Integrand size = 38, antiderivative size = 106 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=\frac {54 \left (2 e^2+e f+b f^2\right ) x}{f^3}-\frac {27 (2 e+f) x^2}{f^2}+\frac {36 x^3}{f}-\frac {\left (108 e^3+54 e^2 f+54 b e f^2-\left (1-\sqrt {1-6 b}-\left (9-6 \sqrt {1-6 b}\right ) b\right ) f^3\right ) \log (e+f x)}{f^4} \] Output:
54*(b*f^2+2*e^2+e*f)*x/f^3-27*(2*e+f)*x^2/f^2+36*x^3/f-(108*e^3+54*e^2*f+5 4*b*e*f^2-(1-(1-6*b)^(1/2)-(9-6*(1-6*b)^(1/2))*b)*f^3)*ln(f*x+e)/f^4
Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=\frac {9 f x \left (12 e^2-6 e f (-1+x)+f^2 (6 b+x (-3+4 x))\right )-\left (108 e^3+54 e^2 f+54 b e f^2+\left (-1+\sqrt {1-6 b}+9 b-6 \sqrt {1-6 b} b\right ) f^3\right ) \log (e+f x)}{f^4} \] Input:
Integrate[(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)/(e + f*x ),x]
Output:
(9*f*x*(12*e^2 - 6*e*f*(-1 + x) + f^2*(6*b + x*(-3 + 4*x))) - (108*e^3 + 5 4*e^2*f + 54*b*e*f^2 + (-1 + Sqrt[1 - 6*b] + 9*b - 6*Sqrt[1 - 6*b]*b)*f^3) *Log[e + f*x])/f^4
Time = 0.51 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2389, 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.
\(\displaystyle \int \frac {54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1}{e+f x} \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {54 \left (b f^2+2 e^2+e f\right )}{f^3}+\frac {-54 b e f^2-(1-6 b)^{3/2} f^3-9 b f^3-108 e^3-54 e^2 f+f^3}{f^3 (e+f x)}-\frac {54 x (2 e+f)}{f^2}+\frac {108 x^2}{f}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {54 x \left (b f^2+2 e^2+e f\right )}{f^3}-\frac {\left (54 b e f^2+(1-6 b)^{3/2} f^3+9 b f^3+108 e^3+54 e^2 f-f^3\right ) \log (e+f x)}{f^4}-\frac {27 x^2 (2 e+f)}{f^2}+\frac {36 x^3}{f}\) |
Input:
Int[(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)/(e + f*x),x]
Output:
(54*(2*e^2 + e*f + b*f^2)*x)/f^3 - (27*(2*e + f)*x^2)/f^2 + (36*x^3)/f - ( (108*e^3 + 54*e^2*f + 54*b*e*f^2 - f^3 + (1 - 6*b)^(3/2)*f^3 + 9*b*f^3)*Lo g[e + f*x])/f^4
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {36 f^{2} x^{3}+54 b x \,f^{2}-54 e f \,x^{2}-27 f^{2} x^{2}+108 e^{2} x +54 e f x}{f^{3}}+\frac {\left (-\left (1-6 b \right )^{\frac {3}{2}} f^{3}-54 b e \,f^{2}-9 b \,f^{3}-108 e^{3}-54 e^{2} f +f^{3}\right ) \ln \left (f x +e \right )}{f^{4}}\) | \(97\) |
norman | \(\frac {36 x^{3}}{f}-\frac {27 \left (2 e +f \right ) x^{2}}{f^{2}}+\frac {54 \left (b \,f^{2}+2 e^{2}+e f \right ) x}{f^{3}}-\frac {\left (-6 \sqrt {1-6 b}\, f^{3} b +54 b e \,f^{2}+9 b \,f^{3}+\sqrt {1-6 b}\, f^{3}+108 e^{3}+54 e^{2} f -f^{3}\right ) \ln \left (f x +e \right )}{f^{4}}\) | \(108\) |
parallelrisch | \(\frac {6 \ln \left (f x +e \right ) \sqrt {1-6 b}\, b \,f^{3}+36 x^{3} f^{3}-\ln \left (f x +e \right ) \sqrt {1-6 b}\, f^{3}-54 \ln \left (f x +e \right ) b e \,f^{2}-9 \ln \left (f x +e \right ) b \,f^{3}-54 x^{2} f^{2} e -27 x^{2} f^{3}+54 x b \,f^{3}-108 \ln \left (f x +e \right ) e^{3}-54 \ln \left (f x +e \right ) e^{2} f +\ln \left (f x +e \right ) f^{3}+108 x \,e^{2} f +54 x e \,f^{2}}{f^{4}}\) | \(147\) |
Input:
int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)/(f*x+e),x,method=_RETURNVE RBOSE)
Output:
54/f^3*(2/3*f^2*x^3+b*x*f^2-e*f*x^2-1/2*f^2*x^2+2*e^2*x+e*f*x)+1/f^4*(-(1- 6*b)^(3/2)*f^3-54*b*e*f^2-9*b*f^3-108*e^3-54*e^2*f+f^3)*ln(f*x+e)
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=\frac {36 \, f^{3} x^{3} + {\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} f^{3} \log \left (f x + e\right ) - 27 \, {\left (2 \, e f^{2} + f^{3}\right )} x^{2} + 54 \, {\left (b f^{3} + 2 \, e^{2} f + e f^{2}\right )} x - {\left (54 \, b e f^{2} + {\left (9 \, b - 1\right )} f^{3} + 108 \, e^{3} + 54 \, e^{2} f\right )} \log \left (f x + e\right )}{f^{4}} \] Input:
integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)/(f*x+e),x, algorithm ="fricas")
Output:
(36*f^3*x^3 + (6*b - 1)*sqrt(-6*b + 1)*f^3*log(f*x + e) - 27*(2*e*f^2 + f^ 3)*x^2 + 54*(b*f^3 + 2*e^2*f + e*f^2)*x - (54*b*e*f^2 + (9*b - 1)*f^3 + 10 8*e^3 + 54*e^2*f)*log(f*x + e))/f^4
Time = 0.55 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=x^{2} \left (- \frac {54 e}{f^{2}} - \frac {27}{f}\right ) + x \left (\frac {54 b}{f} + \frac {108 e^{2}}{f^{3}} + \frac {54 e}{f^{2}}\right ) + \left (- \frac {54 b e}{f^{2}} + \frac {6 b \sqrt {1 - 6 b}}{f} - \frac {9 b}{f} - \frac {108 e^{3}}{f^{4}} - \frac {54 e^{2}}{f^{3}} - \frac {\sqrt {1 - 6 b}}{f} + \frac {1}{f}\right ) \log {\left (e + f x \right )} + \frac {36 x^{3}}{f} \] Input:
integrate((1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)/(f*x+e),x)
Output:
x**2*(-54*e/f**2 - 27/f) + x*(54*b/f + 108*e**2/f**3 + 54*e/f**2) + (-54*b *e/f**2 + 6*b*sqrt(1 - 6*b)/f - 9*b/f - 108*e**3/f**4 - 54*e**2/f**3 - sqr t(1 - 6*b)/f + 1/f)*log(e + f*x) + 36*x**3/f
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=\frac {9 \, {\left (4 \, f^{2} x^{3} - 3 \, {\left (2 \, e f + f^{2}\right )} x^{2} + 6 \, {\left (b f^{2} + 2 \, e^{2} + e f\right )} x\right )}}{f^{3}} - \frac {{\left (54 \, b e f^{2} + {\left ({\left (-6 \, b + 1\right )}^{\frac {3}{2}} + 9 \, b - 1\right )} f^{3} + 108 \, e^{3} + 54 \, e^{2} f\right )} \log \left (f x + e\right )}{f^{4}} \] Input:
integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)/(f*x+e),x, algorithm ="maxima")
Output:
9*(4*f^2*x^3 - 3*(2*e*f + f^2)*x^2 + 6*(b*f^2 + 2*e^2 + e*f)*x)/f^3 - (54* b*e*f^2 + ((-6*b + 1)^(3/2) + 9*b - 1)*f^3 + 108*e^3 + 54*e^2*f)*log(f*x + e)/f^4
Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=\frac {9 \, {\left (4 \, f^{2} x^{3} + 6 \, b f^{2} x - 6 \, e f x^{2} - 3 \, f^{2} x^{2} + 12 \, e^{2} x + 6 \, e f x\right )}}{f^{3}} - \frac {{\left (54 \, b e f^{2} - {\left ({\left (6 \, b - 1\right )} \sqrt {-6 \, b + 1} - 9 \, b + 1\right )} f^{3} + 108 \, e^{3} + 54 \, e^{2} f\right )} \log \left ({\left | f x + e \right |}\right )}{f^{4}} \] Input:
integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)/(f*x+e),x, algorithm ="giac")
Output:
9*(4*f^2*x^3 + 6*b*f^2*x - 6*e*f*x^2 - 3*f^2*x^2 + 12*e^2*x + 6*e*f*x)/f^3 - (54*b*e*f^2 - ((6*b - 1)*sqrt(-6*b + 1) - 9*b + 1)*f^3 + 108*e^3 + 54*e ^2*f)*log(abs(f*x + e))/f^4
Time = 11.97 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=x\,\left (\frac {54\,b}{f}+\frac {e\,\left (\frac {108\,e}{f^2}+\frac {54}{f}\right )}{f}\right )-x^2\,\left (\frac {54\,e}{f^2}+\frac {27}{f}\right )+\frac {36\,x^3}{f}-\frac {\ln \left (x+\frac {e}{f}\right )\,\left (9\,b\,f^3+54\,e^2\,f+f^3\,{\left (1-6\,b\right )}^{3/2}+108\,e^3-f^3+54\,b\,e\,f^2\right )}{f^4} \] Input:
int(-(9*b - 54*b*x + (1 - 6*b)^(3/2) + 54*x^2 - 108*x^3 - 1)/(e + f*x),x)
Output:
x*((54*b)/f + (e*((108*e)/f^2 + 54/f))/f) - x^2*((54*e)/f^2 + 27/f) + (36* x^3)/f - (log(x + e/f)*(9*b*f^3 + 54*e^2*f + f^3*(1 - 6*b)^(3/2) + 108*e^3 - f^3 + 54*b*e*f^2))/f^4
Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.36 \[ \int \frac {1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3}{e+f x} \, dx=\frac {6 \sqrt {-6 b +1}\, \mathrm {log}\left (f x +e \right ) b \,f^{3}-\sqrt {-6 b +1}\, \mathrm {log}\left (f x +e \right ) f^{3}-54 \,\mathrm {log}\left (f x +e \right ) b e \,f^{2}-9 \,\mathrm {log}\left (f x +e \right ) b \,f^{3}-108 \,\mathrm {log}\left (f x +e \right ) e^{3}-54 \,\mathrm {log}\left (f x +e \right ) e^{2} f +\mathrm {log}\left (f x +e \right ) f^{3}+54 b \,f^{3} x +108 e^{2} f x -54 e \,f^{2} x^{2}+54 e \,f^{2} x +36 f^{3} x^{3}-27 f^{3} x^{2}}{f^{4}} \] Input:
int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)/(f*x+e),x)
Output:
(6*sqrt( - 6*b + 1)*log(e + f*x)*b*f**3 - sqrt( - 6*b + 1)*log(e + f*x)*f* *3 - 54*log(e + f*x)*b*e*f**2 - 9*log(e + f*x)*b*f**3 - 108*log(e + f*x)*e **3 - 54*log(e + f*x)*e**2*f + log(e + f*x)*f**3 + 54*b*f**3*x + 108*e**2* f*x - 54*e*f**2*x**2 + 54*e*f**2*x + 36*f**3*x**3 - 27*f**3*x**2)/f**4