\(\int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx\) [1]

Optimal result

Integrand size = 29, antiderivative size = 141 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {d^2 \left (7 e^2 f^2+16 d e f g+8 d^2 g^2\right ) x}{e^2}-\frac {d \left (2 e^2 f^2+7 d e f g+4 d^2 g^2\right ) x^2}{e}-\frac {1}{3} (e f+d g) (e f+7 d g) x^3-\frac {1}{2} e g (e f+2 d g) x^4-\frac {1}{5} e^2 g^2 x^5-\frac {8 d^3 (e f+d g)^2 \log (d-e x)}{e^3} \] Output:

-d^2*(8*d^2*g^2+16*d*e*f*g+7*e^2*f^2)*x/e^2-d*(4*d^2*g^2+7*d*e*f*g+2*e^2*f 
^2)*x^2/e-1/3*(d*g+e*f)*(7*d*g+e*f)*x^3-1/2*e*g*(2*d*g+e*f)*x^4-1/5*e^2*g^ 
2*x^5-8*d^3*(d*g+e*f)^2*ln(-e*x+d)/e^3
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {x \left (240 d^4 g^2+120 d^3 e g (4 f+g x)+70 d^2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+10 d e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )+e^4 x^2 \left (10 f^2+15 f g x+6 g^2 x^2\right )\right )}{30 e^2}-\frac {8 d^3 (e f+d g)^2 \log (d-e x)}{e^3} \] Input:

Integrate[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2),x]
 

Output:

-1/30*(x*(240*d^4*g^2 + 120*d^3*e*g*(4*f + g*x) + 70*d^2*e^2*(3*f^2 + 3*f* 
g*x + g^2*x^2) + 10*d*e^3*x*(6*f^2 + 8*f*g*x + 3*g^2*x^2) + e^4*x^2*(10*f^ 
2 + 15*f*g*x + 6*g^2*x^2)))/e^2 - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {639, 99, 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.

\(\displaystyle \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx\)

\(\Big \downarrow \) 639

\(\displaystyle \int \frac {(d+e x)^3 (f+g x)^2}{d-e x}dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (-\frac {8 d^3 (d g+e f)^2}{e^2 (e x-d)}-\frac {2 d x \left (4 d^2 g^2+7 d e f g+2 e^2 f^2\right )}{e}-\frac {d^2 \left (8 d^2 g^2+16 d e f g+7 e^2 f^2\right )}{e^2}-2 e g x^3 (2 d g+e f)+x^2 (-7 d g-e f) (d g+e f)-e^2 g^2 x^4\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8 d^3 (d g+e f)^2 \log (d-e x)}{e^3}-\frac {d x^2 \left (4 d^2 g^2+7 d e f g+2 e^2 f^2\right )}{e}-\frac {d^2 x \left (8 d^2 g^2+16 d e f g+7 e^2 f^2\right )}{e^2}-\frac {1}{2} e g x^4 (2 d g+e f)-\frac {1}{3} x^3 (d g+e f) (7 d g+e f)-\frac {1}{5} e^2 g^2 x^5\)

Input:

Int[((d + e*x)^4*(f + g*x)^2)/(d^2 - e^2*x^2),x]
 

Output:

-((d^2*(7*e^2*f^2 + 16*d*e*f*g + 8*d^2*g^2)*x)/e^2) - (d*(2*e^2*f^2 + 7*d* 
e*f*g + 4*d^2*g^2)*x^2)/e - ((e*f + d*g)*(e*f + 7*d*g)*x^3)/3 - (e*g*(e*f 
+ 2*d*g)*x^4)/2 - (e^2*g^2*x^5)/5 - (8*d^3*(e*f + d*g)^2*Log[d - e*x])/e^3
 

Defintions of rubi rules used

rule 99

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

rule 639

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Int[(c + d*x)^(m + p)*(e + f*x)^n*(a/c + (b/d)*x)^p, 
 x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (I 
ntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] &&  !IntegerQ[m]))
 

rule 2009

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

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.10

method	result	size
norman	\(\left (-\frac {7}{3} d^{2} g^{2}-\frac {8}{3} d e f g -\frac {1}{3} e^{2} f^{2}\right ) x^{3}-\frac {e^{2} g^{2} x^{5}}{5}-\frac {d \left (4 d^{2} g^{2}+7 d e f g +2 e^{2} f^{2}\right ) x^{2}}{e}-\frac {d^{2} \left (8 d^{2} g^{2}+16 d e f g +7 e^{2} f^{2}\right ) x}{e^{2}}-\frac {e g \left (2 d g +e f \right ) x^{4}}{2}-\frac {8 d^{3} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(155\)
default	\(-\frac {\frac {1}{5} g^{2} e^{4} x^{5}+d \,e^{3} g^{2} x^{4}+\frac {1}{2} e^{4} f g \,x^{4}+\frac {7}{3} d^{2} e^{2} g^{2} x^{3}+\frac {8}{3} d \,e^{3} f g \,x^{3}+\frac {1}{3} e^{4} f^{2} x^{3}+4 d^{3} e \,g^{2} x^{2}+7 d^{2} e^{2} f g \,x^{2}+2 d \,e^{3} f^{2} x^{2}+8 d^{4} g^{2} x +16 d^{3} e f g x +7 d^{2} e^{2} f^{2} x}{e^{2}}-\frac {8 d^{3} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \ln \left (-e x +d \right )}{e^{3}}\)	\(179\)
risch	\(-\frac {e^{2} g^{2} x^{5}}{5}-e d \,g^{2} x^{4}-\frac {e^{2} f g \,x^{4}}{2}-\frac {7 d^{2} g^{2} x^{3}}{3}-\frac {8 e d f g \,x^{3}}{3}-\frac {e^{2} f^{2} x^{3}}{3}-\frac {4 d^{3} g^{2} x^{2}}{e}-7 d^{2} f g \,x^{2}-2 e d \,f^{2} x^{2}-\frac {8 d^{4} g^{2} x}{e^{2}}-\frac {16 d^{3} f g x}{e}-7 d^{2} f^{2} x -\frac {8 d^{5} \ln \left (-e x +d \right ) g^{2}}{e^{3}}-\frac {16 d^{4} \ln \left (-e x +d \right ) f g}{e^{2}}-\frac {8 d^{3} \ln \left (-e x +d \right ) f^{2}}{e}\)	\(183\)
parallelrisch	\(-\frac {6 g^{2} e^{5} x^{5}+30 x^{4} d \,e^{4} g^{2}+15 x^{4} e^{5} f g +70 x^{3} d^{2} e^{3} g^{2}+80 x^{3} d \,e^{4} f g +10 x^{3} e^{5} f^{2}+120 x^{2} d^{3} e^{2} g^{2}+210 x^{2} d^{2} e^{3} f g +60 x^{2} d \,e^{4} f^{2}+240 \ln \left (e x -d \right ) d^{5} g^{2}+480 \ln \left (e x -d \right ) d^{4} e f g +240 \ln \left (e x -d \right ) d^{3} e^{2} f^{2}+240 x \,d^{4} e \,g^{2}+480 x \,d^{3} e^{2} f g +210 x \,d^{2} e^{3} f^{2}}{30 e^{3}}\)	\(199\)

Input:

int((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2),x,method=_RETURNVERBOSE)
 

Output:

(-7/3*d^2*g^2-8/3*d*e*f*g-1/3*e^2*f^2)*x^3-1/5*e^2*g^2*x^5-d*(4*d^2*g^2+7* 
d*e*f*g+2*e^2*f^2)*x^2/e-d^2*(8*d^2*g^2+16*d*e*f*g+7*e^2*f^2)*x/e^2-1/2*e* 
g*(2*d*g+e*f)*x^4-8*d^3*(d^2*g^2+2*d*e*f*g+e^2*f^2)/e^3*ln(-e*x+d)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.25 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {6 \, e^{5} g^{2} x^{5} + 15 \, {\left (e^{5} f g + 2 \, d e^{4} g^{2}\right )} x^{4} + 10 \, {\left (e^{5} f^{2} + 8 \, d e^{4} f g + 7 \, d^{2} e^{3} g^{2}\right )} x^{3} + 30 \, {\left (2 \, d e^{4} f^{2} + 7 \, d^{2} e^{3} f g + 4 \, d^{3} e^{2} g^{2}\right )} x^{2} + 30 \, {\left (7 \, d^{2} e^{3} f^{2} + 16 \, d^{3} e^{2} f g + 8 \, d^{4} e g^{2}\right )} x + 240 \, {\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )} \log \left (e x - d\right )}{30 \, e^{3}} \] Input:

integrate((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2),x, algorithm="fricas")
 

Output:

-1/30*(6*e^5*g^2*x^5 + 15*(e^5*f*g + 2*d*e^4*g^2)*x^4 + 10*(e^5*f^2 + 8*d* 
e^4*f*g + 7*d^2*e^3*g^2)*x^3 + 30*(2*d*e^4*f^2 + 7*d^2*e^3*f*g + 4*d^3*e^2 
*g^2)*x^2 + 30*(7*d^2*e^3*f^2 + 16*d^3*e^2*f*g + 8*d^4*e*g^2)*x + 240*(d^3 
*e^2*f^2 + 2*d^4*e*f*g + d^5*g^2)*log(e*x - d))/e^3
 

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=- \frac {8 d^{3} \left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} - \frac {e^{2} g^{2} x^{5}}{5} - x^{4} \left (d e g^{2} + \frac {e^{2} f g}{2}\right ) - x^{3} \cdot \left (\frac {7 d^{2} g^{2}}{3} + \frac {8 d e f g}{3} + \frac {e^{2} f^{2}}{3}\right ) - x^{2} \cdot \left (\frac {4 d^{3} g^{2}}{e} + 7 d^{2} f g + 2 d e f^{2}\right ) - x \left (\frac {8 d^{4} g^{2}}{e^{2}} + \frac {16 d^{3} f g}{e} + 7 d^{2} f^{2}\right ) \] Input:

integrate((e*x+d)**4*(g*x+f)**2/(-e**2*x**2+d**2),x)
 

Output:

-8*d**3*(d*g + e*f)**2*log(-d + e*x)/e**3 - e**2*g**2*x**5/5 - x**4*(d*e*g 
**2 + e**2*f*g/2) - x**3*(7*d**2*g**2/3 + 8*d*e*f*g/3 + e**2*f**2/3) - x** 
2*(4*d**3*g**2/e + 7*d**2*f*g + 2*d*e*f**2) - x*(8*d**4*g**2/e**2 + 16*d** 
3*f*g/e + 7*d**2*f**2)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {6 \, e^{4} g^{2} x^{5} + 15 \, {\left (e^{4} f g + 2 \, d e^{3} g^{2}\right )} x^{4} + 10 \, {\left (e^{4} f^{2} + 8 \, d e^{3} f g + 7 \, d^{2} e^{2} g^{2}\right )} x^{3} + 30 \, {\left (2 \, d e^{3} f^{2} + 7 \, d^{2} e^{2} f g + 4 \, d^{3} e g^{2}\right )} x^{2} + 30 \, {\left (7 \, d^{2} e^{2} f^{2} + 16 \, d^{3} e f g + 8 \, d^{4} g^{2}\right )} x}{30 \, e^{2}} - \frac {8 \, {\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \] Input:

integrate((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2),x, algorithm="maxima")
 

Output:

-1/30*(6*e^4*g^2*x^5 + 15*(e^4*f*g + 2*d*e^3*g^2)*x^4 + 10*(e^4*f^2 + 8*d* 
e^3*f*g + 7*d^2*e^2*g^2)*x^3 + 30*(2*d*e^3*f^2 + 7*d^2*e^2*f*g + 4*d^3*e*g 
^2)*x^2 + 30*(7*d^2*e^2*f^2 + 16*d^3*e*f*g + 8*d^4*g^2)*x)/e^2 - 8*(d^3*e^ 
2*f^2 + 2*d^4*e*f*g + d^5*g^2)*log(e*x - d)/e^3
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=-\frac {8 \, {\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )} \log \left ({\left | e x - d \right |}\right )}{e^{3}} - \frac {6 \, e^{7} g^{2} x^{5} + 15 \, e^{7} f g x^{4} + 30 \, d e^{6} g^{2} x^{4} + 10 \, e^{7} f^{2} x^{3} + 80 \, d e^{6} f g x^{3} + 70 \, d^{2} e^{5} g^{2} x^{3} + 60 \, d e^{6} f^{2} x^{2} + 210 \, d^{2} e^{5} f g x^{2} + 120 \, d^{3} e^{4} g^{2} x^{2} + 210 \, d^{2} e^{5} f^{2} x + 480 \, d^{3} e^{4} f g x + 240 \, d^{4} e^{3} g^{2} x}{30 \, e^{5}} \] Input:

integrate((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2),x, algorithm="giac")
 

Output:

-8*(d^3*e^2*f^2 + 2*d^4*e*f*g + d^5*g^2)*log(abs(e*x - d))/e^3 - 1/30*(6*e 
^7*g^2*x^5 + 15*e^7*f*g*x^4 + 30*d*e^6*g^2*x^4 + 10*e^7*f^2*x^3 + 80*d*e^6 
*f*g*x^3 + 70*d^2*e^5*g^2*x^3 + 60*d*e^6*f^2*x^2 + 210*d^2*e^5*f*g*x^2 + 1 
20*d^3*e^4*g^2*x^2 + 210*d^2*e^5*f^2*x + 480*d^3*e^4*f*g*x + 240*d^4*e^3*g 
^2*x)/e^5
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=-x^2\,\left (\frac {d^3\,g^2+6\,d^2\,e\,f\,g+3\,d\,e^2\,f^2}{2\,e}+\frac {d\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{e}+\frac {d\,\left (e\,g\,\left (3\,d\,g+2\,e\,f\right )+d\,e\,g^2\right )}{e}\right )}{2\,e}\right )-x^3\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{3\,e}+\frac {d\,\left (e\,g\,\left (3\,d\,g+2\,e\,f\right )+d\,e\,g^2\right )}{3\,e}\right )-x^4\,\left (\frac {e\,g\,\left (3\,d\,g+2\,e\,f\right )}{4}+\frac {d\,e\,g^2}{4}\right )-x\,\left (\frac {d\,\left (\frac {d^3\,g^2+6\,d^2\,e\,f\,g+3\,d\,e^2\,f^2}{e}+\frac {d\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{e}+\frac {d\,\left (e\,g\,\left (3\,d\,g+2\,e\,f\right )+d\,e\,g^2\right )}{e}\right )}{e}\right )}{e}+\frac {d^2\,f\,\left (2\,d\,g+3\,e\,f\right )}{e}\right )-\frac {\ln \left (e\,x-d\right )\,\left (8\,d^5\,g^2+16\,d^4\,e\,f\,g+8\,d^3\,e^2\,f^2\right )}{e^3}-\frac {e^2\,g^2\,x^5}{5} \] Input:

int(((f + g*x)^2*(d + e*x)^4)/(d^2 - e^2*x^2),x)
 

Output:

- x^2*((d^3*g^2 + 3*d*e^2*f^2 + 6*d^2*e*f*g)/(2*e) + (d*((e^3*f^2 + 3*d^2* 
e*g^2 + 6*d*e^2*f*g)/e + (d*(e*g*(3*d*g + 2*e*f) + d*e*g^2))/e))/(2*e)) - 
x^3*((e^3*f^2 + 3*d^2*e*g^2 + 6*d*e^2*f*g)/(3*e) + (d*(e*g*(3*d*g + 2*e*f) 
 + d*e*g^2))/(3*e)) - x^4*((e*g*(3*d*g + 2*e*f))/4 + (d*e*g^2)/4) - x*((d* 
((d^3*g^2 + 3*d*e^2*f^2 + 6*d^2*e*f*g)/e + (d*((e^3*f^2 + 3*d^2*e*g^2 + 6* 
d*e^2*f*g)/e + (d*(e*g*(3*d*g + 2*e*f) + d*e*g^2))/e))/e))/e + (d^2*f*(2*d 
*g + 3*e*f))/e) - (log(e*x - d)*(8*d^5*g^2 + 8*d^3*e^2*f^2 + 16*d^4*e*f*g) 
)/e^3 - (e^2*g^2*x^5)/5
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^4 (f+g x)^2}{d^2-e^2 x^2} \, dx=\frac {-240 \,\mathrm {log}\left (-e x +d \right ) d^{5} g^{2}-480 \,\mathrm {log}\left (-e x +d \right ) d^{4} e f g -240 \,\mathrm {log}\left (-e x +d \right ) d^{3} e^{2} f^{2}-240 d^{4} e \,g^{2} x -480 d^{3} e^{2} f g x -120 d^{3} e^{2} g^{2} x^{2}-210 d^{2} e^{3} f^{2} x -210 d^{2} e^{3} f g \,x^{2}-70 d^{2} e^{3} g^{2} x^{3}-60 d \,e^{4} f^{2} x^{2}-80 d \,e^{4} f g \,x^{3}-30 d \,e^{4} g^{2} x^{4}-10 e^{5} f^{2} x^{3}-15 e^{5} f g \,x^{4}-6 e^{5} g^{2} x^{5}}{30 e^{3}} \] Input:

int((e*x+d)^4*(g*x+f)^2/(-e^2*x^2+d^2),x)
 

Output:

( - 240*log(d - e*x)*d**5*g**2 - 480*log(d - e*x)*d**4*e*f*g - 240*log(d - 
 e*x)*d**3*e**2*f**2 - 240*d**4*e*g**2*x - 480*d**3*e**2*f*g*x - 120*d**3* 
e**2*g**2*x**2 - 210*d**2*e**3*f**2*x - 210*d**2*e**3*f*g*x**2 - 70*d**2*e 
**3*g**2*x**3 - 60*d*e**4*f**2*x**2 - 80*d*e**4*f*g*x**3 - 30*d*e**4*g**2* 
x**4 - 10*e**5*f**2*x**3 - 15*e**5*f*g*x**4 - 6*e**5*g**2*x**5)/(30*e**3)