\(\int \frac {(d+e x)^3 (f+g x)^3}{(d^2-e^2 x^2)^{7/2}} \, dx\) [48]

Optimal result

Integrand size = 31, antiderivative size = 183 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^3 (d+e x)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-13 d g) (e f+d g)^2 (d+e x)^2}{15 d^2 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g) \left (2 e^2 f^2-11 d e f g+32 d^2 g^2\right ) (d+e x)}{15 d^3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \] Output:

1/5*(d*g+e*f)^3*(e*x+d)^3/d/e^4/(-e^2*x^2+d^2)^(5/2)+1/15*(-13*d*g+2*e*f)* 
(d*g+e*f)^2*(e*x+d)^2/d^2/e^4/(-e^2*x^2+d^2)^(3/2)+1/15*(d*g+e*f)*(32*d^2* 
g^2-11*d*e*f*g+2*e^2*f^2)*(e*x+d)/d^3/e^4/(-e^2*x^2+d^2)^(1/2)-g^3*arctan( 
e*x/(-e^2*x^2+d^2)^(1/2))/e^4
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {(e f+d g) \sqrt {d^2-e^2 x^2} \left (22 d^4 g^2+2 e^4 f^2 x^2-d e^3 f x (6 f+11 g x)-d^3 e g (16 f+51 g x)+d^2 e^2 \left (7 f^2+33 f g x+32 g^2 x^2\right )\right )}{d^3 (d-e x)^3}+30 g^3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{15 e^4} \] Input:

Integrate[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]
 

Output:

(((e*f + d*g)*Sqrt[d^2 - e^2*x^2]*(22*d^4*g^2 + 2*e^4*f^2*x^2 - d*e^3*f*x* 
(6*f + 11*g*x) - d^3*e*g*(16*f + 51*g*x) + d^2*e^2*(7*f^2 + 33*f*g*x + 32* 
g^2*x^2)))/(d^3*(d - e*x)^3) + 30*g^3*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - 
 e^2*x^2])])/(15*e^4)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {691, 25, 2166, 25, 27, 665, 27, 224, 216}

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.

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

\(\Big \downarrow \) 691

\(\displaystyle \frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {(d+e x)^2 \left (-\frac {5 d x^2 g^3}{e}-\frac {5 d (3 e f+d g) x g^2}{e^2}+\frac {2 e^3 f^3-9 d e^2 g f^2-9 d^2 e g^2 f-3 d^3 g^3}{e^3}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(d+e x)^2 \left (-\frac {5 d x^2 g^3}{e}-\frac {5 d (3 e f+d g) x g^2}{e^2}+\frac {2 e^3 f^3-9 d e^2 g f^2-9 d^2 e g^2 f-3 d^3 g^3}{e^3}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {(d+e x) \left (2 e^3 f^3-9 d e^2 g f^2+21 d^2 e g^2 f+17 d^3 g^3+15 d^2 e g^3 x\right )}{e^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (2 e^3 f^3-9 d e^2 g f^2+21 d^2 e g^2 f+17 d^3 g^3+15 d^2 e g^3 x\right )}{e^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (2 e^3 f^3-9 d e^2 g f^2+21 d^2 e g^2 f+17 d^3 g^3+15 d^2 e g^3 x\right )}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d e^3}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 665

\(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{d e \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^2 e g^3}{\sqrt {d^2-e^2 x^2}}dx}{e}}{3 d e^3}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{d e \sqrt {d^2-e^2 x^2}}-15 d^2 g^3 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx}{3 d e^3}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{d e \sqrt {d^2-e^2 x^2}}-15 d^2 g^3 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}}{3 d e^3}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f) \left (32 d^2 g^2-11 d e f g+2 e^2 f^2\right )}{d e \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 g^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}}{3 d e^3}+\frac {(d+e x)^2 (2 e f-13 d g) (d g+e f)^2}{3 d e^4 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^3}{5 d e^4 \left (d^2-e^2 x^2\right )^{5/2}}\)

Input:

Int[((d + e*x)^3*(f + g*x)^3)/(d^2 - e^2*x^2)^(7/2),x]
 

Output:

((e*f + d*g)^3*(d + e*x)^3)/(5*d*e^4*(d^2 - e^2*x^2)^(5/2)) + (((2*e*f - 1 
3*d*g)*(e*f + d*g)^2*(d + e*x)^2)/(3*d*e^4*(d^2 - e^2*x^2)^(3/2)) + (((e*f 
 + d*g)*(2*e^2*f^2 - 11*d*e*f*g + 32*d^2*g^2)*(d + e*x))/(d*e*Sqrt[d^2 - e 
^2*x^2]) - (15*d^2*g^3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e)/(3*d*e^3))/(5 
*d)
 

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 216

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

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 665

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( 
x_)^2)^(3/2), x_Symbol] :> Simp[(-2^(m - 1))*d^(m - 2)*(e*f + d*g)^n*((d + 
e*x)/(c*e^(n - 1)*Sqrt[a + c*x^2])), x] + Simp[1/(c*e^(n - 2))   Int[Expand 
ToSum[(2^(m - 1)*d^(m - 1)*(e*f + d*g)^n - e^n*(d + e*x)^(m - 1)*(f + g*x)^ 
n)/(d - e*x), x]/Sqrt[a + c*x^2], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && 
 EqQ[c*d^2 + a*e^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
 

rule 691

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d* 
x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, Simp[(-d)*R*( 
d + e*x)^m*((a + c*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[d/(2*a*(p + 1)) 
   Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q + 
R*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[n, 1] 
&& IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 + a*e^2, 0]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(169)=338\).

Time = 1.01 (sec) , antiderivative size = 667, normalized size of antiderivative = 3.64

method	result	size
default	\(d^{3} f^{3} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+3 g^{2} e^{2} \left (d g +e f \right ) \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+\frac {3 d^{2} f^{2} \left (d g +e f \right )}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+3 e g \left (d^{2} g^{2}+3 d e f g +e^{2} f^{2}\right ) \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+3 d f \left (d^{2} g^{2}+3 d e f g +e^{2} f^{2}\right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\left (d^{3} g^{3}+9 d^{2} e f \,g^{2}+9 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+e^{3} g^{3} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )\)	\(667\)

Input:

int((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

d^3*f^3*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^ 
(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)))+3*g^2*e^2*(d*g+e*f)*(x^4/e^2/(-e^2* 
x^2+d^2)^(5/2)-4*d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(- 
e^2*x^2+d^2)^(5/2)))+3/5*d^2*f^2*(d*g+e*f)/e^2/(-e^2*x^2+d^2)^(5/2)+3*e*g* 
(d^2*g^2+3*d*e*f*g+e^2*f^2)*(1/2*x^3/e^2/(-e^2*x^2+d^2)^(5/2)-3/2*d^2/e^2* 
(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2 
)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2))) 
))+3*d*f*(d^2*g^2+3*d*e*f*g+e^2*f^2)*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d 
^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^( 
3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2))))+(d^3*g^3+9*d^2*e*f*g^2+9*d*e^2*f^2* 
g+e^3*f^3)*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^( 
5/2))+e^3*g^3*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/e^2*(1/3*x^3/e^2/(-e^2*x 
^2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan(( 
e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (169) = 338\).

Time = 0.11 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 \, d^{3} e^{3} f^{3} - 9 \, d^{4} e^{2} f^{2} g + 6 \, d^{5} e f g^{2} + 22 \, d^{6} g^{3} - {\left (7 \, e^{6} f^{3} - 9 \, d e^{5} f^{2} g + 6 \, d^{2} e^{4} f g^{2} + 22 \, d^{3} e^{3} g^{3}\right )} x^{3} + 3 \, {\left (7 \, d e^{5} f^{3} - 9 \, d^{2} e^{4} f^{2} g + 6 \, d^{3} e^{3} f g^{2} + 22 \, d^{4} e^{2} g^{3}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{4} f^{3} - 9 \, d^{3} e^{3} f^{2} g + 6 \, d^{4} e^{2} f g^{2} + 22 \, d^{5} e g^{3}\right )} x - 30 \, {\left (d^{3} e^{3} g^{3} x^{3} - 3 \, d^{4} e^{2} g^{3} x^{2} + 3 \, d^{5} e g^{3} x - d^{6} g^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (7 \, d^{2} e^{3} f^{3} - 9 \, d^{3} e^{2} f^{2} g + 6 \, d^{4} e f g^{2} + 22 \, d^{5} g^{3} + {\left (2 \, e^{5} f^{3} - 9 \, d e^{4} f^{2} g + 21 \, d^{2} e^{3} f g^{2} + 32 \, d^{3} e^{2} g^{3}\right )} x^{2} - 3 \, {\left (2 \, d e^{4} f^{3} - 9 \, d^{2} e^{3} f^{2} g + 6 \, d^{3} e^{2} f g^{2} + 17 \, d^{4} e g^{3}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{7} x^{3} - 3 \, d^{4} e^{6} x^{2} + 3 \, d^{5} e^{5} x - d^{6} e^{4}\right )}} \] Input:

integrate((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
 

Output:

-1/15*(7*d^3*e^3*f^3 - 9*d^4*e^2*f^2*g + 6*d^5*e*f*g^2 + 22*d^6*g^3 - (7*e 
^6*f^3 - 9*d*e^5*f^2*g + 6*d^2*e^4*f*g^2 + 22*d^3*e^3*g^3)*x^3 + 3*(7*d*e^ 
5*f^3 - 9*d^2*e^4*f^2*g + 6*d^3*e^3*f*g^2 + 22*d^4*e^2*g^3)*x^2 - 3*(7*d^2 
*e^4*f^3 - 9*d^3*e^3*f^2*g + 6*d^4*e^2*f*g^2 + 22*d^5*e*g^3)*x - 30*(d^3*e 
^3*g^3*x^3 - 3*d^4*e^2*g^3*x^2 + 3*d^5*e*g^3*x - d^6*g^3)*arctan(-(d - sqr 
t(-e^2*x^2 + d^2))/(e*x)) + (7*d^2*e^3*f^3 - 9*d^3*e^2*f^2*g + 6*d^4*e*f*g 
^2 + 22*d^5*g^3 + (2*e^5*f^3 - 9*d*e^4*f^2*g + 21*d^2*e^3*f*g^2 + 32*d^3*e 
^2*g^3)*x^2 - 3*(2*d*e^4*f^3 - 9*d^2*e^3*f^2*g + 6*d^3*e^2*f*g^2 + 17*d^4* 
e*g^3)*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^7*x^3 - 3*d^4*e^6*x^2 + 3*d^5*e^5*x 
 - d^6*e^4)
 

Sympy [F]

\[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((e*x+d)**3*(g*x+f)**3/(-e**2*x**2+d**2)**(7/2),x)
 

Output:

Integral((d + e*x)**3*(f + g*x)**3/(-(-d + e*x)*(d + e*x))**(7/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (169) = 338\).

Time = 0.16 (sec) , antiderivative size = 903, normalized size of antiderivative = 4.93 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
 

Output:

1/15*e^3*g^3*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^ 
2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 1/3*e*g^3*x*(3 
*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4)) + 
1/5*d*f^3*x/(-e^2*x^2 + d^2)^(5/2) + 3/5*d^2*f^3/((-e^2*x^2 + d^2)^(5/2)*e 
) + 3/5*d^3*f^2*g/((-e^2*x^2 + d^2)^(5/2)*e^2) + 4/15*f^3*x/((-e^2*x^2 + d 
^2)^(3/2)*d) + 4/15*d^2*g^3*x/((-e^2*x^2 + d^2)^(3/2)*e^3) + 8/15*f^3*x/(s 
qrt(-e^2*x^2 + d^2)*d^3) - 7/15*g^3*x/(sqrt(-e^2*x^2 + d^2)*e^3) + 3*(e^3* 
f*g^2 + d*e^2*g^3)*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - g^3*arcsin(e^2*x/(d* 
sqrt(e^2)))/(sqrt(e^2)*e^3) + 3/2*(e^3*f^2*g + 3*d*e^2*f*g^2 + d^2*e*g^3)* 
x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 4*(e^3*f*g^2 + d*e^2*g^3)*d^2*x^2/((-e^ 
2*x^2 + d^2)^(5/2)*e^4) + 1/3*(e^3*f^3 + 9*d*e^2*f^2*g + 9*d^2*e*f*g^2 + d 
^3*g^3)*x^2/((-e^2*x^2 + d^2)^(5/2)*e^2) - 9/10*(e^3*f^2*g + 3*d*e^2*f*g^2 
 + d^2*e*g^3)*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/5*(d*e^2*f^3 + 3*d^2* 
e*f^2*g + d^3*f*g^2)*x/((-e^2*x^2 + d^2)^(5/2)*e^2) + 8/5*(e^3*f*g^2 + d*e 
^2*g^3)*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6) - 2/15*(e^3*f^3 + 9*d*e^2*f^2*g + 
 9*d^2*e*f*g^2 + d^3*g^3)*d^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/10*(e^3*f^2 
*g + 3*d*e^2*f*g^2 + d^2*e*g^3)*x/((-e^2*x^2 + d^2)^(3/2)*e^4) - 1/5*(d*e^ 
2*f^3 + 3*d^2*e*f^2*g + d^3*f*g^2)*x/((-e^2*x^2 + d^2)^(3/2)*d^2*e^2) + 3/ 
5*(e^3*f^2*g + 3*d*e^2*f*g^2 + d^2*e*g^3)*x/(sqrt(-e^2*x^2 + d^2)*d^2*e^4) 
 - 2/5*(d*e^2*f^3 + 3*d^2*e*f^2*g + d^3*f*g^2)*x/(sqrt(-e^2*x^2 + d^2)*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (169) = 338\).

Time = 0.15 (sec) , antiderivative size = 560, normalized size of antiderivative = 3.06 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {g^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{e^{3} {\left | e \right |}} + \frac {2 \, {\left (7 \, e^{3} f^{3} - 9 \, d e^{2} f^{2} g + 6 \, d^{2} e f g^{2} + 22 \, d^{3} g^{3} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e f^{3}}{x} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d f^{2} g}{x} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} f g^{2}}{e x} - \frac {95 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{3} g^{3}}{e^{2} x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} f^{3}}{e x^{2}} - \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d f^{2} g}{e^{2} x^{2}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} f g^{2}}{e^{3} x^{2}} + \frac {145 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{3} g^{3}}{e^{4} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} f^{3}}{e^{3} x^{3}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d f^{2} g}{e^{4} x^{3}} - \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} g^{3}}{e^{6} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} f^{3}}{e^{5} x^{4}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} g^{3}}{e^{8} x^{4}}\right )}}{15 \, d^{3} e^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \] Input:

integrate((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
 

Output:

-g^3*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^3*abs(e)) + 2/15*(7*e^3*f^3 - 9*d*e^2* 
f^2*g + 6*d^2*e*f*g^2 + 22*d^3*g^3 - 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e) 
)*e*f^3/x + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*f^2*g/x - 30*(d*e + s 
qrt(-e^2*x^2 + d^2)*abs(e))*d^2*f*g^2/(e*x) - 95*(d*e + sqrt(-e^2*x^2 + d^ 
2)*abs(e))*d^3*g^3/(e^2*x) + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*f^3/ 
(e*x^2) - 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*f^2*g/(e^2*x^2) + 60* 
(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*f*g^2/(e^3*x^2) + 145*(d*e + sqr 
t(-e^2*x^2 + d^2)*abs(e))^2*d^3*g^3/(e^4*x^2) - 30*(d*e + sqrt(-e^2*x^2 + 
d^2)*abs(e))^3*f^3/(e^3*x^3) + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d* 
f^2*g/(e^4*x^3) - 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^3*g^3/(e^6*x^ 
3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*f^3/(e^5*x^4) + 15*(d*e + sq 
rt(-e^2*x^2 + d^2)*abs(e))^4*d^3*g^3/(e^8*x^4))/(d^3*e^3*((d*e + sqrt(-e^2 
*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \] Input:

int(((f + g*x)^3*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x)
 

Output:

int(((f + g*x)^3*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.76 \[ \int \frac {(d+e x)^3 (f+g x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^3*(g*x+f)^3/(-e^2*x^2+d^2)^(7/2),x)
 

Output:

( - 15*asin((e*x)/d)*tan(asin((e*x)/d)/2)**5*d**3*g**3 + 75*asin((e*x)/d)* 
tan(asin((e*x)/d)/2)**4*d**3*g**3 - 150*asin((e*x)/d)*tan(asin((e*x)/d)/2) 
**3*d**3*g**3 + 150*asin((e*x)/d)*tan(asin((e*x)/d)/2)**2*d**3*g**3 - 75*a 
sin((e*x)/d)*tan(asin((e*x)/d)/2)*d**3*g**3 + 15*asin((e*x)/d)*d**3*g**3 - 
 6*tan(asin((e*x)/d)/2)**5*d**3*g**3 - 6*tan(asin((e*x)/d)/2)**5*e**3*f**3 
 + 90*tan(asin((e*x)/d)/2)**3*d**3*g**3 - 90*tan(asin((e*x)/d)/2)**3*d*e** 
2*f**2*g - 230*tan(asin((e*x)/d)/2)**2*d**3*g**3 - 120*tan(asin((e*x)/d)/2 
)**2*d**2*e*f*g**2 + 90*tan(asin((e*x)/d)/2)**2*d*e**2*f**2*g - 20*tan(asi 
n((e*x)/d)/2)**2*e**3*f**3 + 160*tan(asin((e*x)/d)/2)*d**3*g**3 + 60*tan(a 
sin((e*x)/d)/2)*d**2*e*f*g**2 - 90*tan(asin((e*x)/d)/2)*d*e**2*f**2*g + 10 
*tan(asin((e*x)/d)/2)*e**3*f**3 - 38*d**3*g**3 - 12*d**2*e*f*g**2 + 18*d*e 
**2*f**2*g - 8*e**3*f**3)/(15*d**3*e**4*(tan(asin((e*x)/d)/2)**5 - 5*tan(a 
sin((e*x)/d)/2)**4 + 10*tan(asin((e*x)/d)/2)**3 - 10*tan(asin((e*x)/d)/2)* 
*2 + 5*tan(asin((e*x)/d)/2) - 1))