\(\int (a+b \log (c (d (e+f x)^p)^q))^{3/2} \, dx\) [486]

Optimal result

Integrand size = 22, antiderivative size = 176 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {3 b^{3/2} e^{-\frac {a}{b p q}} p^{3/2} \sqrt {\pi } q^{3/2} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f}-\frac {3 b p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f}+\frac {(e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f} \] Output:

3/4*b^(3/2)*p^(3/2)*Pi^(1/2)*q^(3/2)*(f*x+e)*erfi((a+b*ln(c*(d*(f*x+e)^p)^ 
q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))/exp(a/b/p/q)/f/((c*(d*(f*x+e)^p)^q)^(1/ 
p/q))-3/2*b*p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/f+(f*x+e)*(a+b*l 
n(c*(d*(f*x+e)^p)^q))^(3/2)/f
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.91 \[ \int \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {(e+f x) \left (3 b^{3/2} e^{-\frac {a}{b p q}} p^{3/2} \sqrt {\pi } q^{3/2} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )+2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \left (2 a-3 b p q+2 b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{4 f} \] Input:

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

Output:

((e + f*x)*((3*b^(3/2)*p^(3/2)*Sqrt[Pi]*q^(3/2)*Erfi[Sqrt[a + b*Log[c*(d*( 
e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(E^(a/(b*p*q))*(c*(d*(e + f*x) 
^p)^q)^(1/(p*q))) + 2*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]*(2*a - 3*b*p*q 
+ 2*b*Log[c*(d*(e + f*x)^p)^q])))/(4*f)
 

Rubi [A] (warning: unable to verify)

Time = 1.23 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2895, 2836, 2733, 2733, 2737, 2611, 2633}

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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2836

\(\displaystyle \frac {\int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^{3/2}d(e+f x)}{f}\)

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2611

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

((e + f*x)*(a + b*Log[c*d^q*(e + f*x)^(p*q)])^(3/2) - (3*b*p*q*(-1/2*(Sqrt 
[b]*Sqrt[p]*Sqrt[Pi]*Sqrt[q]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*d^q*(e + f*x) 
^(p*q)]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(E^(a/(b*p*q))*(c*d^q*(e + f*x)^(p*q) 
)^(1/(p*q))) + (e + f*x)*Sqrt[a + b*Log[c*d^q*(e + f*x)^(p*q)]]))/2)/f
 

Defintions of rubi rules used

rule 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[2/d   Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d 
*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[b]
 

rule 2733

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b 
*Log[c*x^n])^p, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; 
 FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 2737

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x 
^n)^(1/n))   Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ 
[{a, b, c, n, p}, x]
 

rule 2836

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

rule 2895

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. 
)*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], 
 c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ 
IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c)*a 
*b*e + 4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q 
)*c)*a*b*f*x + 2*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f* 
x)**(p*q)*c)*b**2*f*p*q*x + 4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a** 
2*e + 4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a**2*f*x - 4*sqrt(log(d** 
q*(e + f*x)**(p*q)*c)*b + a)*a*b*f*p*q*x - 6*int((sqrt(log(d**q*(e + f*x)* 
*(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c)*x)/(2*log(d**q*(e + f*x)**(p 
*q)*c)*a*b*e + 2*log(d**q*(e + f*x)**(p*q)*c)*a*b*f*x + log(d**q*(e + f*x) 
**(p*q)*c)*b**2*e*p*q + log(d**q*(e + f*x)**(p*q)*c)*b**2*f*p*q*x + 2*a**2 
*e + 2*a**2*f*x + a*b*e*p*q + a*b*f*p*q*x),x)*a*b**3*f**2*p**2*q**2 - 3*in 
t((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c)*x 
)/(2*log(d**q*(e + f*x)**(p*q)*c)*a*b*e + 2*log(d**q*(e + f*x)**(p*q)*c)*a 
*b*f*x + log(d**q*(e + f*x)**(p*q)*c)*b**2*e*p*q + log(d**q*(e + f*x)**(p* 
q)*c)*b**2*f*p*q*x + 2*a**2*e + 2*a**2*f*x + a*b*e*p*q + a*b*f*p*q*x),x)*b 
**4*f**2*p**3*q**3)/(2*f*(2*a + b*p*q))