\(\int \frac {x^7}{(c+d x)^2 (c^2-d^2 x^2)^{7/2}} \, dx\) [498]

Optimal result

Integrand size = 27, antiderivative size = 150 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {2 c^8}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}-\frac {2 d x^9}{9 c \left (c^2-d^2 x^2\right )^{9/2}}-\frac {c^6}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}+\frac {9 c^4}{5 d^8 \left (c^2-d^2 x^2\right )^{5/2}}-\frac {5 c^2}{3 d^8 \left (c^2-d^2 x^2\right )^{3/2}}+\frac {1}{d^8 \sqrt {c^2-d^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.77 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (16 c^7+32 c^6 d x-24 c^5 d^2 x^2-80 c^4 d^3 x^3-10 c^3 d^4 x^4+60 c^2 d^5 x^5+25 c d^6 x^6-10 d^7 x^7\right )}{45 c d^8 (c-d x)^3 (c+d x)^5} \] Input:

Integrate[x^7/((c + d*x)^2*(c^2 - d^2*x^2)^(7/2)),x]
 

Output:

(Sqrt[c^2 - d^2*x^2]*(16*c^7 + 32*c^6*d*x - 24*c^5*d^2*x^2 - 80*c^4*d^3*x^ 
3 - 10*c^3*d^4*x^4 + 60*c^2*d^5*x^5 + 25*c*d^6*x^6 - 10*d^7*x^7))/(45*c*d^ 
8*(c - d*x)^3*(c + d*x)^5)
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {570, 529, 25, 2166, 27, 2345, 27, 2345, 27, 453}

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.

\(\displaystyle \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 570

\(\displaystyle \int \frac {x^7 (c-d x)^2}{\left (c^2-d^2 x^2\right )^{11/2}}dx\)

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2166

\(\displaystyle \frac {-\frac {\int \frac {21 \left (\frac {2 c^7}{d^7}-\frac {15 x c^6}{d^6}+\frac {12 x^2 c^5}{d^5}-\frac {9 x^3 c^4}{d^4}+\frac {6 x^4 c^3}{d^3}-\frac {3 x^5 c^2}{d^2}\right )}{\left (c^2-d^2 x^2\right )^{7/2}}dx}{7 c}-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}}{9 c}+\frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \int \frac {\frac {2 c^7}{d^7}-\frac {15 x c^6}{d^6}+\frac {12 x^2 c^5}{d^5}-\frac {9 x^3 c^4}{d^4}+\frac {6 x^4 c^3}{d^3}-\frac {3 x^5 c^2}{d^2}}{\left (c^2-d^2 x^2\right )^{7/2}}dx}{c}-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}}{9 c}+\frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {5 \left (\frac {2 c^7}{d^7}-\frac {12 x c^6}{d^6}+\frac {6 x^2 c^5}{d^5}-\frac {3 x^3 c^4}{d^4}\right )}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{5 c^2}-\frac {c^5 (27 c-20 d x)}{5 d^8 \left (c^2-d^2 x^2\right )^{5/2}}\right )}{c}-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}}{9 c}+\frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (-\frac {\int \frac {\frac {2 c^7}{d^7}-\frac {12 x c^6}{d^6}+\frac {6 x^2 c^5}{d^5}-\frac {3 x^3 c^4}{d^4}}{\left (c^2-d^2 x^2\right )^{5/2}}dx}{c^2}-\frac {c^5 (27 c-20 d x)}{5 d^8 \left (c^2-d^2 x^2\right )^{5/2}}\right )}{c}-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}}{9 c}+\frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {\int \frac {c^6 (2 c-9 d x)}{d^7 \left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c^2}-\frac {c^5 (15 c-8 d x)}{3 d^8 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}-\frac {c^5 (27 c-20 d x)}{5 d^8 \left (c^2-d^2 x^2\right )^{5/2}}\right )}{c}-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}}{9 c}+\frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 \left (-\frac {-\frac {c^4 \int \frac {2 c-9 d x}{\left (c^2-d^2 x^2\right )^{3/2}}dx}{3 d^7}-\frac {c^5 (15 c-8 d x)}{3 d^8 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}-\frac {c^5 (27 c-20 d x)}{5 d^8 \left (c^2-d^2 x^2\right )^{5/2}}\right )}{c}-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}}{9 c}+\frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}\)

\(\Big \downarrow \) 453

\(\displaystyle \frac {c^6 (c-d x)^2}{9 d^8 \left (c^2-d^2 x^2\right )^{9/2}}+\frac {-\frac {8 c^6 (c-d x)}{d^8 \left (c^2-d^2 x^2\right )^{7/2}}-\frac {3 \left (-\frac {c^5 (27 c-20 d x)}{5 d^8 \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\frac {c^3 (9 c-2 d x)}{3 d^8 \sqrt {c^2-d^2 x^2}}-\frac {c^5 (15 c-8 d x)}{3 d^8 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}\right )}{c}}{9 c}\)

Input:

Int[x^7/((c + d*x)^2*(c^2 - d^2*x^2)^(7/2)),x]
 

Output:

(c^6*(c - d*x)^2)/(9*d^8*(c^2 - d^2*x^2)^(9/2)) + ((-8*c^6*(c - d*x))/(d^8 
*(c^2 - d^2*x^2)^(7/2)) - (3*(-1/5*(c^5*(27*c - 20*d*x))/(d^8*(c^2 - d^2*x 
^2)^(5/2)) - (-1/3*(c^5*(15*c - 8*d*x))/(d^8*(c^2 - d^2*x^2)^(3/2)) + (c^3 
*(9*c - 2*d*x))/(3*d^8*Sqrt[c^2 - d^2*x^2]))/c^2))/c)/(9*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 453

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* 
d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
 

rule 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.73

method	result	size
gosper	\(\frac {\left (-d x +c \right ) \left (-10 d^{7} x^{7}+25 c \,d^{6} x^{6}+60 c^{2} d^{5} x^{5}-10 d^{4} c^{3} x^{4}-80 d^{3} c^{4} x^{3}-24 c^{5} d^{2} x^{2}+32 c^{6} d x +16 c^{7}\right )}{45 \left (d x +c \right ) c \,d^{8} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}\)	\(110\)
orering	\(\frac {\left (-d x +c \right ) \left (-10 d^{7} x^{7}+25 c \,d^{6} x^{6}+60 c^{2} d^{5} x^{5}-10 d^{4} c^{3} x^{4}-80 d^{3} c^{4} x^{3}-24 c^{5} d^{2} x^{2}+32 c^{6} d x +16 c^{7}\right )}{45 \left (d x +c \right ) c \,d^{8} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}\)	\(110\)
trager	\(\frac {\left (-10 d^{7} x^{7}+25 c \,d^{6} x^{6}+60 c^{2} d^{5} x^{5}-10 d^{4} c^{3} x^{4}-80 d^{3} c^{4} x^{3}-24 c^{5} d^{2} x^{2}+32 c^{6} d x +16 c^{7}\right ) \sqrt {-d^{2} x^{2}+c^{2}}}{45 c \,d^{8} \left (d x +c \right )^{5} \left (-d x +c \right )^{3}}\)	\(112\)
default	\(\text {Expression too large to display}\)	\(971\)

Input:

int(x^7/(d*x+c)^2/(-d^2*x^2+c^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/45*(-d*x+c)*(-10*d^7*x^7+25*c*d^6*x^6+60*c^2*d^5*x^5-10*c^3*d^4*x^4-80*c 
^4*d^3*x^3-24*c^5*d^2*x^2+32*c^6*d*x+16*c^7)/(d*x+c)/c/d^8/(-d^2*x^2+c^2)^ 
(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.65 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {16 \, d^{8} x^{8} + 32 \, c d^{7} x^{7} - 32 \, c^{2} d^{6} x^{6} - 96 \, c^{3} d^{5} x^{5} + 96 \, c^{5} d^{3} x^{3} + 32 \, c^{6} d^{2} x^{2} - 32 \, c^{7} d x - 16 \, c^{8} + {\left (10 \, d^{7} x^{7} - 25 \, c d^{6} x^{6} - 60 \, c^{2} d^{5} x^{5} + 10 \, c^{3} d^{4} x^{4} + 80 \, c^{4} d^{3} x^{3} + 24 \, c^{5} d^{2} x^{2} - 32 \, c^{6} d x - 16 \, c^{7}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{45 \, {\left (c d^{16} x^{8} + 2 \, c^{2} d^{15} x^{7} - 2 \, c^{3} d^{14} x^{6} - 6 \, c^{4} d^{13} x^{5} + 6 \, c^{6} d^{11} x^{3} + 2 \, c^{7} d^{10} x^{2} - 2 \, c^{8} d^{9} x - c^{9} d^{8}\right )}} \] Input:

integrate(x^7/(d*x+c)^2/(-d^2*x^2+c^2)^(7/2),x, algorithm="fricas")
 

Output:

1/45*(16*d^8*x^8 + 32*c*d^7*x^7 - 32*c^2*d^6*x^6 - 96*c^3*d^5*x^5 + 96*c^5 
*d^3*x^3 + 32*c^6*d^2*x^2 - 32*c^7*d*x - 16*c^8 + (10*d^7*x^7 - 25*c*d^6*x 
^6 - 60*c^2*d^5*x^5 + 10*c^3*d^4*x^4 + 80*c^4*d^3*x^3 + 24*c^5*d^2*x^2 - 3 
2*c^6*d*x - 16*c^7)*sqrt(-d^2*x^2 + c^2))/(c*d^16*x^8 + 2*c^2*d^15*x^7 - 2 
*c^3*d^14*x^6 - 6*c^4*d^13*x^5 + 6*c^6*d^11*x^3 + 2*c^7*d^10*x^2 - 2*c^8*d 
^9*x - c^9*d^8)
 

Sympy [F]

\[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\int \frac {x^{7}}{\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {7}{2}} \left (c + d x\right )^{2}}\, dx \] Input:

integrate(x**7/(d*x+c)**2/(-d**2*x**2+c**2)**(7/2),x)
 

Output:

Integral(x**7/((-(-c + d*x)*(c + d*x))**(7/2)*(c + d*x)**2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (130) = 260\).

Time = 0.06 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.84 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {c^{6}}{9 \, {\left ({\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{10} x^{2} + 2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c d^{9} x + {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c^{2} d^{8}\right )}} - \frac {8 \, c^{5}}{9 \, {\left ({\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{9} x + {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} c d^{8}\right )}} + \frac {x^{4}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{4}} - \frac {c x^{3}}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{5}} - \frac {c^{2} x^{2}}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{6}} - \frac {c^{3} x}{3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{7}} + \frac {17 \, c^{4}}{15 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {5}{2}} d^{8}} - \frac {c x}{9 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{7}} - \frac {2 \, x}{9 \, \sqrt {-d^{2} x^{2} + c^{2}} c d^{7}} \] Input:

integrate(x^7/(d*x+c)^2/(-d^2*x^2+c^2)^(7/2),x, algorithm="maxima")
 

Output:

1/9*c^6/((-d^2*x^2 + c^2)^(5/2)*d^10*x^2 + 2*(-d^2*x^2 + c^2)^(5/2)*c*d^9* 
x + (-d^2*x^2 + c^2)^(5/2)*c^2*d^8) - 8/9*c^5/((-d^2*x^2 + c^2)^(5/2)*d^9* 
x + (-d^2*x^2 + c^2)^(5/2)*c*d^8) + x^4/((-d^2*x^2 + c^2)^(5/2)*d^4) - c*x 
^3/((-d^2*x^2 + c^2)^(5/2)*d^5) - 1/3*c^2*x^2/((-d^2*x^2 + c^2)^(5/2)*d^6) 
 - 1/3*c^3*x/((-d^2*x^2 + c^2)^(5/2)*d^7) + 17/15*c^4/((-d^2*x^2 + c^2)^(5 
/2)*d^8) - 1/9*c*x/((-d^2*x^2 + c^2)^(3/2)*d^7) - 2/9*x/(sqrt(-d^2*x^2 + c 
^2)*c*d^7)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.95 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=-\frac {\frac {1280 i \, \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )}{c d^{7}} - \frac {3 \, {\left (315 \, {\left (\frac {2 \, c}{d x + c} - 1\right )}^{2} - \frac {70 \, c}{d x + c} + 38\right )}}{c d^{7} {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{d x + c}\right ) \mathrm {sgn}\left (d\right )} - \frac {5 \, c^{8} d^{56} {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{8} \mathrm {sgn}\left (d\right )^{8} - 45 \, c^{8} d^{56} {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{8} \mathrm {sgn}\left (d\right )^{8} + 189 \, c^{8} d^{56} {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{8} \mathrm {sgn}\left (d\right )^{8} - 525 \, c^{8} d^{56} {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{8} \mathrm {sgn}\left (d\right )^{8} + 1575 \, c^{8} d^{56} \sqrt {\frac {2 \, c}{d x + c} - 1} \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{8} \mathrm {sgn}\left (d\right )^{8}}{c^{9} d^{63} \mathrm {sgn}\left (\frac {1}{d x + c}\right )^{9} \mathrm {sgn}\left (d\right )^{9}}}{5760 \, {\left | d \right |}} \] Input:

integrate(x^7/(d*x+c)^2/(-d^2*x^2+c^2)^(7/2),x, algorithm="giac")
 

Output:

-1/5760*(1280*I*sgn(1/(d*x + c))*sgn(d)/(c*d^7) - 3*(315*(2*c/(d*x + c) - 
1)^2 - 70*c/(d*x + c) + 38)/(c*d^7*(2*c/(d*x + c) - 1)^(5/2)*sgn(1/(d*x + 
c))*sgn(d)) - (5*c^8*d^56*(2*c/(d*x + c) - 1)^(9/2)*sgn(1/(d*x + c))^8*sgn 
(d)^8 - 45*c^8*d^56*(2*c/(d*x + c) - 1)^(7/2)*sgn(1/(d*x + c))^8*sgn(d)^8 
+ 189*c^8*d^56*(2*c/(d*x + c) - 1)^(5/2)*sgn(1/(d*x + c))^8*sgn(d)^8 - 525 
*c^8*d^56*(2*c/(d*x + c) - 1)^(3/2)*sgn(1/(d*x + c))^8*sgn(d)^8 + 1575*c^8 
*d^56*sqrt(2*c/(d*x + c) - 1)*sgn(1/(d*x + c))^8*sgn(d)^8)/(c^9*d^63*sgn(1 
/(d*x + c))^9*sgn(d)^9))/abs(d)
 

Mupad [B] (verification not implemented)

Time = 7.34 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.31 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {1}{d^8}-\frac {2\,x}{9\,c\,d^7}\right )}{\left (c+d\,x\right )\,\left (c-d\,x\right )}+\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {121\,c^4}{120\,d^8}-\frac {23\,c^3\,x}{24\,d^7}\right )}{{\left (c+d\,x\right )}^3\,{\left (c-d\,x\right )}^3}-\frac {13\,c^2\,\sqrt {c^2-d^2\,x^2}}{144\,d^8\,{\left (c+d\,x\right )}^4}+\frac {c^3\,\sqrt {c^2-d^2\,x^2}}{72\,d^8\,{\left (c+d\,x\right )}^5}-\frac {\sqrt {c^2-d^2\,x^2}\,\left (\frac {227\,c^2}{144\,d^8}-\frac {8\,c\,x}{9\,d^7}\right )}{{\left (c+d\,x\right )}^2\,{\left (c-d\,x\right )}^2} \] Input:

int(x^7/((c^2 - d^2*x^2)^(7/2)*(c + d*x)^2),x)
 

Output:

((c^2 - d^2*x^2)^(1/2)*(1/d^8 - (2*x)/(9*c*d^7)))/((c + d*x)*(c - d*x)) + 
((c^2 - d^2*x^2)^(1/2)*((121*c^4)/(120*d^8) - (23*c^3*x)/(24*d^7)))/((c + 
d*x)^3*(c - d*x)^3) - (13*c^2*(c^2 - d^2*x^2)^(1/2))/(144*d^8*(c + d*x)^4) 
 + (c^3*(c^2 - d^2*x^2)^(1/2))/(72*d^8*(c + d*x)^5) - ((c^2 - d^2*x^2)^(1/ 
2)*((227*c^2)/(144*d^8) - (8*c*x)/(9*d^7)))/((c + d*x)^2*(c - d*x)^2)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.08 \[ \int \frac {x^7}{(c+d x)^2 \left (c^2-d^2 x^2\right )^{7/2}} \, dx=\frac {16 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6}+32 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d x -16 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{2} x^{2}-64 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{3} x^{3}-16 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{4} x^{4}+32 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{5} x^{5}+16 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{6} x^{6}+16 c^{7}+32 c^{6} d x -24 c^{5} d^{2} x^{2}-80 c^{4} d^{3} x^{3}-10 c^{3} d^{4} x^{4}+60 c^{2} d^{5} x^{5}+25 c \,d^{6} x^{6}-10 d^{7} x^{7}}{45 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{8} \left (d^{6} x^{6}+2 c \,d^{5} x^{5}-c^{2} d^{4} x^{4}-4 c^{3} d^{3} x^{3}-c^{4} d^{2} x^{2}+2 c^{5} d x +c^{6}\right )} \] Input:

int(x^7/(d*x+c)^2/(-d^2*x^2+c^2)^(7/2),x)
 

Output:

(16*sqrt(c**2 - d**2*x**2)*c**6 + 32*sqrt(c**2 - d**2*x**2)*c**5*d*x - 16* 
sqrt(c**2 - d**2*x**2)*c**4*d**2*x**2 - 64*sqrt(c**2 - d**2*x**2)*c**3*d** 
3*x**3 - 16*sqrt(c**2 - d**2*x**2)*c**2*d**4*x**4 + 32*sqrt(c**2 - d**2*x* 
*2)*c*d**5*x**5 + 16*sqrt(c**2 - d**2*x**2)*d**6*x**6 + 16*c**7 + 32*c**6* 
d*x - 24*c**5*d**2*x**2 - 80*c**4*d**3*x**3 - 10*c**3*d**4*x**4 + 60*c**2* 
d**5*x**5 + 25*c*d**6*x**6 - 10*d**7*x**7)/(45*sqrt(c**2 - d**2*x**2)*c*d* 
*8*(c**6 + 2*c**5*d*x - c**4*d**2*x**2 - 4*c**3*d**3*x**3 - c**2*d**4*x**4 
 + 2*c*d**5*x**5 + d**6*x**6))