\(\int (d+e x)^m \log (c (a+b x^3)^p) \, dx\) [206]

Optimal result

Integrand size = 20, antiderivative size = 301 \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {\sqrt [3]{b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {\sqrt [3]{b} p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{e \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right ) (1+m) (2+m)}+\frac {(d+e x)^{1+m} \log \left (c \left (a+b x^3\right )^p\right )}{e (1+m)} \] Output:

b^(1/3)*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],b^(1/3)*(e*x+d)/(b^(1/3)* 
d-a^(1/3)*e))/e/(b^(1/3)*d-a^(1/3)*e)/(1+m)/(2+m)+b^(1/3)*p*(e*x+d)^(2+m)* 
hypergeom([1, 2+m],[3+m],b^(1/3)*(e*x+d)/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e)) 
/e/(b^(1/3)*d+(-1)^(1/3)*a^(1/3)*e)/(1+m)/(2+m)+b^(1/3)*p*(e*x+d)^(2+m)*hy 
pergeom([1, 2+m],[3+m],b^(1/3)*(e*x+d)/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e))/e 
/(b^(1/3)*d-(-1)^(2/3)*a^(1/3)*e)/(1+m)/(2+m)+(e*x+d)^(1+m)*ln(c*(b*x^3+a) 
^p)/e/(1+m)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.79 \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (-\frac {\sqrt [3]{b} p (d+e x) \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-\sqrt [3]{a} e}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}-\frac {\operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{2+m}+\log \left (c \left (a+b x^3\right )^p\right )\right )}{e (1+m)} \] Input:

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

Output:

((d + e*x)^(1 + m)*(-((b^(1/3)*p*(d + e*x)*(-(Hypergeometric2F1[1, 2 + m, 
3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)]/(b^(1/3)*d - a^(1/3)*e 
)) - Hypergeometric2F1[1, 2 + m, 3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d + ( 
-1)^(1/3)*a^(1/3)*e)]/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e) - Hypergeometric2 
F1[1, 2 + m, 3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e) 
]/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)))/(2 + m)) + Log[c*(a + b*x^3)^p]))/( 
e*(1 + m))
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2913, 7276, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2913

\(\displaystyle \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}-\frac {3 b p \int \frac {x^2 (d+e x)^{m+1}}{b x^3+a}dx}{e (m+1)}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}-\frac {3 b p \int \left (\frac {(d+e x)^{m+1}}{3 b^{2/3} \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}+\frac {(d+e x)^{m+1}}{3 b^{2/3} \left (\sqrt [3]{b} x-\sqrt [3]{-1} \sqrt [3]{a}\right )}+\frac {(d+e x)^{m+1}}{3 b^{2/3} \left (\sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a}\right )}\right )dx}{e (m+1)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^{m+1} \log \left (c \left (a+b x^3\right )^p\right )}{e (m+1)}-\frac {3 b p \left (-\frac {(d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-\sqrt [3]{a} e}\right )}{3 b^{2/3} (m+2) \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right )}-\frac {(d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d+\sqrt [3]{-1} \sqrt [3]{a} e}\right )}{3 b^{2/3} (m+2) \left (\sqrt [3]{-1} \sqrt [3]{a} e+\sqrt [3]{b} d\right )}-\frac {(d+e x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,\frac {\sqrt [3]{b} (d+e x)}{\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e}\right )}{3 b^{2/3} (m+2) \left (\sqrt [3]{b} d-(-1)^{2/3} \sqrt [3]{a} e\right )}\right )}{e (m+1)}\)

Input:

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

Output:

(-3*b*p*(-1/3*((d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b^(1/ 
3)*(d + e*x))/(b^(1/3)*d - a^(1/3)*e)])/(b^(2/3)*(b^(1/3)*d - a^(1/3)*e)*( 
2 + m)) - ((d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (b^(1/3)*( 
d + e*x))/(b^(1/3)*d + (-1)^(1/3)*a^(1/3)*e)])/(3*b^(2/3)*(b^(1/3)*d + (-1 
)^(1/3)*a^(1/3)*e)*(2 + m)) - ((d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + 
m, 3 + m, (b^(1/3)*(d + e*x))/(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)])/(3*b^(2 
/3)*(b^(1/3)*d - (-1)^(2/3)*a^(1/3)*e)*(2 + m))))/(e*(1 + m)) + ((d + e*x) 
^(1 + m)*Log[c*(a + b*x^3)^p])/(e*(1 + m))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2913

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_. 
)*(x_))^(r_.), x_Symbol] :> Simp[(f + g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n 
)^p])/(g*(r + 1))), x] - Simp[b*e*n*(p/(g*(r + 1)))   Int[x^(n - 1)*((f + g 
*x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x 
] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

\[\int \left (e x +d \right )^{m} \ln \left (c \left (b \,x^{3}+a \right )^{p}\right )d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \,d x } \] Input:

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

Output:

integral((e*x + d)^m*log((b*x^3 + a)^p*c), x)
 

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**m*ln(c*(b*x**3+a)**p),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \,d x } \] Input:

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

Output:

(e*x + d)*(e*x + d)^m*log((b*x^3 + a)^p)/(e*(m + 1)) + integrate(-(3*b*d*p 
*x^2 - (e*(m + 1)*log(c) - 3*e*p)*b*x^3 - a*e*(m + 1)*log(c))*(e*x + d)^m/ 
(b*e*(m + 1)*x^3 + a*e*(m + 1)), x)
 

Giac [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int { {\left (e x + d\right )}^{m} \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \,d x } \] Input:

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

Output:

integrate((e*x + d)^m*log((b*x^3 + a)^p*c), x)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\int \ln \left (c\,{\left (b\,x^3+a\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \] Input:

int(log(c*(a + b*x^3)^p)*(d + e*x)^m,x)
 

Output:

int(log(c*(a + b*x^3)^p)*(d + e*x)^m, x)
 

Reduce [F]

\[ \int (d+e x)^m \log \left (c \left (a+b x^3\right )^p\right ) \, dx=\frac {\left (e x +d \right )^{m} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) d \,m^{2}+\left (e x +d \right )^{m} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) d m +\left (e x +d \right )^{m} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) e \,m^{2} x +\left (e x +d \right )^{m} \mathrm {log}\left (\left (b \,x^{3}+a \right )^{p} c \right ) e m x -6 \left (e x +d \right )^{m} d m p -3 \left (e x +d \right )^{m} d p -3 \left (e x +d \right )^{m} e m p x +6 \left (\int \frac {\left (e x +d \right )^{m}}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) a d e \,m^{3} p +12 \left (\int \frac {\left (e x +d \right )^{m}}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) a d e \,m^{2} p +6 \left (\int \frac {\left (e x +d \right )^{m}}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) a d e m p -3 \left (\int \frac {\left (e x +d \right )^{m} x^{2}}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) b \,d^{2} m^{3} p -6 \left (\int \frac {\left (e x +d \right )^{m} x^{2}}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) b \,d^{2} m^{2} p -3 \left (\int \frac {\left (e x +d \right )^{m} x^{2}}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) b \,d^{2} m p +3 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) a \,e^{2} m^{3} p +6 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) a \,e^{2} m^{2} p +3 \left (\int \frac {\left (e x +d \right )^{m} x}{b e m \,x^{4}+b d m \,x^{3}+b e \,x^{4}+b d \,x^{3}+a e m x +a d m +a e x +a d}d x \right ) a \,e^{2} m p}{e m \left (m^{2}+2 m +1\right )} \] Input:

int((e*x+d)^m*log(c*(b*x^3+a)^p),x)
 

Output:

((d + e*x)**m*log((a + b*x**3)**p*c)*d*m**2 + (d + e*x)**m*log((a + b*x**3 
)**p*c)*d*m + (d + e*x)**m*log((a + b*x**3)**p*c)*e*m**2*x + (d + e*x)**m* 
log((a + b*x**3)**p*c)*e*m*x - 6*(d + e*x)**m*d*m*p - 3*(d + e*x)**m*d*p - 
 3*(d + e*x)**m*e*m*p*x + 6*int((d + e*x)**m/(a*d*m + a*d + a*e*m*x + a*e* 
x + b*d*m*x**3 + b*d*x**3 + b*e*m*x**4 + b*e*x**4),x)*a*d*e*m**3*p + 12*in 
t((d + e*x)**m/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x**3 + b*d*x**3 + b* 
e*m*x**4 + b*e*x**4),x)*a*d*e*m**2*p + 6*int((d + e*x)**m/(a*d*m + a*d + a 
*e*m*x + a*e*x + b*d*m*x**3 + b*d*x**3 + b*e*m*x**4 + b*e*x**4),x)*a*d*e*m 
*p - 3*int(((d + e*x)**m*x**2)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x**3 
 + b*d*x**3 + b*e*m*x**4 + b*e*x**4),x)*b*d**2*m**3*p - 6*int(((d + e*x)** 
m*x**2)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x**3 + b*d*x**3 + b*e*m*x** 
4 + b*e*x**4),x)*b*d**2*m**2*p - 3*int(((d + e*x)**m*x**2)/(a*d*m + a*d + 
a*e*m*x + a*e*x + b*d*m*x**3 + b*d*x**3 + b*e*m*x**4 + b*e*x**4),x)*b*d**2 
*m*p + 3*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x**3 
+ b*d*x**3 + b*e*m*x**4 + b*e*x**4),x)*a*e**2*m**3*p + 6*int(((d + e*x)**m 
*x)/(a*d*m + a*d + a*e*m*x + a*e*x + b*d*m*x**3 + b*d*x**3 + b*e*m*x**4 + 
b*e*x**4),x)*a*e**2*m**2*p + 3*int(((d + e*x)**m*x)/(a*d*m + a*d + a*e*m*x 
 + a*e*x + b*d*m*x**3 + b*d*x**3 + b*e*m*x**4 + b*e*x**4),x)*a*e**2*m*p)/( 
e*m*(m**2 + 2*m + 1))