 + 10*b*d^3*m^2 + 5*b*d^3*m + b*d^3)*n*r^2 + 12*(b*d^3*m^6 + 6*b*d^3*m
^5 + 15*b*d^3*m^4 + 20*b*d^3*m^3 + 15*b*d^3*m^2 + 6*b*d^3*m + b*d^3)*n*r + (b*d^3*m^7 + 7*b*d^3*m^6 + 21*b*d^3
*m^5 + 35*b*d^3*m^4 + 35*b*d^3*m^3 + 21*b*d^3*m^2 + 7*b*d^3*m + b*d^3)*n)*x*log(x) + (a*d^3*m^7 + 7*a*d^3*m^6
+ 21*a*d^3*m^5 + 35*a*d^3*m^4 + 35*a*d^3*m^3 + 36*(a*d^3*m - b*d^3*n + a*d^3)*r^6 + 21*a*d^3*m^2 + 132*(a*d^3*
m^2 + 2*a*d^3*m + a*d^3 - (b*d^3*m + b*d^3)*n)*r^5 + 7*a*d^3*m + 193*(a*d^3*m^3 + 3*a*d^3*m^2 + 3*a*d^3*m + a*
d^3 - (b*d^3*m^2 + 2*b*d^3*m + b*d^3)*n)*r^4 + a*d^3 + 144*(a*d^3*m^4 + 4*a*d^3*m^3 + 6*a*d^3*m^2 + 4*a*d^3*m
+ a*d^3 - (b*d^3*m^3 + 3*b*d^3*m^2 + 3*b*d^3*m + b*d^3)*n)*r^3 + 58*(a*d^3*m^5 + 5*a*d^3*m^4 + 10*a*d^3*m^3 +
10*a*d^3*m^2 + 5*a*d^3*m + a*d^3 - (b*d^3*m^4 + 4*b*d^3*m^3 + 6*b*d^3*m^2 + 4*b*d^3*m + b*d^3)*n)*r^2 - (b*d^3
*m^6 + 6*b*d^3*m^5 + 15*b*d^3*m^4 + 20*b*d^3*m^3 + 15*b*d^3*m^2 + 6*b*d^3*m + b*d^3)*n + 12*(a*d^3*m^6 + 6*a*d
^3*m^5 + 15*a*d^3*m^4 + 20*a*d^3*m^3 + 15*a*d^3*m^2 + 6*a*d^3*m + a*d^3 - (b*d^3*m^5 + 5*b*d^3*m^4 + 10*b*d^3*
m^3 + 10*b*d^3*m^2 + 5*b*d^3*m + b*d^3)*n)*r)*x)*e^(m*log(f) + m*log(x)))/(m^8 + 8*m^7 + 36*(m^2 + 2*m + 1)*r^
6 + 28*m^6 + 132*(m^3 + 3*m^2 + 3*m + 1)*r^5 + 56*m^5 + 193*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*r^4 + 70*m^4 + 144
*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*r^3 + 56*m^3 + 58*(m^6 + 6*m^5 + 15*m^4 + 20*m^3 + 15*m^2 + 6*m + 1
)*r^2 + 28*m^2 + 12*(m^7 + 7*m^6 + 21*m^5 + 35*m^4 + 35*m^3 + 21*m^2 + 7*m + 1)*r + 8*m + 1)

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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((f*x)**m*(d+e*x**r)**3*(a+b*ln(c*x**n)),x)

[Out]

Timed out

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Giac [B]  time = 1.40581, size = 1034, normalized size = 4.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*d^2*f^m*m*n*x*x^m*x^r*e*log(x)/(m^2 + 2*m*r + r^2 + 2*m + 2*r + 1) + 3*b*d^2*f^m*n*r*x*x^m*x^r*e*log(x)/(m
^2 + 2*m*r + r^2 + 2*m + 2*r + 1) + b*d^3*f^m*m*n*x*x^m*log(x)/(m^2 + 2*m + 1) + 3*b*d*f^m*m*n*x*x^m*x^(2*r)*e
^2*log(x)/(m^2 + 4*m*r + 4*r^2 + 2*m + 4*r + 1) + 6*b*d*f^m*n*r*x*x^m*x^(2*r)*e^2*log(x)/(m^2 + 4*m*r + 4*r^2
+ 2*m + 4*r + 1) + 3*b*d^2*f^m*n*x*x^m*x^r*e*log(x)/(m^2 + 2*m*r + r^2 + 2*m + 2*r + 1) - 3*b*d^2*f^m*n*x*x^m*
x^r*e/(m^2 + 2*m*r + r^2 + 2*m + 2*r + 1) + 3*b*d^2*f^m*x*x^m*x^r*e*log(c)/(m + r + 1) + b*d^3*f^m*n*x*x^m*log
(x)/(m^2 + 2*m + 1) + b*f^m*m*n*x*x^m*x^(3*r)*e^3*log(x)/(m^2 + 6*m*r + 9*r^2 + 2*m + 6*r + 1) + 3*b*f^m*n*r*x
*x^m*x^(3*r)*e^3*log(x)/(m^2 + 6*m*r + 9*r^2 + 2*m + 6*r + 1) + 3*b*d*f^m*n*x*x^m*x^(2*r)*e^2*log(x)/(m^2 + 4*
m*r + 4*r^2 + 2*m + 4*r + 1) - b*d^3*f^m*n*x*x^m/(m^2 + 2*m + 1) - 3*b*d*f^m*n*x*x^m*x^(2*r)*e^2/(m^2 + 4*m*r
+ 4*r^2 + 2*m + 4*r + 1) + 3*a*d^2*f^m*x*x^m*x^r*e/(m + r + 1) + 3*b*d*f^m*x*x^m*x^(2*r)*e^2*log(c)/(m + 2*r +
1) + b*f^m*n*x*x^m*x^(3*r)*e^3*log(x)/(m^2 + 6*m*r + 9*r^2 + 2*m + 6*r + 1) - b*f^m*n*x*x^m*x^(3*r)*e^3/(m^2
+ 6*m*r + 9*r^2 + 2*m + 6*r + 1) + 3*a*d*f^m*x*x^m*x^(2*r)*e^2/(m + 2*r + 1) + (f*x)^m*b*d^3*x*log(c)/(m + 1)
+ b*f^m*x*x^m*x^(3*r)*e^3*log(c)/(m + 3*r + 1) + (f*x)^m*a*d^3*x/(m + 1) + a*f^m*x*x^m*x^(3*r)*e^3/(m + 3*r +
1)