3.2.29 \(\int \frac {(f+g x)^2}{(a+b \log (c (d+e x)^n))^{3/2}} \, dx\) [129]

3.2.29.1 Optimal result

Integrand size = 26, antiderivative size = 325 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {4 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}+\frac {2 e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^3 n^{3/2}}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

output

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

3.2.29.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(828\) vs. \(2(325)=650\).

Time = 0.70 (sec) , antiderivative size = 828, normalized size of antiderivative = 2.55 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 \left (-\sqrt {b} d e^2 f^2 \sqrt {n}-\sqrt {b} e^3 f^2 \sqrt {n} x-2 \sqrt {b} d e^2 f g \sqrt {n} x-2 \sqrt {b} e^3 f g \sqrt {n} x^2-\sqrt {b} d e^2 g^2 \sqrt {n} x^2-\sqrt {b} e^3 g^2 \sqrt {n} x^3-4 d e e^{-\frac {a}{b n}} f g \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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+d^2 e^{-\frac {a}{b n}} g^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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+2 e e^{-\frac {2 a}{b n}} f g \sqrt {2 \pi } (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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}-2 d e^{-\frac {2 a}{b n}} g^2 \sqrt {2 \pi } (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 ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+\sqrt {b} e^2 e^{-\frac {a}{b n}} f^2 \sqrt {n} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}+2 \sqrt {b} d e e^{-\frac {a}{b n}} f g \sqrt {n} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )}{b^{3/2} e^3 n^{3/2} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

input

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

output

(2*(-(Sqrt[b]*d*e^2*f^2*Sqrt[n]) - Sqrt[b]*e^3*f^2*Sqrt[n]*x - 2*Sqrt[b]*d 
*e^2*f*g*Sqrt[n]*x - 2*Sqrt[b]*e^3*f*g*Sqrt[n]*x^2 - Sqrt[b]*d*e^2*g^2*Sqr 
t[n]*x^2 - Sqrt[b]*e^3*g^2*Sqrt[n]*x^3 - (4*d*e*f*g*Sqrt[Pi]*(d + e*x)*Erf 
i[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + 
e*x)^n]])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (d^2*g^2*Sqrt[Pi]*(d + e* 
x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c 
*(d + e*x)^n]])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (2*e*f*g*Sqrt[2*Pi] 
*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n 
])]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^((2*a)/(b*n))*(c*(d + e*x)^n)^(2/n) 
) - (2*d*g^2*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e* 
x)^n]])/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(E^((2*a)/(b*n) 
)*(c*(d + e*x)^n)^(2/n)) + (g^2*Sqrt[3*Pi]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[ 
a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n 
]])/(E^((3*a)/(b*n))*(c*(d + e*x)^n)^(3/n)) + (Sqrt[b]*e^2*f^2*Sqrt[n]*(d 
+ e*x)*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*Sqrt[-((a + b*Log[c 
*(d + e*x)^n])/(b*n))])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)) + (2*Sqrt[b]* 
d*e*f*g*Sqrt[n]*(d + e*x)*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]* 
Sqrt[-((a + b*Log[c*(d + e*x)^n])/(b*n))])/(E^(a/(b*n))*(c*(d + e*x)^n)^n^ 
(-1))))/(b^(3/2)*e^3*n^(3/2)*Sqrt[a + b*Log[c*(d + e*x)^n]])
 

3.2.29.3 Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.64, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {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.

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

\(\Big \downarrow \) 2847

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

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

input

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

output

(-4*(e*f - d*g)*(((e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d 
+ e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^2*E^(a/(b*n))*Sqrt[n]*(c*(d + e* 
x)^n)^n^(-1)) + (g*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*( 
d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^2*E^((2*a)/(b*n))*Sqrt[n]*(c* 
(d + e*x)^n)^(2/n))))/(b*e*n) + (6*(((e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi 
[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^(a/(b*n 
))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2 
*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b 
]*e^3*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(2/n)) + (g^2*Sqrt[Pi/3]*(d 
+ e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]) 
/(Sqrt[b]*e^3*E^((3*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(3/n))))/(b*n) - (2* 
(d + e*x)*(f + g*x)^2)/(b*e*n*Sqrt[a + b*Log[c*(d + e*x)^n]])
 

3.2.29.3.1 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]
 

3.2.29.4 Maple [F]

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

input

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

output

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

3.2.29.5 Fricas [F(-2)]

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

input

integrate((g*x+f)^2/(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)
 

3.2.29.6 Sympy [F]

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

input

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

output

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

3.2.29.7 Maxima [F]

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

input

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

output

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

3.2.29.8 Giac [F]

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

input

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

output

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

3.2.29.9 Mupad [F(-1)]

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

input

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

output

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