\(\int (f+g x)^{3/2} (a+b \log (c (d+e x)^n)) \, dx\) [138]

Optimal result

Integrand size = 24, antiderivative size = 163 \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {4 b (e f-d g)^2 n \sqrt {f+g x}}{5 e^2 g}-\frac {4 b (e f-d g) n (f+g x)^{3/2}}{15 e g}-\frac {4 b n (f+g x)^{5/2}}{25 g}+\frac {4 b (e f-d g)^{5/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{5 e^{5/2} g}+\frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g} \] Output:

-4/5*b*(-d*g+e*f)^2*n*(g*x+f)^(1/2)/e^2/g-4/15*b*(-d*g+e*f)*n*(g*x+f)^(3/2 
)/e/g-4/25*b*n*(g*x+f)^(5/2)/g+4/5*b*(-d*g+e*f)^(5/2)*n*arctanh(e^(1/2)*(g 
*x+f)^(1/2)/(-d*g+e*f)^(1/2))/e^(5/2)/g+2/5*(g*x+f)^(5/2)*(a+b*ln(c*(e*x+d 
)^n))/g
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.84 \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {2 \left (-\frac {2}{5} b n (f+g x)^{5/2}-\frac {2 b (e f-d g) n \left (\sqrt {e} \sqrt {f+g x} (4 e f-3 d g+e g x)-3 (e f-d g)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )}{3 e^{5/2}}+(f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{5 g} \] Input:

Integrate[(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]),x]
 

Output:

(2*((-2*b*n*(f + g*x)^(5/2))/5 - (2*b*(e*f - d*g)*n*(Sqrt[e]*Sqrt[f + g*x] 
*(4*e*f - 3*d*g + e*g*x) - 3*(e*f - d*g)^(3/2)*ArcTanh[(Sqrt[e]*Sqrt[f + g 
*x])/Sqrt[e*f - d*g]]))/(3*e^(5/2)) + (f + g*x)^(5/2)*(a + b*Log[c*(d + e* 
x)^n])))/(5*g)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2842, 60, 60, 60, 73, 221}

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.

\(\displaystyle \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \int \frac {(f+g x)^{5/2}}{d+e x}dx}{5 g}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {(e f-d g) \int \frac {(f+g x)^{3/2}}{d+e x}dx}{e}+\frac {2 (f+g x)^{5/2}}{5 e}\right )}{5 g}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {(e f-d g) \left (\frac {(e f-d g) \int \frac {\sqrt {f+g x}}{d+e x}dx}{e}+\frac {2 (f+g x)^{3/2}}{3 e}\right )}{e}+\frac {2 (f+g x)^{5/2}}{5 e}\right )}{5 g}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {(e f-d g) \left (\frac {(e f-d g) \left (\frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x}}dx}{e}+\frac {2 \sqrt {f+g x}}{e}\right )}{e}+\frac {2 (f+g x)^{3/2}}{3 e}\right )}{e}+\frac {2 (f+g x)^{5/2}}{5 e}\right )}{5 g}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {(e f-d g) \left (\frac {(e f-d g) \left (\frac {2 (e f-d g) \int \frac {1}{d+\frac {e (f+g x)}{g}-\frac {e f}{g}}d\sqrt {f+g x}}{e g}+\frac {2 \sqrt {f+g x}}{e}\right )}{e}+\frac {2 (f+g x)^{3/2}}{3 e}\right )}{e}+\frac {2 (f+g x)^{5/2}}{5 e}\right )}{5 g}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{5 g}-\frac {2 b e n \left (\frac {(e f-d g) \left (\frac {(e f-d g) \left (\frac {2 \sqrt {f+g x}}{e}-\frac {2 \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2}}\right )}{e}+\frac {2 (f+g x)^{3/2}}{3 e}\right )}{e}+\frac {2 (f+g x)^{5/2}}{5 e}\right )}{5 g}\)

Input:

Int[(f + g*x)^(3/2)*(a + b*Log[c*(d + e*x)^n]),x]
 

Output:

(-2*b*e*n*((2*(f + g*x)^(5/2))/(5*e) + ((e*f - d*g)*((2*(f + g*x)^(3/2))/( 
3*e) + ((e*f - d*g)*((2*Sqrt[f + g*x])/e - (2*Sqrt[e*f - d*g]*ArcTanh[(Sqr 
t[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/e^(3/2)))/e))/e))/(5*g) + (2*(f + g* 
x)^(5/2)*(a + b*Log[c*(d + e*x)^n]))/(5*g)
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 2842

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

Maple [F]

\[\int \left (g x +f \right )^{\frac {3}{2}} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )d x\]

Input:

int((g*x+f)^(3/2)*(a+b*ln(c*(e*x+d)^n)),x)
 

Output:

int((g*x+f)^(3/2)*(a+b*ln(c*(e*x+d)^n)),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 538, normalized size of antiderivative = 3.30 \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\left [\frac {2 \, {\left (15 \, {\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} n \sqrt {\frac {e f - d g}{e}} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {g x + f} e \sqrt {\frac {e f - d g}{e}}}{e x + d}\right ) + {\left (15 \, a e^{2} f^{2} - 3 \, {\left (2 \, b e^{2} g^{2} n - 5 \, a e^{2} g^{2}\right )} x^{2} - 2 \, {\left (23 \, b e^{2} f^{2} - 35 \, b d e f g + 15 \, b d^{2} g^{2}\right )} n + 2 \, {\left (15 \, a e^{2} f g - {\left (11 \, b e^{2} f g - 5 \, b d e g^{2}\right )} n\right )} x + 15 \, {\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + b e^{2} f^{2} n\right )} \log \left (e x + d\right ) + 15 \, {\left (b e^{2} g^{2} x^{2} + 2 \, b e^{2} f g x + b e^{2} f^{2}\right )} \log \left (c\right )\right )} \sqrt {g x + f}\right )}}{75 \, e^{2} g}, \frac {2 \, {\left (30 \, {\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} n \sqrt {-\frac {e f - d g}{e}} \arctan \left (-\frac {\sqrt {g x + f} e \sqrt {-\frac {e f - d g}{e}}}{e f - d g}\right ) + {\left (15 \, a e^{2} f^{2} - 3 \, {\left (2 \, b e^{2} g^{2} n - 5 \, a e^{2} g^{2}\right )} x^{2} - 2 \, {\left (23 \, b e^{2} f^{2} - 35 \, b d e f g + 15 \, b d^{2} g^{2}\right )} n + 2 \, {\left (15 \, a e^{2} f g - {\left (11 \, b e^{2} f g - 5 \, b d e g^{2}\right )} n\right )} x + 15 \, {\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + b e^{2} f^{2} n\right )} \log \left (e x + d\right ) + 15 \, {\left (b e^{2} g^{2} x^{2} + 2 \, b e^{2} f g x + b e^{2} f^{2}\right )} \log \left (c\right )\right )} \sqrt {g x + f}\right )}}{75 \, e^{2} g}\right ] \] Input:

integrate((g*x+f)^(3/2)*(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")
 

Output:

[2/75*(15*(b*e^2*f^2 - 2*b*d*e*f*g + b*d^2*g^2)*n*sqrt((e*f - d*g)/e)*log( 
(e*g*x + 2*e*f - d*g + 2*sqrt(g*x + f)*e*sqrt((e*f - d*g)/e))/(e*x + d)) + 
 (15*a*e^2*f^2 - 3*(2*b*e^2*g^2*n - 5*a*e^2*g^2)*x^2 - 2*(23*b*e^2*f^2 - 3 
5*b*d*e*f*g + 15*b*d^2*g^2)*n + 2*(15*a*e^2*f*g - (11*b*e^2*f*g - 5*b*d*e* 
g^2)*n)*x + 15*(b*e^2*g^2*n*x^2 + 2*b*e^2*f*g*n*x + b*e^2*f^2*n)*log(e*x + 
 d) + 15*(b*e^2*g^2*x^2 + 2*b*e^2*f*g*x + b*e^2*f^2)*log(c))*sqrt(g*x + f) 
)/(e^2*g), 2/75*(30*(b*e^2*f^2 - 2*b*d*e*f*g + b*d^2*g^2)*n*sqrt(-(e*f - d 
*g)/e)*arctan(-sqrt(g*x + f)*e*sqrt(-(e*f - d*g)/e)/(e*f - d*g)) + (15*a*e 
^2*f^2 - 3*(2*b*e^2*g^2*n - 5*a*e^2*g^2)*x^2 - 2*(23*b*e^2*f^2 - 35*b*d*e* 
f*g + 15*b*d^2*g^2)*n + 2*(15*a*e^2*f*g - (11*b*e^2*f*g - 5*b*d*e*g^2)*n)* 
x + 15*(b*e^2*g^2*n*x^2 + 2*b*e^2*f*g*n*x + b*e^2*f^2*n)*log(e*x + d) + 15 
*(b*e^2*g^2*x^2 + 2*b*e^2*f*g*x + b*e^2*f^2)*log(c))*sqrt(g*x + f))/(e^2*g 
)]
 

Sympy [F]

\[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g x\right )^{\frac {3}{2}}\, dx \] Input:

integrate((g*x+f)**(3/2)*(a+b*ln(c*(e*x+d)**n)),x)
 

Output:

Integral((a + b*log(c*(d + e*x)**n))*(f + g*x)**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)^(3/2)*(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?` f 
or more de
 

Giac [F]

\[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int { {\left (g x + f\right )}^{\frac {3}{2}} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \,d x } \] Input:

integrate((g*x+f)^(3/2)*(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")
 

Output:

integrate((g*x + f)^(3/2)*(b*log((e*x + d)^n*c) + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int {\left (f+g\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \] Input:

int((f + g*x)^(3/2)*(a + b*log(c*(d + e*x)^n)),x)
 

Output:

int((f + g*x)^(3/2)*(a + b*log(c*(d + e*x)^n)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.42 \[ \int (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {\frac {4 \sqrt {e}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) b \,d^{2} g^{2} n}{5}-\frac {8 \sqrt {e}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) b d e f g n}{5}+\frac {4 \sqrt {e}\, \sqrt {d g -e f}\, \mathit {atan} \left (\frac {\sqrt {g x +f}\, e}{\sqrt {e}\, \sqrt {d g -e f}}\right ) b \,e^{2} f^{2} n}{5}+\frac {2 \sqrt {g x +f}\, \mathrm {log}\left (\frac {\left (e g x +d g \right )^{n} c}{g^{n}}\right ) b \,e^{3} f^{2}}{5}+\frac {4 \sqrt {g x +f}\, \mathrm {log}\left (\frac {\left (e g x +d g \right )^{n} c}{g^{n}}\right ) b \,e^{3} f g x}{5}+\frac {2 \sqrt {g x +f}\, \mathrm {log}\left (\frac {\left (e g x +d g \right )^{n} c}{g^{n}}\right ) b \,e^{3} g^{2} x^{2}}{5}+\frac {2 \sqrt {g x +f}\, a \,e^{3} f^{2}}{5}+\frac {4 \sqrt {g x +f}\, a \,e^{3} f g x}{5}+\frac {2 \sqrt {g x +f}\, a \,e^{3} g^{2} x^{2}}{5}-\frac {4 \sqrt {g x +f}\, b \,d^{2} e \,g^{2} n}{5}+\frac {28 \sqrt {g x +f}\, b d \,e^{2} f g n}{15}+\frac {4 \sqrt {g x +f}\, b d \,e^{2} g^{2} n x}{15}-\frac {92 \sqrt {g x +f}\, b \,e^{3} f^{2} n}{75}-\frac {44 \sqrt {g x +f}\, b \,e^{3} f g n x}{75}-\frac {4 \sqrt {g x +f}\, b \,e^{3} g^{2} n \,x^{2}}{25}}{e^{3} g} \] Input:

int((g*x+f)^(3/2)*(a+b*log(c*(e*x+d)^n)),x)
 

Output:

(2*(30*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - 
e*f)))*b*d**2*g**2*n - 60*sqrt(e)*sqrt(d*g - e*f)*atan((sqrt(f + g*x)*e)/( 
sqrt(e)*sqrt(d*g - e*f)))*b*d*e*f*g*n + 30*sqrt(e)*sqrt(d*g - e*f)*atan((s 
qrt(f + g*x)*e)/(sqrt(e)*sqrt(d*g - e*f)))*b*e**2*f**2*n + 15*sqrt(f + g*x 
)*log(((d*g + e*g*x)**n*c)/g**n)*b*e**3*f**2 + 30*sqrt(f + g*x)*log(((d*g 
+ e*g*x)**n*c)/g**n)*b*e**3*f*g*x + 15*sqrt(f + g*x)*log(((d*g + e*g*x)**n 
*c)/g**n)*b*e**3*g**2*x**2 + 15*sqrt(f + g*x)*a*e**3*f**2 + 30*sqrt(f + g* 
x)*a*e**3*f*g*x + 15*sqrt(f + g*x)*a*e**3*g**2*x**2 - 30*sqrt(f + g*x)*b*d 
**2*e*g**2*n + 70*sqrt(f + g*x)*b*d*e**2*f*g*n + 10*sqrt(f + g*x)*b*d*e**2 
*g**2*n*x - 46*sqrt(f + g*x)*b*e**3*f**2*n - 22*sqrt(f + g*x)*b*e**3*f*g*n 
*x - 6*sqrt(f + g*x)*b*e**3*g**2*n*x**2))/(75*e**3*g)