\(\int \frac {(g+h x)^2}{(a+b \log (c (d (e+f x)^p)^q))^{3/2}} \, dx\) [493]

Optimal result

Integrand size = 30, antiderivative size = 404 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b p q}} (f g-e h)^2 \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}+\frac {2 e^{-\frac {3 a}{b p q}} h^2 \sqrt {3 \pi } (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^3 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1040\) vs. \(2(404)=808\).

Time = 2.52 (sec) , antiderivative size = 1040, normalized size of antiderivative = 2.57 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*(-(Sqrt[b]*e*f^2*g^2*Sqrt[p]*Sqrt[q]) - Sqrt[b]*f^3*g^2*Sqrt[p]*Sqrt[q] 
*x - 2*Sqrt[b]*e*f^2*g*h*Sqrt[p]*Sqrt[q]*x - 2*Sqrt[b]*f^3*g*h*Sqrt[p]*Sqr 
t[q]*x^2 - Sqrt[b]*e*f^2*h^2*Sqrt[p]*Sqrt[q]*x^2 - Sqrt[b]*f^3*h^2*Sqrt[p] 
*Sqrt[q]*x^3 - (4*e*f*g*h*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + 
 f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q 
]])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (e^2*h^2*Sqrt[Pi]*(e 
 + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q] 
)]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p) 
^q)^(1/(p*q))) + (2*f*g*h*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b* 
Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d* 
(e + f*x)^p)^q]])/(E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - (2 
*e*h^2*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^ 
p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/( 
E^((2*a)/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (h^2*Sqrt[3*Pi]*(e + 
f*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p 
]*Sqrt[q])]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(E^((3*a)/(b*p*q))*(c*(d 
*(e + f*x)^p)^q)^(3/(p*q))) + (Sqrt[b]*f^2*g^2*Sqrt[p]*Sqrt[q]*(e + f*x)*G 
amma[1/2, -((a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q))]*Sqrt[-((a + b*Log[c 
*(d*(e + f*x)^p)^q])/(b*p*q))])/(E^(a/(b*p*q))*(c*(d*(e + f*x)^p)^q)^(1/(p 
*q))) + (2*Sqrt[b]*e*f*g*h*Sqrt[p]*Sqrt[q]*(e + f*x)*Gamma[1/2, -((a + ...
 

Rubi [A] (verified)

Time = 5.00 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.65, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2895, 2847, 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.

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

\(\Big \downarrow \) 2895

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

\(\Big \downarrow \) 2847

\(\displaystyle -\frac {4 (f g-e h) \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\)

\(\Big \downarrow \) 2848

\(\displaystyle \frac {6 \int \left (\frac {(f g-e h)^2}{f^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {2 h (e+f x) (f g-e h)}{f^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {h^2 (e+f x)^2}{f^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )dx}{b p q}-\frac {4 (f g-e h) \int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {h (e+f x)}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )dx}{b f p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 \left (\frac {\sqrt {2 \pi } h (e+f x)^2 e^{-\frac {2 a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^3 \sqrt {p} \sqrt {q}}+\frac {\sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^3 \sqrt {p} \sqrt {q}}+\frac {\sqrt {\frac {\pi }{3}} h^2 (e+f x)^3 e^{-\frac {3 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^3 \sqrt {p} \sqrt {q}}\right )}{b p q}-\frac {4 (f g-e h) \left (\frac {\sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {\sqrt {\frac {\pi }{2}} h (e+f x)^2 e^{-\frac {2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}\right )}{b f p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\)

Input:

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

Output:

(-4*(f*g - e*h)*(((f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d* 
(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a/(b*p*q))*f^2*Sq 
rt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*Sqrt[Pi/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])])/(Sqrt[b]*E^((2*a)/(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q 
)^(2/(p*q)))))/(b*f*p*q) + (6*(((f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt 
[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a 
/(b*p*q))*f^3*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*(f*g - 
 e*h)*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p 
)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/(b*p*q))*f^3*Sqrt[p]* 
Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (h^2*Sqrt[Pi/3]*(e + f*x)^3*Erf 
i[(Sqrt[3]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q]) 
])/(Sqrt[b]*E^((3*a)/(b*p*q))*f^3*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(3 
/(p*q)))))/(b*p*q) - (2*(e + f*x)*(g + h*x)^2)/(b*f*p*q*Sqrt[a + b*Log[c*( 
d*(e + f*x)^p)^q]])
 

Defintions of rubi rules used

rule 2009

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

rule 2847

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

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*e*g**2 + int((sqrt(log(d**q 
*(e + f*x)**(p*q)*c)*b + a)*x**3)/(log(d**q*(e + f*x)**(p*q)*c)**2*b**2*e 
+ log(d**q*(e + f*x)**(p*q)*c)**2*b**2*f*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 + a**2*e + a**2*f*x),x)*l 
og(d**q*(e + f*x)**(p*q)*c)*b**2*f**2*h**2*p*q + int((sqrt(log(d**q*(e + f 
*x)**(p*q)*c)*b + a)*x**3)/(log(d**q*(e + f*x)**(p*q)*c)**2*b**2*e + log(d 
**q*(e + f*x)**(p*q)*c)**2*b**2*f*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 + a**2*e + a**2*f*x),x)*a*b*f**2 
*h**2*p*q + int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x**2)/(log(d**q* 
(e + f*x)**(p*q)*c)**2*b**2*e + log(d**q*(e + f*x)**(p*q)*c)**2*b**2*f*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 + a**2*e + a**2*f*x),x)*log(d**q*(e + f*x)**(p*q)*c)*b**2*e*f*h**2*p* 
q + 2*int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x**2)/(log(d**q*(e + f 
*x)**(p*q)*c)**2*b**2*e + log(d**q*(e + f*x)**(p*q)*c)**2*b**2*f*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 + 
 a**2*e + a**2*f*x),x)*log(d**q*(e + f*x)**(p*q)*c)*b**2*f**2*g*h*p*q + in 
t((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x**2)/(log(d**q*(e + f*x)**(p* 
q)*c)**2*b**2*e + log(d**q*(e + f*x)**(p*q)*c)**2*b**2*f*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 + a**2*e 
+ a**2*f*x),x)*a*b*e*f*h**2*p*q + 2*int((sqrt(log(d**q*(e + f*x)**(p*q)...