3.2.26 \(\int (d+e x)^3 \sqrt {a+b \log (c x^n)} \, dx\) [126]

3.2.26.1 Optimal result

Integrand size = 22, antiderivative size = 402 \[ \int (d+e x)^3 \sqrt {a+b \log \left (c x^n\right )} \, dx=-\frac {1}{2} \sqrt {b} d^3 e^{-\frac {a}{b n}} \sqrt {n} \sqrt {\pi } x \left (c x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{16} \sqrt {b} e^3 e^{-\frac {4 a}{b n}} \sqrt {n} \sqrt {\pi } x^4 \left (c x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {3}{4} \sqrt {b} d^2 e e^{-\frac {2 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-\frac {1}{2} \sqrt {b} d e^2 e^{-\frac {3 a}{b n}} \sqrt {n} \sqrt {\frac {\pi }{3}} x^3 \left (c x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+d^3 x \sqrt {a+b \log \left (c x^n\right )}+\frac {3}{2} d^2 e x^2 \sqrt {a+b \log \left (c x^n\right )}+d e^2 x^3 \sqrt {a+b \log \left (c x^n\right )}+\frac {1}{4} e^3 x^4 \sqrt {a+b \log \left (c x^n\right )} \]

output

-1/6*d*e^2*x^3*erfi(3^(1/2)*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2) 
*n^(1/2)*3^(1/2)*Pi^(1/2)/exp(3*a/b/n)/((c*x^n)^(3/n))-3/8*d^2*e*x^2*erfi( 
2^(1/2)*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*2^(1/2)*Pi^ 
(1/2)/exp(2*a/b/n)/((c*x^n)^(2/n))-1/2*d^3*x*erfi((a+b*ln(c*x^n))^(1/2)/b^ 
(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*Pi^(1/2)/exp(a/b/n)/((c*x^n)^(1/n))-1/16*e^ 
3*x^4*erfi(2*(a+b*ln(c*x^n))^(1/2)/b^(1/2)/n^(1/2))*b^(1/2)*n^(1/2)*Pi^(1/ 
2)/exp(4*a/b/n)/((c*x^n)^(4/n))+d^3*x*(a+b*ln(c*x^n))^(1/2)+3/2*d^2*e*x^2* 
(a+b*ln(c*x^n))^(1/2)+d*e^2*x^3*(a+b*ln(c*x^n))^(1/2)+1/4*e^3*x^4*(a+b*ln( 
c*x^n))^(1/2)
 

3.2.26.2 Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.91 \[ \int (d+e x)^3 \sqrt {a+b \log \left (c x^n\right )} \, dx=\frac {1}{48} e^{-\frac {4 a}{b n}} x \left (c x^n\right )^{-4/n} \left (-24 \sqrt {b} d^3 e^{\frac {3 a}{b n}} \sqrt {n} \sqrt {\pi } \left (c x^n\right )^{3/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-3 \sqrt {b} e^3 \sqrt {n} \sqrt {\pi } x^3 \text {erfi}\left (\frac {2 \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 e^{\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \left (-9 \sqrt {b} d^2 e e^{\frac {a}{b n}} \sqrt {n} \sqrt {2 \pi } x \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )-4 \sqrt {b} d e^2 \sqrt {n} \sqrt {3 \pi } x^2 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c x^n\right )}}{\sqrt {b} \sqrt {n}}\right )+6 e^{\frac {3 a}{b n}} \left (c x^n\right )^{3/n} \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \sqrt {a+b \log \left (c x^n\right )}\right )\right ) \]

input

Integrate[(d + e*x)^3*Sqrt[a + b*Log[c*x^n]],x]
 

output

(x*(-24*Sqrt[b]*d^3*E^((3*a)/(b*n))*Sqrt[n]*Sqrt[Pi]*(c*x^n)^(3/n)*Erfi[Sq 
rt[a + b*Log[c*x^n]]/(Sqrt[b]*Sqrt[n])] - 3*Sqrt[b]*e^3*Sqrt[n]*Sqrt[Pi]*x 
^3*Erfi[(2*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])] + 2*E^(a/(b*n))*(c*x 
^n)^n^(-1)*(-9*Sqrt[b]*d^2*e*E^(a/(b*n))*Sqrt[n]*Sqrt[2*Pi]*x*(c*x^n)^n^(- 
1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]*Sqrt[n])] - 4*Sqrt[b]*d* 
e^2*Sqrt[n]*Sqrt[3*Pi]*x^2*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*x^n]])/(Sqrt[b]* 
Sqrt[n])] + 6*E^((3*a)/(b*n))*(c*x^n)^(3/n)*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x 
^2 + e^3*x^3)*Sqrt[a + b*Log[c*x^n]])))/(48*E^((4*a)/(b*n))*(c*x^n)^(4/n))
 

3.2.26.3 Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2767, 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.

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

\(\Big \downarrow \) 2767

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

\(\Big \downarrow \) 2009

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

input

Int[(d + e*x)^3*Sqrt[a + b*Log[c*x^n]],x]
 

output

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

3.2.26.3.1 Defintions of rubi rules used

rule 2009

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

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

3.2.26.4 Maple [F]

\[\int \left (e x +d \right )^{3} \sqrt {a +b \ln \left (c \,x^{n}\right )}d x\]

input

int((e*x+d)^3*(a+b*ln(c*x^n))^(1/2),x)
 

output

int((e*x+d)^3*(a+b*ln(c*x^n))^(1/2),x)
 

3.2.26.5 Fricas [F(-2)]

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

input

integrate((e*x+d)^3*(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")
 

output

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

3.2.26.6 Sympy [F]

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

input

integrate((e*x+d)**3*(a+b*ln(c*x**n))**(1/2),x)
 

output

Integral(sqrt(a + b*log(c*x**n))*(d + e*x)**3, x)
 

3.2.26.7 Maxima [F]

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

input

integrate((e*x+d)^3*(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")
 

output

integrate((e*x + d)^3*sqrt(b*log(c*x^n) + a), x)
 

3.2.26.8 Giac [F]

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

input

integrate((e*x+d)^3*(a+b*log(c*x^n))^(1/2),x, algorithm="giac")
 

output

integrate((e*x + d)^3*sqrt(b*log(c*x^n) + a), x)
 

3.2.26.9 Mupad [F(-1)]

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

input

int((a + b*log(c*x^n))^(1/2)*(d + e*x)^3,x)
 

output

int((a + b*log(c*x^n))^(1/2)*(d + e*x)^3, x)