\(\int \frac {1}{(e+f x)^2 (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3)^{3/2}} \, dx\) [164]

Optimal result

Integrand size = 42, antiderivative size = 1153 \[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

1/3*2^(1/2)*(1080*(1-6*b)^(1/2)*e^3-108*(3-5*(1-6*b)^(1/2)-18*b)*e^2*f-54* 
(2+(1-6*b)^(1/2)-4*(3+4*(1-6*b)^(1/2))*b)*e*f^2-(89+19*(1-6*b)^(1/2)-6*(16 
9+24*(1-6*b)^(1/2))*b+2880*b^2)*f^3)*(1-(1-6*x)/(1-6*b)^(1/2))^3*(2+(1-6*x 
)/(1-6*b)^(1/2))/(6*e+f-(1-6*b)^(1/2)*f)^3/(6*e+f+2*(1-6*b)^(1/2)*f)^2/(-2 
*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)+1/3*2^(1/2)*(1-(1-6*x)/( 
1-6*b)^(1/2))*(2+(1-6*x)/(1-6*b)^(1/2))/(f-(6*e+f)/(1-6*b)^(1/2))/(-2*(1-6 
*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)/(f*x+e)-1/18*(1-6*b)^(1/2)*(3 
0*e+(5-23*(1-6*b)^(1/2))*f)*(1-(1-6*x)/(1-6*b)^(1/2))^2*(2+(1-6*x)/(1-6*b) 
^(1/2))*2^(1/2)/(6*e+f-(1-6*b)^(1/2)*f)^2/(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6 
*x)-(1-6*x)^3)^(3/2)/(f*x+e)-1/18*(1-6*b)*f*(180*e^2+6*(10-19*(1-6*b)^(1/2 
))*e*f-(89+19*(1-6*b)^(1/2)-564*b)*f^2)*(1-(1-6*x)/(1-6*b)^(1/2))^3*(2+(1- 
6*x)/(1-6*b)^(1/2))*2^(1/2)/(6*e+f-(1-6*b)^(1/2)*f)^3/(6*e+f+2*(1-6*b)^(1/ 
2)*f)/(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)/(f*x+e)-2/9*6^( 
1/2)*(1-6*b)^(1/2)*(3240*e^4+216*(10-7*(1-6*b)^(1/2))*e^3*f+54*(5-14*(1-6* 
b)^(1/2)+30*b)*e^2*f^2-6*(5-113*(1-6*b)^(1/2)-(90-804*(1-6*b)^(1/2))*b)*e* 
f^3+(197+127*(1-6*b)^(1/2)-3*(793+268*(1-6*b)^(1/2))*b+7272*b^2)*f^4)*(1-( 
1-6*x)/(1-6*b)^(1/2))^3*(2+(1-6*x)/(1-6*b)^(1/2))^(3/2)*arctanh(1/3*(2+(1- 
6*x)/(1-6*b)^(1/2))^(1/2)*3^(1/2))/(6*e+f-(1-6*b)^(1/2)*f)^4/(6*e+f+2*(1-6 
*b)^(1/2)*f)^2/(-2*(1-6*b)^(3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^(3/2)+108*2^ 
(1/2)*(1-6*b)^(9/4)*f^(7/2)*(6*e+f+(1-6*b)^(1/2)*f)*(1-(1-6*x)/(1-6*b)^...
 

Mathematica [F]

\[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx \] Input:

Integrate[1/((e + f*x)^2*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 10 
8*x^3)^(3/2)),x]
 

Output:

Integrate[1/((e + f*x)^2*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 10 
8*x^3)^(3/2)), x]
 

Rubi [A] (warning: unable to verify)

Time = 3.45 (sec) , antiderivative size = 1113, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2489, 27, 114, 27, 168, 27, 168, 27, 169, 27, 174, 73, 217, 218}

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.

\(\displaystyle \int \frac {1}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2} (e+f x)^2} \, dx\)

\(\Big \downarrow \) 2489

\(\displaystyle -\frac {7808611824626688 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \int -\frac {1}{7808611824626688 \sqrt {2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)^2}dx}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)^2}dx}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (\frac {\int -\frac {6 \left (\left (15 e-8 \sqrt {1-6 b} f-f\right ) (1-6 b)^2+21 f x (1-6 b)^2\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)^2}dx}{36 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {\int \frac {\left (15 e-8 \sqrt {1-6 b} f-f\right ) (1-6 b)^2+21 f x (1-6 b)^2}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)^2}dx}{6 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {\frac {\int -\frac {3 (1-6 b)^4 \left (270 e^2+3 \left (5-77 \sqrt {1-6 b}\right ) f e+\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2+15 f \left (30 e+\left (5-23 \sqrt {1-6 b}\right ) f\right ) x\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)^2}dx}{18 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {\left (5-23 \sqrt {1-6 b}\right ) f+30 e}{6 \sqrt {1-6 b} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}}{6 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {\sqrt {1-6 b} \int \frac {270 e^2+3 \left (5-77 \sqrt {1-6 b}\right ) f e+\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2+15 f \left (30 e+\left (5-23 \sqrt {1-6 b}\right ) f\right ) x}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)^2}dx}{6 \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {\left (5-23 \sqrt {1-6 b}\right ) f+30 e}{6 \sqrt {1-6 b} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}}{6 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {\sqrt {1-6 b} \left (\frac {\int \frac {54 (1-6 b)^2 \left (180 e^3+6 \left (10-19 \sqrt {1-6 b}\right ) f e^2+\left (-84 b-19 \sqrt {1-6 b}+19\right ) f^2 e+18 \left (\sqrt {1-6 b}+1\right ) (1-6 b) f^3+f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right ) x\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)}dx}{(1-6 b)^2 \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}-\frac {f \left (6 \left (10-19 \sqrt {1-6 b}\right ) e f-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2+180 e^2\right )}{(1-6 b)^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{6 \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {\left (5-23 \sqrt {1-6 b}\right ) f+30 e}{6 \sqrt {1-6 b} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}}{6 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {\sqrt {1-6 b} \left (\frac {54 \int \frac {180 e^3+6 \left (10-19 \sqrt {1-6 b}\right ) f e^2+\left (-84 b-19 \sqrt {1-6 b}+19\right ) f^2 e+18 \left (\sqrt {1-6 b}+1\right ) (1-6 b) f^3+f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right ) x}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} (e+f x)}dx}{\left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right )}-\frac {f \left (6 \left (10-19 \sqrt {1-6 b}\right ) e f-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2+180 e^2\right )}{(1-6 b)^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (-\sqrt {1-6 b} f+6 e+f\right ) \left (2 \sqrt {1-6 b} f+6 e+f\right ) (e+f x)}\right )}{6 \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {\left (5-23 \sqrt {1-6 b}\right ) f+30 e}{6 \sqrt {1-6 b} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}}{6 (1-6 b)^{7/2} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} \left (\left (1-\sqrt {1-6 b}\right ) f+6 e\right ) (e+f x)}\right )}{\left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {30 e+\left (5-23 \sqrt {1-6 b}\right ) f}{6 \sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}-\frac {\sqrt {1-6 b} \left (\frac {54 \left (\frac {1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3}{9 (1-6 b)^{5/2} \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}+\frac {\int -\frac {3 (1-6 b)^2 \left (1080 e^4+108 \left (5-3 \sqrt {1-6 b}\right ) f e^3-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e^2-\left (-24 \left (6-61 \sqrt {1-6 b}\right ) b-235 \sqrt {1-6 b}+19\right ) f^3 e+54 (1-6 b) \left (-6 b+\sqrt {1-6 b}+1\right ) f^4+f \left (1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3\right ) x\right )}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}dx}{9 (1-6 b)^{7/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )}\right )}{\left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right )}-\frac {f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right )}{(1-6 b)^2 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{6 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}}{6 (1-6 b)^{7/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{\left (108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {30 e+\left (5-23 \sqrt {1-6 b}\right ) f}{6 \sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}-\frac {\sqrt {1-6 b} \left (\frac {54 \left (\frac {1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3}{9 (1-6 b)^{5/2} \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\int \frac {1080 e^4+108 \left (5-3 \sqrt {1-6 b}\right ) f e^3-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e^2-\left (-24 \left (6-61 \sqrt {1-6 b}\right ) b-235 \sqrt {1-6 b}+19\right ) f^3 e+54 (1-6 b) \left (-6 b+\sqrt {1-6 b}+1\right ) f^4+f \left (1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3\right ) x}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}dx}{3 (1-6 b)^{3/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )}\right )}{\left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right )}-\frac {f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right )}{(1-6 b)^2 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{6 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}}{6 (1-6 b)^{7/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{\left (108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1\right )^{3/2}}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {30 e+\left (5-23 \sqrt {1-6 b}\right ) f}{6 \sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}-\frac {\sqrt {1-6 b} \left (\frac {54 \left (\frac {1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3}{9 (1-6 b)^{5/2} \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\frac {54 \left (6 \sqrt {1-6 b} e+\left (-6 b+\sqrt {1-6 b}+1\right ) f\right ) \int \frac {1}{\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}dx f^4}{6 e+\left (1-\sqrt {1-6 b}\right ) f}+\frac {2 \left (3240 e^4+216 \left (10-7 \sqrt {1-6 b}\right ) f e^3+54 \left (30 b-14 \sqrt {1-6 b}+5\right ) f^2 e^2-6 \left (-\left (\left (90-804 \sqrt {1-6 b}\right ) b\right )-113 \sqrt {1-6 b}+5\right ) f^3 e+\left (7272 b^2-3 \left (268 \sqrt {1-6 b}+793\right ) b+127 \sqrt {1-6 b}+197\right ) f^4\right ) \int \frac {1}{\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}dx}{6 e-\sqrt {1-6 b} f+f}}{3 (1-6 b)^{3/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )}\right )}{\left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right )}-\frac {f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right )}{(1-6 b)^2 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{6 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}}{6 (1-6 b)^{7/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{\left (108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1\right )^{3/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {30 e+\left (5-23 \sqrt {1-6 b}\right ) f}{6 \sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}-\frac {\sqrt {1-6 b} \left (\frac {54 \left (\frac {1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3}{9 (1-6 b)^{5/2} \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\frac {18 \left (6 \sqrt {1-6 b} e+\left (-6 b+\sqrt {1-6 b}+1\right ) f\right ) \int \frac {1}{\frac {1}{6} \left (6 e+2 \sqrt {1-6 b} f+f\right )+\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )}{6 (1-6 b)}}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} f^4}{(1-6 b) \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}+\frac {2 \left (3240 e^4+216 \left (10-7 \sqrt {1-6 b}\right ) f e^3+54 \left (30 b-14 \sqrt {1-6 b}+5\right ) f^2 e^2-6 \left (-\left (\left (90-804 \sqrt {1-6 b}\right ) b\right )-113 \sqrt {1-6 b}+5\right ) f^3 e+\left (7272 b^2-3 \left (268 \sqrt {1-6 b}+793\right ) b+127 \sqrt {1-6 b}+197\right ) f^4\right ) \int \frac {1}{-3 (1-6 b)^{3/2}+\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 x (1-6 b)}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{3 (1-6 b) \left (6 e-\sqrt {1-6 b} f+f\right )}}{3 (1-6 b)^{3/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )}\right )}{\left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right )}-\frac {f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right )}{(1-6 b)^2 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{6 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}}{6 (1-6 b)^{7/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{\left (108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1\right )^{3/2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {30 e+\left (5-23 \sqrt {1-6 b}\right ) f}{6 \sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}-\frac {\sqrt {1-6 b} \left (\frac {54 \left (\frac {1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3}{9 (1-6 b)^{5/2} \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\frac {18 f^4 \left (6 \sqrt {1-6 b} e+\left (-6 b+\sqrt {1-6 b}+1\right ) f\right ) \int \frac {1}{\frac {1}{6} \left (6 e+2 \sqrt {1-6 b} f+f\right )+\frac {f \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )}{6 (1-6 b)}}d\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{(1-6 b) \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {2 \left (3240 e^4+216 \left (10-7 \sqrt {1-6 b}\right ) f e^3+54 \left (30 b-14 \sqrt {1-6 b}+5\right ) f^2 e^2-6 \left (-\left (\left (90-804 \sqrt {1-6 b}\right ) b\right )-113 \sqrt {1-6 b}+5\right ) f^3 e+\left (7272 b^2-3 \left (268 \sqrt {1-6 b}+793\right ) b+127 \sqrt {1-6 b}+197\right ) f^4\right ) \arctan \left (\frac {\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {3} (1-6 b)^{3/4}}\right )}{3 \sqrt {3} (1-6 b)^{7/4} \left (6 e-\sqrt {1-6 b} f+f\right )}}{3 (1-6 b)^{3/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )}\right )}{\left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right )}-\frac {f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right )}{(1-6 b)^2 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{6 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}}{6 (1-6 b)^{7/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{\left (108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1\right )^{3/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^3 \left (6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{3/2} \left (-\frac {-\frac {30 e+\left (5-23 \sqrt {1-6 b}\right ) f}{6 \sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}-\frac {\sqrt {1-6 b} \left (\frac {54 \left (\frac {1080 e^3+108 \left (5-3 \sqrt {1-6 b}\right ) f e^2-54 \left (-16 b+2 \sqrt {1-6 b}+1\right ) f^2 e-\left (-48 \left (10 \sqrt {1-6 b}+3\right ) b+89 \sqrt {1-6 b}+19\right ) f^3}{9 (1-6 b)^{5/2} \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}-\frac {\frac {108 f^{7/2} \left (6 \sqrt {1-6 b} e+\left (-6 b+\sqrt {1-6 b}+1\right ) f\right ) \arctan \left (\frac {\sqrt {f} \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {1-6 b} \sqrt {6 e+2 \sqrt {1-6 b} f+f}}\right )}{\sqrt {1-6 b} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \sqrt {6 e+2 \sqrt {1-6 b} f+f}}-\frac {2 \left (3240 e^4+216 \left (10-7 \sqrt {1-6 b}\right ) f e^3+54 \left (30 b-14 \sqrt {1-6 b}+5\right ) f^2 e^2-6 \left (-\left (\left (90-804 \sqrt {1-6 b}\right ) b\right )-113 \sqrt {1-6 b}+5\right ) f^3 e+\left (7272 b^2-3 \left (268 \sqrt {1-6 b}+793\right ) b+127 \sqrt {1-6 b}+197\right ) f^4\right ) \arctan \left (\frac {\sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)}}{\sqrt {3} (1-6 b)^{3/4}}\right )}{3 \sqrt {3} (1-6 b)^{7/4} \left (6 e-\sqrt {1-6 b} f+f\right )}}{3 (1-6 b)^{3/2} \left (6 e+2 \sqrt {1-6 b} f+f\right )}\right )}{\left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right )}-\frac {f \left (180 e^2+6 \left (10-19 \sqrt {1-6 b}\right ) f e-\left (-564 b+19 \sqrt {1-6 b}+89\right ) f^2\right )}{(1-6 b)^2 \left (6 e-\sqrt {1-6 b} f+f\right ) \left (6 e+2 \sqrt {1-6 b} f+f\right ) \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{6 \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}}{6 (1-6 b)^{7/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right )}-\frac {1}{6 (1-6 b)^{5/2} \left (6 e+\left (1-\sqrt {1-6 b}\right ) f\right ) \left (\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x\right )^2 \sqrt {6 (1-6 b) x-\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)} (e+f x)}\right )}{\left (108 x^3-54 x^2+54 b x-(1-6 b)^{3/2}-9 b+1\right )^{3/2}}\)

Input:

Int[1/((e + f*x)^2*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3) 
^(3/2)),x]
 

Output:

(((1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)^3*(-((1 + 2*Sqrt[1 - 6*b] 
)*(1 - 6*b)) + 6*(1 - 6*b)*x)^(3/2)*(-1/6*1/((1 - 6*b)^(5/2)*(6*e + (1 - S 
qrt[1 - 6*b])*f)*((1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)^2*Sqrt[-( 
(1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x]*(e + f*x)) - (-1/6*(30*e 
 + (5 - 23*Sqrt[1 - 6*b])*f)/(Sqrt[1 - 6*b]*(6*e + (1 - Sqrt[1 - 6*b])*f)* 
((1 - Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)*Sqrt[-((1 + 2*Sqrt[1 - 6*b 
])*(1 - 6*b)) + 6*(1 - 6*b)*x]*(e + f*x)) - (Sqrt[1 - 6*b]*(-((f*(180*e^2 
+ 6*(10 - 19*Sqrt[1 - 6*b])*e*f - (89 + 19*Sqrt[1 - 6*b] - 564*b)*f^2))/(( 
1 - 6*b)^2*(6*e + f - Sqrt[1 - 6*b]*f)*(6*e + f + 2*Sqrt[1 - 6*b]*f)*Sqrt[ 
-((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b)) + 6*(1 - 6*b)*x]*(e + f*x))) + (54*((10 
80*e^3 + 108*(5 - 3*Sqrt[1 - 6*b])*e^2*f - 54*(1 + 2*Sqrt[1 - 6*b] - 16*b) 
*e*f^2 - (19 + 89*Sqrt[1 - 6*b] - 48*(3 + 10*Sqrt[1 - 6*b])*b)*f^3)/(9*(1 
- 6*b)^(5/2)*(6*e + f + 2*Sqrt[1 - 6*b]*f)*Sqrt[-((1 + 2*Sqrt[1 - 6*b])*(1 
 - 6*b)) + 6*(1 - 6*b)*x]) - ((-2*(3240*e^4 + 216*(10 - 7*Sqrt[1 - 6*b])*e 
^3*f + 54*(5 - 14*Sqrt[1 - 6*b] + 30*b)*e^2*f^2 - 6*(5 - 113*Sqrt[1 - 6*b] 
 - (90 - 804*Sqrt[1 - 6*b])*b)*e*f^3 + (197 + 127*Sqrt[1 - 6*b] - 3*(793 + 
 268*Sqrt[1 - 6*b])*b + 7272*b^2)*f^4)*ArcTan[Sqrt[-((1 + 2*Sqrt[1 - 6*b]) 
*(1 - 6*b)) + 6*(1 - 6*b)*x]/(Sqrt[3]*(1 - 6*b)^(3/4))])/(3*Sqrt[3]*(1 - 6 
*b)^(7/4)*(6*e + f - Sqrt[1 - 6*b]*f)) + (108*f^(7/2)*(6*Sqrt[1 - 6*b]*e + 
 (1 + Sqrt[1 - 6*b] - 6*b)*f)*ArcTan[(Sqrt[f]*Sqrt[-((1 + 2*Sqrt[1 - 6*...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 2489

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*( 
x_)^3)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2 + d*x^3)^p/((c^3 - 4*b*c*d 
+ 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p)) 
 Int[(e + f*x)^m*(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a 
*d + 2*(c^2 - 3*b*d)*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] 
 && NeQ[c^2 - 3*b*d, 0] && EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 2 
7*a^2*d^2, 0] &&  !IntegerQ[p]
 

Maple [F]

\[\int \frac {1}{\left (f x +e \right )^{2} \left (1-\left (1-6 b \right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}\right )^{\frac {3}{2}}}d x\]

Input:

int(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x)
 

Output:

int(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19567 vs. \(2 (990) = 1980\).

Time = 129.87 (sec) , antiderivative size = 81003, normalized size of antiderivative = 70.25 \[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x, 
 algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\int \frac {1}{\left (e + f x\right )^{2} \left (54 b x + 6 b \sqrt {1 - 6 b} - 9 b + 108 x^{3} - 54 x^{2} - \sqrt {1 - 6 b} + 1\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(f*x+e)**2/(1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)**(3/ 
2),x)
 

Output:

Integral(1/((e + f*x)**2*(54*b*x + 6*b*sqrt(1 - 6*b) - 9*b + 108*x**3 - 54 
*x**2 - sqrt(1 - 6*b) + 1)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1\right )}^{\frac {3}{2}} {\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x, 
 algorithm="maxima")
 

Output:

integrate(1/((108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1)^(3/2 
)*(f*x + e)^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x, 
 algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\text {Hanged} \] Input:

int(1/((e + f*x)^2*(54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1) 
^(3/2)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{(e+f x)^2 \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^{3/2}} \, dx=\int \frac {1}{\left (f x +e \right )^{2} \left (1-\left (-6 b +1\right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}\right )^{\frac {3}{2}}}d x \] Input:

int(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x)
 

Output:

int(1/(f*x+e)^2/(1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^(3/2),x)