\(\int (g+h x) (a+b \log (c (d (e+f x)^p)^q))^{3/2} \, dx\) [485]

Optimal result

Integrand size = 28, antiderivative size = 396 \[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {3 b^{3/2} e^{-\frac {a}{b p q}} (f g-e h) p^{3/2} \sqrt {\pi } q^{3/2} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{4 f^2}+\frac {3 b^{3/2} e^{-\frac {2 a}{b p q}} h p^{3/2} \sqrt {\frac {\pi }{2}} q^{3/2} (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{16 f^2}-\frac {3 b (f g-e h) p q (e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{2 f^2}-\frac {3 b h p q (e+f x)^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{8 f^2}+\frac {(f g-e h) (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{f^2}+\frac {h (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{2 f^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.88 \[ \int (g+h x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2} \, dx=\frac {(e+f x) \left (32 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+16 h (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}+3 b h p q (e+f x) \left (\sqrt {b} e^{-\frac {2 a}{b p q}} \sqrt {p} \sqrt {2 \pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-4 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )+24 b (f g-e h) p q \left (\sqrt {b} e^{-\frac {a}{b p q}} \sqrt {p} \sqrt {\pi } \sqrt {q} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )-2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}\right )\right )}{32 f^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.65 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2895, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((h*x+g)*(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]

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

integrate((h*x+g)*(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)*(g + h*x), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 8*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c 
)*a*b*e**2*h + 16*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f 
*x)**(p*q)*c)*a*b*e*f*g + 16*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log( 
d**q*(e + f*x)**(p*q)*c)*a*b*f**2*g*x + 8*sqrt(log(d**q*(e + f*x)**(p*q)*c 
)*b + a)*log(d**q*(e + f*x)**(p*q)*c)*a*b*f**2*h*x**2 + 4*sqrt(log(d**q*(e 
 + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c)*b**2*e*f*g*p*q + 4*s 
qrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c)*b**2* 
f**2*g*p*q*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**2*h*p*q*x**2 - 8*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b 
+ a)*a**2*e**2*h + 16*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a**2*e*f*g 
+ 16*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a**2*f**2*g*x + 8*sqrt(log(d 
**q*(e + f*x)**(p*q)*c)*b + a)*a**2*f**2*h*x**2 + 4*sqrt(log(d**q*(e + f*x 
)**(p*q)*c)*b + a)*a*b*e*f*g*p*q + 12*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b 
+ a)*a*b*e*f*h*p*q*x - 20*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a*b*f** 
2*g*p*q*x - 4*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*a*b*f**2*h*p*q*x**2 
 - 6*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*b**2*f**2*g*p**2*q**2*x - 12 
*int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*log(d**q*(e + f*x)**(p*q)*c 
)*x**2)/(4*log(d**q*(e + f*x)**(p*q)*c)*a*b*e + 4*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 + 4*a**2*e + 4*a**2*f*x + a*b*e*p*q + a*b*f*p*...