\(\int (f x)^m \log (c (d+e \sqrt {x})^p) \, dx\) [61]

Optimal result

Integrand size = 20, antiderivative size = 83 \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=-\frac {e p x^{3/2} (f x)^m \operatorname {Hypergeometric2F1}\left (1,3+2 m,2 (2+m),-\frac {e \sqrt {x}}{d}\right )}{d \left (3+5 m+2 m^2\right )}+\frac {(f x)^{1+m} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (1+m)} \] Output:

-e*p*x^(3/2)*(f*x)^m*hypergeom([1, 3+2*m],[4+2*m],-e*x^(1/2)/d)/d/(2*m^2+5 
*m+3)+(f*x)^(1+m)*ln(c*(d+e*x^(1/2))^p)/f/(1+m)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92 \[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\frac {x (f x)^m \left (-e p \sqrt {x} \operatorname {Hypergeometric2F1}\left (1,3+2 m,4+2 m,-\frac {e \sqrt {x}}{d}\right )+d (3+2 m) \log \left (c \left (d+e \sqrt {x}\right )^p\right )\right )}{d (1+m) (3+2 m)} \] Input:

Integrate[(f*x)^m*Log[c*(d + e*Sqrt[x])^p],x]
 

Output:

(x*(f*x)^m*(-(e*p*Sqrt[x]*Hypergeometric2F1[1, 3 + 2*m, 4 + 2*m, -((e*Sqrt 
[x])/d)]) + d*(3 + 2*m)*Log[c*(d + e*Sqrt[x])^p]))/(d*(1 + m)*(3 + 2*m))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2905, 30, 864, 74}

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.

\(\displaystyle \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2905

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 864

\(\displaystyle \frac {(f x)^{m+1} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f (m+1)}-\frac {e p x^{-m} (f x)^m \int \frac {x^{m+1}}{d+e \sqrt {x}}d\sqrt {x}}{m+1}\)

\(\Big \downarrow \) 74

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

Input:

Int[(f*x)^m*Log[c*(d + e*Sqrt[x])^p],x]
 

Output:

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

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 864

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi 
nator[n]}, Simp[k   Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x 
^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
 

rule 2905

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

Maple [F]

\[\int \left (f x \right )^{m} \ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )d x\]

Input:

int((f*x)^m*ln(c*(d+e*x^(1/2))^p),x)
 

Output:

int((f*x)^m*ln(c*(d+e*x^(1/2))^p),x)
 

Fricas [F]

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

integrate((f*x)^m*log(c*(d+e*x^(1/2))^p),x, algorithm="fricas")
 

Output:

integral((f*x)^m*log((e*sqrt(x) + d)^p*c), x)
 

Sympy [F]

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

integrate((f*x)**m*ln(c*(d+e*x**(1/2))**p),x)
 

Output:

Integral((f*x)**m*log(c*(d + e*sqrt(x))**p), x)
 

Maxima [F]

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

integrate((f*x)^m*log(c*(d+e*x^(1/2))^p),x, algorithm="maxima")
 

Output:

e^2*f^m*p*integrate(1/2*x*x^m/(d*e*(m + 1)*sqrt(x) + d^2*(m + 1)), x) + (d 
*f^m*(2*m + 3)*x*x^m*log((e*sqrt(x) + d)^p) + d*f^m*(2*m + 3)*x*x^m*log(c) 
 - e*f^m*p*x^(3/2)*x^m)/((2*m^2 + 5*m + 3)*d)
 

Giac [F]

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

integrate((f*x)^m*log(c*(d+e*x^(1/2))^p),x, algorithm="giac")
 

Output:

integrate((f*x)^m*log((e*sqrt(x) + d)^p*c), x)
 

Mupad [F(-1)]

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

int(log(c*(d + e*x^(1/2))^p)*(f*x)^m,x)
 

Output:

int(log(c*(d + e*x^(1/2))^p)*(f*x)^m, x)
 

Reduce [F]

\[ \int (f x)^m \log \left (c \left (d+e \sqrt {x}\right )^p\right ) \, dx=\frac {f^{m} \left (2 x^{m +\frac {1}{2}} d e \,m^{2} p +2 x^{m +\frac {1}{2}} d e m p +4 x^{m} \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{p} c \right ) e^{2} m^{3} x +6 x^{m} \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{p} c \right ) e^{2} m^{2} x +2 x^{m} \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{p} c \right ) e^{2} m x -2 x^{m} d^{2} m^{2} p -3 x^{m} d^{2} m p -x^{m} d^{2} p -2 x^{m} e^{2} m^{2} p x -x^{m} e^{2} m p x -2 \left (\int \frac {x^{m +\frac {1}{2}}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{3} e \,m^{4} p -5 \left (\int \frac {x^{m +\frac {1}{2}}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{3} e \,m^{3} p -4 \left (\int \frac {x^{m +\frac {1}{2}}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{3} e \,m^{2} p -\left (\int \frac {x^{m +\frac {1}{2}}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{3} e m p +2 \left (\int \frac {x^{m}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{4} m^{4} p +5 \left (\int \frac {x^{m}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{4} m^{3} p +4 \left (\int \frac {x^{m}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{4} m^{2} p +\left (\int \frac {x^{m}}{-e^{2} m \,x^{2}+d^{2} m x -e^{2} x^{2}+d^{2} x}d x \right ) d^{4} m p \right )}{2 e^{2} m \left (2 m^{3}+5 m^{2}+4 m +1\right )} \] Input:

int((f*x)^m*log(c*(d+e*x^(1/2))^p),x)
 

Output:

(f**m*(2*x**((2*m + 1)/2)*d*e*m**2*p + 2*x**((2*m + 1)/2)*d*e*m*p + 4*x**m 
*log((sqrt(x)*e + d)**p*c)*e**2*m**3*x + 6*x**m*log((sqrt(x)*e + d)**p*c)* 
e**2*m**2*x + 2*x**m*log((sqrt(x)*e + d)**p*c)*e**2*m*x - 2*x**m*d**2*m**2 
*p - 3*x**m*d**2*m*p - x**m*d**2*p - 2*x**m*e**2*m**2*p*x - x**m*e**2*m*p* 
x - 2*int(x**((2*m + 1)/2)/(d**2*m*x + d**2*x - e**2*m*x**2 - e**2*x**2),x 
)*d**3*e*m**4*p - 5*int(x**((2*m + 1)/2)/(d**2*m*x + d**2*x - e**2*m*x**2 
- e**2*x**2),x)*d**3*e*m**3*p - 4*int(x**((2*m + 1)/2)/(d**2*m*x + d**2*x 
- e**2*m*x**2 - e**2*x**2),x)*d**3*e*m**2*p - int(x**((2*m + 1)/2)/(d**2*m 
*x + d**2*x - e**2*m*x**2 - e**2*x**2),x)*d**3*e*m*p + 2*int(x**m/(d**2*m* 
x + d**2*x - e**2*m*x**2 - e**2*x**2),x)*d**4*m**4*p + 5*int(x**m/(d**2*m* 
x + d**2*x - e**2*m*x**2 - e**2*x**2),x)*d**4*m**3*p + 4*int(x**m/(d**2*m* 
x + d**2*x - e**2*m*x**2 - e**2*x**2),x)*d**4*m**2*p + int(x**m/(d**2*m*x 
+ d**2*x - e**2*m*x**2 - e**2*x**2),x)*d**4*m*p))/(2*e**2*m*(2*m**3 + 5*m* 
*2 + 4*m + 1))