\(\int (f+g x) (a+b \log (c (d+e x)^n))^{3/2} \, dx\) [112]

Optimal result

Integrand size = 24, antiderivative size = 330 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\frac {3 b^{3/2} e^{-\frac {a}{b n}} (e f-d g) n^{3/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b n}} g n^{3/2} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{16 e^2}-\frac {3 b (e f-d g) n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac {3 b g n (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{8 e^2}+\frac {(e f-d g) (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2} \] Output:

3/4*b^(3/2)*(-d*g+e*f)*n^(3/2)*Pi^(1/2)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n)) 
^(1/2)/b^(1/2)/n^(1/2))/e^2/exp(a/b/n)/((c*(e*x+d)^n)^(1/n))+3/32*b^(3/2)* 
g*n^(3/2)*2^(1/2)*Pi^(1/2)*(e*x+d)^2*erfi(2^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1 
/2)/b^(1/2)/n^(1/2))/e^2/exp(2*a/b/n)/((c*(e*x+d)^n)^(2/n))-3/2*b*(-d*g+e* 
f)*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(1/2)/e^2-3/8*b*g*n*(e*x+d)^2*(a+b*ln(c 
*(e*x+d)^n))^(1/2)/e^2+(-d*g+e*f)*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(3/2)/e^2+ 
1/2*g*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))^(3/2)/e^2
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.85 \[ \int (f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\frac {(d+e x) \left (32 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+16 g (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+3 b g n (d+e x) \left (\sqrt {b} e^{-\frac {2 a}{b n}} \sqrt {n} \sqrt {2 \pi } \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-4 \sqrt {a+b \log \left (c (d+e x)^n\right )}\right )+24 b (e f-d g) n \left (\sqrt {b} e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-2 \sqrt {a+b \log \left (c (d+e x)^n\right )}\right )\right )}{32 e^2} \] Input:

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

Output:

((d + e*x)*(32*(e*f - d*g)*(a + b*Log[c*(d + e*x)^n])^(3/2) + 16*g*(d + e* 
x)*(a + b*Log[c*(d + e*x)^n])^(3/2) + 3*b*g*n*(d + e*x)*((Sqrt[b]*Sqrt[n]* 
Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n]) 
])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n)) - 4*Sqrt[a + b*Log[c*(d + e*x)^ 
n]]) + 24*b*(e*f - d*g)*n*((Sqrt[b]*Sqrt[n]*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[c 
*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) - 
2*Sqrt[a + b*Log[c*(d + e*x)^n]])))/(32*e^2)
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2848, 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.

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt {\pi } b^{3/2} n^{3/2} e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e^2}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} g n^{3/2} e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{16 e^2}+\frac {(d+e x) (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e^2}-\frac {3 b n (d+e x) (e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2}+\frac {g (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 e^2}-\frac {3 b g n (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{8 e^2}\)

Input:

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

Output:

(3*b^(3/2)*(e*f - d*g)*n^(3/2)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d 
 + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(4*e^2*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) 
 + (3*b^(3/2)*g*n^(3/2)*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Lo 
g[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(16*e^2*E^((2*a)/(b*n))*(c*(d + e*x 
)^n)^(2/n)) - (3*b*(e*f - d*g)*n*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]) 
/(2*e^2) - (3*b*g*n*(d + e*x)^2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(8*e^2) + 
((e*f - d*g)*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/e^2 + (g*(d + e*x 
)^2*(a + b*Log[c*(d + e*x)^n])^(3/2))/(2*e^2)
 

Defintions of rubi rules used

rule 2009

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

rule 2848

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 8*sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b*d**2*g + 16* 
sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b*d*e*f + 16*sqrt(lo 
g((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b*e**2*f*x + 8*sqrt(log((d 
+ e*x)**n*c)*b + a)*log((d + e*x)**n*c)*a*b*e**2*g*x**2 + 4*sqrt(log((d + 
e*x)**n*c)*b + a)*log((d + e*x)**n*c)*b**2*d*e*f*n + 4*sqrt(log((d + e*x)* 
*n*c)*b + a)*log((d + e*x)**n*c)*b**2*e**2*f*n*x + 2*sqrt(log((d + e*x)**n 
*c)*b + a)*log((d + e*x)**n*c)*b**2*e**2*g*n*x**2 - 8*sqrt(log((d + e*x)** 
n*c)*b + a)*a**2*d**2*g + 16*sqrt(log((d + e*x)**n*c)*b + a)*a**2*d*e*f + 
16*sqrt(log((d + e*x)**n*c)*b + a)*a**2*e**2*f*x + 8*sqrt(log((d + e*x)**n 
*c)*b + a)*a**2*e**2*g*x**2 + 4*sqrt(log((d + e*x)**n*c)*b + a)*a*b*d*e*f* 
n + 12*sqrt(log((d + e*x)**n*c)*b + a)*a*b*d*e*g*n*x - 20*sqrt(log((d + e* 
x)**n*c)*b + a)*a*b*e**2*f*n*x - 4*sqrt(log((d + e*x)**n*c)*b + a)*a*b*e** 
2*g*n*x**2 - 6*sqrt(log((d + e*x)**n*c)*b + a)*b**2*e**2*f*n**2*x - 12*int 
((sqrt(log((d + e*x)**n*c)*b + a)*log((d + e*x)**n*c)*x**2)/(4*log((d + e* 
x)**n*c)*a*b*d + 4*log((d + e*x)**n*c)*a*b*e*x + log((d + e*x)**n*c)*b**2* 
d*n + log((d + e*x)**n*c)*b**2*e*n*x + 4*a**2*d + 4*a**2*e*x + a*b*d*n + a 
*b*e*n*x),x)*a*b**3*e**3*g*n**2 - 3*int((sqrt(log((d + e*x)**n*c)*b + a)*l 
og((d + e*x)**n*c)*x**2)/(4*log((d + e*x)**n*c)*a*b*d + 4*log((d + e*x)**n 
*c)*a*b*e*x + log((d + e*x)**n*c)*b**2*d*n + log((d + e*x)**n*c)*b**2*e*n* 
x + 4*a**2*d + 4*a**2*e*x + a*b*d*n + a*b*e*n*x),x)*b**4*e**3*g*n**3 - ...