3.5.87 \(\int \frac {(a+c x^2)^3}{(d+e x)^{10}} \, dx\) [487]

3.5.87.1 Optimal result

Integrand size = 17, antiderivative size = 190 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\left (c d^2+a e^2\right )^3}{9 e^7 (d+e x)^9}+\frac {3 c d \left (c d^2+a e^2\right )^2}{4 e^7 (d+e x)^8}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{7 e^7 (d+e x)^7}+\frac {2 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^6}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{5 e^7 (d+e x)^5}+\frac {3 c^3 d}{2 e^7 (d+e x)^4}-\frac {c^3}{3 e^7 (d+e x)^3} \]

output

-1/9*(a*e^2+c*d^2)^3/e^7/(e*x+d)^9+3/4*c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)^8-3 
/7*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)/e^7/(e*x+d)^7+2/3*c^2*d*(3*a*e^2+5*c*d^ 
2)/e^7/(e*x+d)^6-3/5*c^2*(a*e^2+5*c*d^2)/e^7/(e*x+d)^5+3/2*c^3*d/e^7/(e*x+ 
d)^4-1/3*c^3/e^7/(e*x+d)^3
 

3.5.87.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {140 a^3 e^6+15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{1260 e^7 (d+e x)^9} \]

input

Integrate[(a + c*x^2)^3/(d + e*x)^10,x]
 

output

-1/1260*(140*a^3*e^6 + 15*a^2*c*e^4*(d^2 + 9*d*e*x + 36*e^2*x^2) + 6*a*c^2 
*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*c 
^3*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 
126*d*e^5*x^5 + 84*e^6*x^6))/(e^7*(d + e*x)^9)
 

3.5.87.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {476, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 476

\(\displaystyle \int \left (\frac {3 c^2 \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^6}-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{e^6 (d+e x)^7}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^6 (d+e x)^8}-\frac {6 c d \left (a e^2+c d^2\right )^2}{e^6 (d+e x)^9}+\frac {\left (a e^2+c d^2\right )^3}{e^6 (d+e x)^{10}}+\frac {c^3}{e^6 (d+e x)^4}-\frac {6 c^3 d}{e^6 (d+e x)^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac {2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac {\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac {c^3}{3 e^7 (d+e x)^3}+\frac {3 c^3 d}{2 e^7 (d+e x)^4}\)

input

Int[(a + c*x^2)^3/(d + e*x)^10,x]
 

output

-1/9*(c*d^2 + a*e^2)^3/(e^7*(d + e*x)^9) + (3*c*d*(c*d^2 + a*e^2)^2)/(4*e^ 
7*(d + e*x)^8) - (3*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2))/(7*e^7*(d + e*x)^ 
7) + (2*c^2*d*(5*c*d^2 + 3*a*e^2))/(3*e^7*(d + e*x)^6) - (3*c^2*(5*c*d^2 + 
 a*e^2))/(5*e^7*(d + e*x)^5) + (3*c^3*d)/(2*e^7*(d + e*x)^4) - c^3/(3*e^7* 
(d + e*x)^3)
 

3.5.87.3.1 Defintions of rubi rules used

rule 476

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, 
 x] && IGtQ[p, 0]
 

rule 2009

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

3.5.87.4 Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.05

method	result	size
risch	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{3} d \,x^{5}}{2 e^{2}}-\frac {c^{2} \left (6 e^{2} a +5 c \,d^{2}\right ) x^{4}}{10 e^{3}}-\frac {c^{2} d \left (6 e^{2} a +5 c \,d^{2}\right ) x^{3}}{15 e^{4}}-\frac {c \left (15 a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{2}}{35 e^{5}}-\frac {d c \left (15 a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x}{140 e^{6}}-\frac {140 e^{6} a^{3}+15 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{1260 e^{7}}}{\left (e x +d \right )^{9}}\)	\(199\)
gosper	\(-\frac {420 x^{6} c^{3} e^{6}+630 x^{5} c^{3} d \,e^{5}+756 x^{4} a \,c^{2} e^{6}+630 x^{4} c^{3} d^{2} e^{4}+504 x^{3} a \,c^{2} d \,e^{5}+420 x^{3} c^{3} d^{3} e^{3}+540 x^{2} a^{2} c \,e^{6}+216 x^{2} a \,c^{2} d^{2} e^{4}+180 x^{2} c^{3} d^{4} e^{2}+135 x \,a^{2} c d \,e^{5}+54 x a \,c^{2} d^{3} e^{3}+45 x \,c^{3} d^{5} e +140 e^{6} a^{3}+15 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}}{1260 e^{7} \left (e x +d \right )^{9}}\)	\(205\)
parallelrisch	\(\frac {-420 c^{3} x^{6} e^{8}-630 c^{3} d \,x^{5} e^{7}-756 a \,c^{2} e^{8} x^{4}-630 c^{3} d^{2} e^{6} x^{4}-504 a \,c^{2} d \,e^{7} x^{3}-420 c^{3} d^{3} e^{5} x^{3}-540 a^{2} c \,e^{8} x^{2}-216 a \,c^{2} d^{2} e^{6} x^{2}-180 c^{3} d^{4} e^{4} x^{2}-135 a^{2} c d \,e^{7} x -54 a \,c^{2} d^{3} e^{5} x -45 c^{3} d^{5} e^{3} x -140 a^{3} e^{8}-15 a^{2} c \,d^{2} e^{6}-6 a \,c^{2} d^{4} e^{4}-5 c^{3} d^{6} e^{2}}{1260 e^{9} \left (e x +d \right )^{9}}\)	\(210\)
default	\(-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{7 e^{7} \left (e x +d \right )^{7}}+\frac {2 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{3 e^{7} \left (e x +d \right )^{6}}-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}}{9 e^{7} \left (e x +d \right )^{9}}-\frac {c^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {3 c^{3} d}{2 e^{7} \left (e x +d \right )^{4}}+\frac {3 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{8}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{5 e^{7} \left (e x +d \right )^{5}}\)	\(218\)
norman	\(\frac {-\frac {c^{3} x^{6}}{3 e}-\frac {c^{3} d \,x^{5}}{2 e^{2}}-\frac {\left (6 e^{4} c^{2} a +5 d^{2} e^{2} c^{3}\right ) x^{4}}{10 e^{5}}-\frac {d \left (6 e^{4} c^{2} a +5 d^{2} e^{2} c^{3}\right ) x^{3}}{15 e^{6}}-\frac {\left (15 e^{6} a^{2} c +6 d^{2} e^{4} c^{2} a +5 d^{4} e^{2} c^{3}\right ) x^{2}}{35 e^{7}}-\frac {d \left (15 e^{6} a^{2} c +6 d^{2} e^{4} c^{2} a +5 d^{4} e^{2} c^{3}\right ) x}{140 e^{8}}-\frac {140 a^{3} e^{8}+15 a^{2} c \,d^{2} e^{6}+6 a \,c^{2} d^{4} e^{4}+5 c^{3} d^{6} e^{2}}{1260 e^{9}}}{\left (e x +d \right )^{9}}\)	\(222\)

input

int((c*x^2+a)^3/(e*x+d)^10,x,method=_RETURNVERBOSE)
 

output

(-1/3*c^3*x^6/e-1/2*c^3*d*x^5/e^2-1/10*c^2*(6*a*e^2+5*c*d^2)/e^3*x^4-1/15* 
c^2*d*(6*a*e^2+5*c*d^2)/e^4*x^3-1/35/e^5*c*(15*a^2*e^4+6*a*c*d^2*e^2+5*c^2 
*d^4)*x^2-1/140*d/e^6*c*(15*a^2*e^4+6*a*c*d^2*e^2+5*c^2*d^4)*x-1/1260/e^7* 
(140*a^3*e^6+15*a^2*c*d^2*e^4+6*a*c^2*d^4*e^2+5*c^3*d^6))/(e*x+d)^9
 

3.5.87.5 Fricas [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \, {\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \, {\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \, {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

input

integrate((c*x^2+a)^3/(e*x+d)^10,x, algorithm="fricas")
 

output

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 
 + 15*a^2*c*d^2*e^4 + 140*a^3*e^6 + 126*(5*c^3*d^2*e^4 + 6*a*c^2*e^6)*x^4 
+ 84*(5*c^3*d^3*e^3 + 6*a*c^2*d*e^5)*x^3 + 36*(5*c^3*d^4*e^2 + 6*a*c^2*d^2 
*e^4 + 15*a^2*c*e^6)*x^2 + 9*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + 15*a^2*c*d*e 
^5)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126* 
d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8 
*e^8*x + d^9*e^7)
 

3.5.87.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=\text {Timed out} \]

input

integrate((c*x**2+a)**3/(e*x+d)**10,x)
 

output

Timed out
 

3.5.87.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \, {\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \, {\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \, {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

input

integrate((c*x^2+a)^3/(e*x+d)^10,x, algorithm="maxima")
 

output

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 
 + 15*a^2*c*d^2*e^4 + 140*a^3*e^6 + 126*(5*c^3*d^2*e^4 + 6*a*c^2*e^6)*x^4 
+ 84*(5*c^3*d^3*e^3 + 6*a*c^2*d*e^5)*x^3 + 36*(5*c^3*d^4*e^2 + 6*a*c^2*d^2 
*e^4 + 15*a^2*c*e^6)*x^2 + 9*(5*c^3*d^5*e + 6*a*c^2*d^3*e^3 + 15*a^2*c*d*e 
^5)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126* 
d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8 
*e^8*x + d^9*e^7)
 

3.5.87.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 630 \, c^{3} d^{2} e^{4} x^{4} + 756 \, a c^{2} e^{6} x^{4} + 420 \, c^{3} d^{3} e^{3} x^{3} + 504 \, a c^{2} d e^{5} x^{3} + 180 \, c^{3} d^{4} e^{2} x^{2} + 216 \, a c^{2} d^{2} e^{4} x^{2} + 540 \, a^{2} c e^{6} x^{2} + 45 \, c^{3} d^{5} e x + 54 \, a c^{2} d^{3} e^{3} x + 135 \, a^{2} c d e^{5} x + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6}}{1260 \, {\left (e x + d\right )}^{9} e^{7}} \]

input

integrate((c*x^2+a)^3/(e*x+d)^10,x, algorithm="giac")
 

output

-1/1260*(420*c^3*e^6*x^6 + 630*c^3*d*e^5*x^5 + 630*c^3*d^2*e^4*x^4 + 756*a 
*c^2*e^6*x^4 + 420*c^3*d^3*e^3*x^3 + 504*a*c^2*d*e^5*x^3 + 180*c^3*d^4*e^2 
*x^2 + 216*a*c^2*d^2*e^4*x^2 + 540*a^2*c*e^6*x^2 + 45*c^3*d^5*e*x + 54*a*c 
^2*d^3*e^3*x + 135*a^2*c*d*e^5*x + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*a^2*c* 
d^2*e^4 + 140*a^3*e^6)/((e*x + d)^9*e^7)
 

3.5.87.9 Mupad [B] (verification not implemented)

Time = 9.57 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\frac {140\,a^3\,e^6+15\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{1260\,e^7}+\frac {c^3\,x^6}{3\,e}+\frac {c^3\,d\,x^5}{2\,e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+6\,a\,e^2\right )}{10\,e^3}+\frac {c\,x^2\,\left (15\,a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{35\,e^5}+\frac {c\,d\,x\,\left (15\,a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{140\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+6\,a\,e^2\right )}{15\,e^4}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]

input

int((a + c*x^2)^3/(d + e*x)^10,x)
 

output

-((140*a^3*e^6 + 5*c^3*d^6 + 6*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4)/(1260*e^7 
) + (c^3*x^6)/(3*e) + (c^3*d*x^5)/(2*e^2) + (c^2*x^4*(6*a*e^2 + 5*c*d^2))/ 
(10*e^3) + (c*x^2*(15*a^2*e^4 + 5*c^2*d^4 + 6*a*c*d^2*e^2))/(35*e^5) + (c* 
d*x*(15*a^2*e^4 + 5*c^2*d^4 + 6*a*c*d^2*e^2))/(140*e^6) + (c^2*d*x^3*(6*a* 
e^2 + 5*c*d^2))/(15*e^4))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 
84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d 
^2*e^7*x^7 + 9*d^8*e*x)