\(\int \frac {(d+e x)^3}{(f+g x)^3 (d^2-e^2 x^2)^{7/2}} \, dx\) [54]

Optimal result

Integrand size = 31, antiderivative size = 398 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5 \sqrt {e^2 f^2-d^2 g^2}} \] Output:

4/5*d*e^2*(e*x+d)/(d*g+e*f)^3/(-e^2*x^2+d^2)^(5/2)-1/15*e^2*(5*d*(-5*d*g+e 
*f)-e*(31*d*g+e*f)*x)/d/(d*g+e*f)^4/(-e^2*x^2+d^2)^(3/2)+1/15*e^2*(90*d^3* 
g^2+e*(107*d^2*g^2+19*d*e*f*g+2*e^2*f^2)*x)/d^3/(d*g+e*f)^5/(-e^2*x^2+d^2) 
^(1/2)+1/2*g^4*(-e^2*x^2+d^2)^(1/2)/(-d*g+e*f)/(d*g+e*f)^4/(g*x+f)^2+3/2*e 
*g^4*(-2*d*g+3*e*f)*(-e^2*x^2+d^2)^(1/2)/(-d*g+e*f)^2/(d*g+e*f)^5/(g*x+f)+ 
1/2*e^2*g^3*(13*d^2*g^2-30*d*e*f*g+20*e^2*f^2)*arctan((e^2*f*x+d^2*g)/(-d^ 
2*g^2+e^2*f^2)^(1/2)/(-e^2*x^2+d^2)^(1/2))/(-d*g+e*f)^2/(d*g+e*f)^5/(-d^2* 
g^2+e^2*f^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.46 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 e^2 (e f+d g)^2}{d (d-e x)^3}+\frac {2 e^2 (e f+d g) (2 e f+17 d g)}{d^2 (d-e x)^2}+\frac {2 e^2 \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right )}{d^3 (d-e x)}+\frac {15 g^4 (e f+d g)}{(e f-d g) (f+g x)^2}+\frac {45 e g^4 (3 e f-2 d g)}{(e f-d g)^2 (f+g x)}\right )-\frac {15 i e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \log \left (\frac {4 (e f-d g)^2 (e f+d g)^5 \left (i d^2 g+i e^2 f x+\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}\right )}{e^2 g^2 \sqrt {e^2 f^2-d^2 g^2} \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) (f+g x)}\right )}{(e f-d g)^2 \sqrt {e^2 f^2-d^2 g^2}}}{30 (e f+d g)^5} \] Input:

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

Output:

(Sqrt[d^2 - e^2*x^2]*((6*e^2*(e*f + d*g)^2)/(d*(d - e*x)^3) + (2*e^2*(e*f 
+ d*g)*(2*e*f + 17*d*g))/(d^2*(d - e*x)^2) + (2*e^2*(2*e^2*f^2 + 19*d*e*f* 
g + 107*d^2*g^2))/(d^3*(d - e*x)) + (15*g^4*(e*f + d*g))/((e*f - d*g)*(f + 
 g*x)^2) + (45*e*g^4*(3*e*f - 2*d*g))/((e*f - d*g)^2*(f + g*x))) - ((15*I) 
*e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*Log[(4*(e*f - d*g)^2*(e*f 
+ d*g)^5*(I*d^2*g + I*e^2*f*x + Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2*x^2 
]))/(e^2*g^2*Sqrt[e^2*f^2 - d^2*g^2]*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2 
)*(f + g*x))])/((e*f - d*g)^2*Sqrt[e^2*f^2 - d^2*g^2]))/(30*(e*f + d*g)^5)
 

Rubi [A] (verified)

Time = 2.95 (sec) , antiderivative size = 438, 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 = {713, 2178, 2178, 27, 2182, 27, 679, 488, 217}

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.

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

\(\Big \downarrow \) 713

\(\displaystyle \frac {\int \frac {\frac {16 d^3 g^3 x^3 e^5}{(e f+d g)^3}+\frac {4 d^3 g^2 (12 e f+5 d g) x^2 e^4}{(e f+d g)^3}-\frac {d^2 \left (5 e^3 f^3-33 d e^2 g f^2-45 d^2 e g^2 f-15 d^3 g^3\right ) x e^3}{(e f+d g)^3}+\frac {d^3 \left (e^3 f^3+15 d e^2 g f^2+15 d^2 e g^2 f+5 d^3 g^3\right ) e^2}{(e f+d g)^3}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {\frac {\int \frac {\frac {2 d^3 g^3 (e f+31 d g) x^3 e^7}{(e f+d g)^4}+\frac {3 d^3 g^2 \left (2 e^2 f^2+57 d e g f+25 d^2 g^2\right ) x^2 e^6}{(e f+d g)^4}+\frac {3 d^3 g \left (2 e^2 f^2+45 d e g f+15 d^2 g^2\right ) x e^5}{(e f+d g)^3}+\frac {d^3 \left (2 e^4 f^4+17 d e^3 g f^3+90 d^2 e^2 g^2 f^2+60 d^3 e g^3 f+15 d^4 g^4\right ) e^4}{(e f+d g)^4}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {\frac {\frac {\int \frac {15 \left (\frac {6 d^6 g^5 x^2 e^8}{(e f+d g)^5}+\frac {3 d^6 g^4 (5 e f+d g) x e^7}{(e f+d g)^5}+\frac {d^6 g^3 \left (10 e^2 f^2+5 d e g f+d^2 g^2\right ) e^6}{(e f+d g)^5}\right )}{(f+g x)^3 \sqrt {d^2-e^2 x^2}}dx}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {15 \int \frac {\frac {6 d^6 g^5 x^2 e^8}{(e f+d g)^5}+\frac {3 d^6 g^4 (5 e f+d g) x e^7}{(e f+d g)^5}+\frac {d^6 g^3 \left (10 e^2 f^2+5 d e g f+d^2 g^2\right ) e^6}{(e f+d g)^5}}{(f+g x)^3 \sqrt {d^2-e^2 x^2}}dx}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {\frac {15 \left (\frac {\int \frac {d^6 e^7 g^3 \left (2 \left (10 e^2 f^2-5 d e g f-3 d^2 g^2\right )+e g (11 e f-13 d g) x\right )}{(e f+d g)^4 (f+g x)^2 \sqrt {d^2-e^2 x^2}}dx}{2 \left (e^2 f^2-d^2 g^2\right )}+\frac {d^6 e^6 g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}\right )}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {15 \left (\frac {d^6 e^7 g^3 \int \frac {2 \left (10 e^2 f^2-5 d e g f-3 d^2 g^2\right )+e g (11 e f-13 d g) x}{(f+g x)^2 \sqrt {d^2-e^2 x^2}}dx}{2 (d g+e f)^4 \left (e^2 f^2-d^2 g^2\right )}+\frac {d^6 e^6 g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}\right )}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {\frac {15 \left (\frac {d^6 e^7 g^3 \left (\frac {e \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}}dx}{e f-d g}+\frac {3 g \sqrt {d^2-e^2 x^2} (3 e f-2 d g)}{(f+g x) (e f-d g)}\right )}{2 (d g+e f)^4 \left (e^2 f^2-d^2 g^2\right )}+\frac {d^6 e^6 g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}\right )}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {15 \left (\frac {d^6 e^7 g^3 \left (\frac {3 g \sqrt {d^2-e^2 x^2} (3 e f-2 d g)}{(f+g x) (e f-d g)}-\frac {e \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \int \frac {1}{-e^2 f^2+d^2 g^2-\frac {\left (g d^2+e^2 f x\right )^2}{d^2-e^2 x^2}}d\frac {g d^2+e^2 f x}{\sqrt {d^2-e^2 x^2}}}{e f-d g}\right )}{2 (d g+e f)^4 \left (e^2 f^2-d^2 g^2\right )}+\frac {d^6 e^6 g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}\right )}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {15 \left (\frac {d^6 e^7 g^3 \left (\frac {e \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )}{(e f-d g) \sqrt {e^2 f^2-d^2 g^2}}+\frac {3 g \sqrt {d^2-e^2 x^2} (3 e f-2 d g)}{(f+g x) (e f-d g)}\right )}{2 (d g+e f)^4 \left (e^2 f^2-d^2 g^2\right )}+\frac {d^6 e^6 g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}\right )}{d^2 e^2}+\frac {d e^6 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{\sqrt {d^2-e^2 x^2} (d g+e f)^5}}{3 d^2 e^2}-\frac {d e^4 (5 d (e f-5 d g)-e x (31 d g+e f))}{3 \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}}{5 d^2 e^2}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}\)

Input:

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

Output:

(4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) + (-1/3*(d*e^4 
*(5*d*(e*f - 5*d*g) - e*(e*f + 31*d*g)*x))/((e*f + d*g)^4*(d^2 - e^2*x^2)^ 
(3/2)) + ((d*e^6*(90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x) 
)/((e*f + d*g)^5*Sqrt[d^2 - e^2*x^2]) + (15*((d^6*e^6*g^4*Sqrt[d^2 - e^2*x 
^2])/(2*(e*f - d*g)*(e*f + d*g)^4*(f + g*x)^2) + (d^6*e^7*g^3*((3*g*(3*e*f 
 - 2*d*g)*Sqrt[d^2 - e^2*x^2])/((e*f - d*g)*(f + g*x)) + (e*(20*e^2*f^2 - 
30*d*e*f*g + 13*d^2*g^2)*ArcTan[(d^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2] 
*Sqrt[d^2 - e^2*x^2])])/((e*f - d*g)*Sqrt[e^2*f^2 - d^2*g^2])))/(2*(e*f + 
d*g)^4*(e^2*f^2 - d^2*g^2))))/(d^2*e^2))/(3*d^2*e^2))/(5*d^2*e^2)
 

Defintions of rubi rules used

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 217

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

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 679

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

rule 713

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x)^m*(f + g*x)^n, 
 a + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x)^m*(f + g*x)^n, a + 
c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[(d + e*x)^m*(f + g*x)^n, a 
+ c*x^2, x], x, 1]}, Simp[(a*S - c*R*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1) 
)), x] + Simp[1/(2*a*c*(p + 1))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Expan 
dToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m + (c*R*(2*p + 3))/(d + e*x)^m, x], x], 
 x]] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[n, 1] && LtQ[p, -1] && ILtQ[m, 
 0] && NeQ[c*d^2 + a*e^2, 0]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6395\) vs. \(2(370)=740\).

Time = 1.19 (sec) , antiderivative size = 6396, normalized size of antiderivative = 16.07

method	result	size
default	\(\text {Expression too large to display}\)	\(6396\)

Input:

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

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2651 vs. \(2 (369) = 738\).

Time = 2.17 (sec) , antiderivative size = 5361, normalized size of antiderivative = 13.47 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1401 vs. \(2 (369) = 738\).

Time = 0.31 (sec) , antiderivative size = 1401, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-(20*e^5*f^2*g^3 - 30*d*e^4*f*g^4 + 13*d^2*e^3*g^5)*arctan((d*g + (d*e + s 
qrt(-e^2*x^2 + d^2)*abs(e))*f/(e*x))/sqrt(e^2*f^2 - d^2*g^2))/((e^7*f^7*ab 
s(e) + 3*d*e^6*f^6*g*abs(e) + d^2*e^5*f^5*g^2*abs(e) - 5*d^3*e^4*f^4*g^3*a 
bs(e) - 5*d^4*e^3*f^3*g^4*abs(e) + d^5*e^2*f^2*g^5*abs(e) + 3*d^6*e*f*g^6* 
abs(e) + d^7*g^7*abs(e))*sqrt(e^2*f^2 - d^2*g^2)) - (10*d*e^5*f^4*g^4 - 6* 
d^2*e^4*f^3*g^5 - d^3*e^3*f^2*g^6 + 29*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) 
*d^2*e^2*f^3*g^5/x - 18*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^3*e*f^2*g^6/ 
x - 2*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*f*g^7/x + 10*(d*e + sqrt(-e^ 
2*x^2 + d^2)*abs(e))^2*d*e*f^4*g^4/x^2 - 6*(d*e + sqrt(-e^2*x^2 + d^2)*abs 
(e))^2*d^2*f^3*g^5/x^2 + 19*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^3*f^2* 
g^6/(e*x^2) - 12*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^4*f*g^7/(e^2*x^2) 
 - 2*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^5*g^8/(e^3*x^2) + 11*(d*e + s 
qrt(-e^2*x^2 + d^2)*abs(e))^3*d^2*f^3*g^5/(e^2*x^3) - 6*(d*e + sqrt(-e^2*x 
^2 + d^2)*abs(e))^3*d^3*f^2*g^6/(e^3*x^3) - 2*(d*e + sqrt(-e^2*x^2 + d^2)* 
abs(e))^3*d^4*f*g^7/(e^4*x^3))/((e^7*f^9*abs(e) + 3*d*e^6*f^8*g*abs(e) + d 
^2*e^5*f^7*g^2*abs(e) - 5*d^3*e^4*f^6*g^3*abs(e) - 5*d^4*e^3*f^5*g^4*abs(e 
) + d^5*e^2*f^4*g^5*abs(e) + 3*d^6*e*f^3*g^6*abs(e) + d^7*f^2*g^7*abs(e))* 
(e*f + 2*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*g/(e^2*x) + (d*e + sqrt(-e^ 
2*x^2 + d^2)*abs(e))^2*f/(e^3*x^2))^2) + 2/15*(7*e^5*f^2 + 44*d*e^4*f*g + 
127*d^2*e^3*g^2 - 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3*f^2/x - 14...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 9329, normalized size of antiderivative = 23.44 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(e*( - 780*sqrt( - d**2*g**2 + e**2*f**2)*atan((tan(asin((e*x)/d)/2)*e*f + 
 d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin((e*x)/d)/2)**9*d**6*e**3*f* 
*3*g**6 + 2775*sqrt( - d**2*g**2 + e**2*f**2)*atan((tan(asin((e*x)/d)/2)*e 
*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin((e*x)/d)/2)**9*d**5*e** 
4*f**4*g**5 - 3450*sqrt( - d**2*g**2 + e**2*f**2)*atan((tan(asin((e*x)/d)/ 
2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin((e*x)/d)/2)**9*d**4 
*e**5*f**5*g**4 + 1500*sqrt( - d**2*g**2 + e**2*f**2)*atan((tan(asin((e*x) 
/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin((e*x)/d)/2)**9* 
d**3*e**6*f**6*g**3 - 3120*sqrt( - d**2*g**2 + e**2*f**2)*atan((tan(asin(( 
e*x)/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin((e*x)/d)/2) 
**8*d**7*e**2*f**2*g**7 + 15000*sqrt( - d**2*g**2 + e**2*f**2)*atan((tan(a 
sin((e*x)/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin((e*x)/ 
d)/2)**8*d**6*e**3*f**3*g**6 - 27675*sqrt( - d**2*g**2 + e**2*f**2)*atan(( 
tan(asin((e*x)/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(asin(( 
e*x)/d)/2)**8*d**5*e**4*f**4*g**5 + 23250*sqrt( - d**2*g**2 + e**2*f**2)*a 
tan((tan(asin((e*x)/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*tan(a 
sin((e*x)/d)/2)**8*d**4*e**5*f**5*g**4 - 7500*sqrt( - d**2*g**2 + e**2*f** 
2)*atan((tan(asin((e*x)/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f**2))*t 
an(asin((e*x)/d)/2)**8*d**3*e**6*f**6*g**3 - 3120*sqrt( - d**2*g**2 + e**2 
*f**2)*atan((tan(asin((e*x)/d)/2)*e*f + d*g)/sqrt( - d**2*g**2 + e**2*f...