\(\int \frac {(d+e x)^3}{(f+g x) (d^2-e^2 x^2)^{7/2}} \, dx\) [52]

Optimal result

Integrand size = 31, antiderivative size = 242 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d^3 g^2+e \left (2 e^2 f^2+9 d e f g+22 d^2 g^2\right ) x}{15 d^3 (e f+d g)^3 \sqrt {d^2-e^2 x^2}}+\frac {g^3 \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{(e f+d g)^3 \sqrt {e^2 f^2-d^2 g^2}} \] Output:

4/5*d*(e*x+d)/(d*g+e*f)/(-e^2*x^2+d^2)^(5/2)-1/15*(5*d*(-d*g+e*f)-e*(11*d* 
g+e*f)*x)/d/(d*g+e*f)^2/(-e^2*x^2+d^2)^(3/2)+1/15*(15*d^3*g^2+e*(22*d^2*g^ 
2+9*d*e*f*g+2*e^2*f^2)*x)/d^3/(d*g+e*f)^3/(-e^2*x^2+d^2)^(1/2)+g^3*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)^3 
/(-d^2*g^2+e^2*f^2)^(1/2)
 

Mathematica [A] (verified)

Time = 10.42 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\left (-e^2 f^2+d^2 g^2\right ) (d+e x) \left (32 d^4 g^2+2 e^4 f^2 x^2+3 d^3 e g (8 f-17 g x)+3 d e^3 f x (-2 f+3 g x)+d^2 e^2 \left (7 f^2-27 f g x+22 g^2 x^2\right )\right )}{d^3 (d-e x)^2 \sqrt {d^2-e^2 x^2}}-15 g^3 \sqrt {e^2 f^2-d^2 g^2} \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{15 (-e f+d g) (e f+d g)^4} \] Input:

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

Output:

(((-(e^2*f^2) + d^2*g^2)*(d + e*x)*(32*d^4*g^2 + 2*e^4*f^2*x^2 + 3*d^3*e*g 
*(8*f - 17*g*x) + 3*d*e^3*f*x*(-2*f + 3*g*x) + d^2*e^2*(7*f^2 - 27*f*g*x + 
 22*g^2*x^2)))/(d^3*(d - e*x)^2*Sqrt[d^2 - e^2*x^2]) - 15*g^3*Sqrt[e^2*f^2 
 - 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])])/(15*(-(e*f) + d*g)*(e*f + d*g)^4)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.34, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {713, 27, 686, 25, 27, 686, 27, 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.

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

\(\Big \downarrow \) 713

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {-\frac {\int -\frac {d e^2 \left ((e f-d g) \left (2 e^2 f^2+7 d e g f+15 d^2 g^2\right )+2 e g (e f-d g) (e f+11 d g) x\right )}{(f+g x) \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2 e^2 \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {d e^2 (e f-d g) \left (2 e^2 f^2+7 d e g f+15 d^2 g^2+2 e g (e f+11 d g) x\right )}{(f+g x) \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2 e^2 \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(e f-d g) \int \frac {2 e^2 f^2+7 d e g f+15 d^2 g^2+2 e g (e f+11 d g) x}{(f+g x) \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

\(\Big \downarrow \) 686

\(\displaystyle \frac {\frac {(e f-d g) \left (\frac {(e f-d g) \left (15 d^3 g^2+e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )\right )}{d^2 \sqrt {d^2-e^2 x^2} \left (e^2 f^2-d^2 g^2\right )}-\frac {\int -\frac {15 d^3 e^2 g^3 (e f-d g)}{(f+g x) \sqrt {d^2-e^2 x^2}}dx}{d^2 e^2 \left (e^2 f^2-d^2 g^2\right )}\right )}{3 d \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(e f-d g) \left (\frac {15 d g^3 (e f-d g) \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}}dx}{e^2 f^2-d^2 g^2}+\frac {(e f-d g) \left (15 d^3 g^2+e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )\right )}{d^2 \sqrt {d^2-e^2 x^2} \left (e^2 f^2-d^2 g^2\right )}\right )}{3 d \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {(e f-d g) \left (\frac {(e f-d g) \left (15 d^3 g^2+e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )\right )}{d^2 \sqrt {d^2-e^2 x^2} \left (e^2 f^2-d^2 g^2\right )}-\frac {15 d g^3 (e f-d g) \int \frac {1}{-e^2 f^2+d^2 g^2-\frac {\left (g d^2+e^2 f x\right )^2}{d^2-e^2 x^2}}d\frac {g d^2+e^2 f x}{\sqrt {d^2-e^2 x^2}}}{e^2 f^2-d^2 g^2}\right )}{3 d \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {(e f-d g) \left (\frac {15 d g^3 (e f-d g) \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )}{\left (e^2 f^2-d^2 g^2\right )^{3/2}}+\frac {(e f-d g) \left (15 d^3 g^2+e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )\right )}{d^2 \sqrt {d^2-e^2 x^2} \left (e^2 f^2-d^2 g^2\right )}\right )}{3 d \left (e^2 f^2-d^2 g^2\right )}-\frac {5 d (e f-d g)^2-e x (e f-d g) (11 d g+e f)}{3 d \left (d^2-e^2 x^2\right )^{3/2} \left (e^2 f^2-d^2 g^2\right )}}{5 (d g+e f)}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}\)

Input:

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

Output:

(4*d*(d + e*x))/(5*(e*f + d*g)*(d^2 - e^2*x^2)^(5/2)) + (-1/3*(5*d*(e*f - 
d*g)^2 - e*(e*f - d*g)*(e*f + 11*d*g)*x)/(d*(e^2*f^2 - d^2*g^2)*(d^2 - e^2 
*x^2)^(3/2)) + ((e*f - d*g)*(((e*f - d*g)*(15*d^3*g^2 + e*(2*e^2*f^2 + 9*d 
*e*f*g + 22*d^2*g^2)*x))/(d^2*(e^2*f^2 - d^2*g^2)*Sqrt[d^2 - e^2*x^2]) + ( 
15*d*g^3*(e*f - d*g)*ArcTan[(d^2*g + e^2*f*x)/(Sqrt[e^2*f^2 - d^2*g^2]*Sqr 
t[d^2 - e^2*x^2])])/(e^2*f^2 - d^2*g^2)^(3/2)))/(3*d*(e^2*f^2 - d^2*g^2))) 
/(5*(e*f + d*g))
 

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 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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1661\) vs. \(2(224)=448\).

Time = 1.07 (sec) , antiderivative size = 1662, normalized size of antiderivative = 6.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1662\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (223) = 446\).

Time = 0.14 (sec) , antiderivative size = 1767, normalized size of antiderivative = 7.30 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/15*(7*d^3*e^4*f^4 + 24*d^4*e^3*f^3*g + 25*d^5*e^2*f^2*g^2 - 24*d^6*e*f* 
g^3 - 32*d^7*g^4 - (7*e^7*f^4 + 24*d*e^6*f^3*g + 25*d^2*e^5*f^2*g^2 - 24*d 
^3*e^4*f*g^3 - 32*d^4*e^3*g^4)*x^3 + 3*(7*d*e^6*f^4 + 24*d^2*e^5*f^3*g + 2 
5*d^3*e^4*f^2*g^2 - 24*d^4*e^3*f*g^3 - 32*d^5*e^2*g^4)*x^2 + 15*(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)*sqrt(-e^2*f^2 + d^2* 
g^2)*log((d*e^2*f*g*x + d^3*g^2 - sqrt(-e^2*f^2 + d^2*g^2)*(e^2*f*x + d^2* 
g + sqrt(-e^2*x^2 + d^2)*d*g) - (e^2*f^2 - d^2*g^2)*sqrt(-e^2*x^2 + d^2))/ 
(g*x + f)) - 3*(7*d^2*e^5*f^4 + 24*d^3*e^4*f^3*g + 25*d^4*e^3*f^2*g^2 - 24 
*d^5*e^2*f*g^3 - 32*d^6*e*g^4)*x + (7*d^2*e^4*f^4 + 24*d^3*e^3*f^3*g + 25* 
d^4*e^2*f^2*g^2 - 24*d^5*e*f*g^3 - 32*d^6*g^4 + (2*e^6*f^4 + 9*d*e^5*f^3*g 
 + 20*d^2*e^4*f^2*g^2 - 9*d^3*e^3*f*g^3 - 22*d^4*e^2*g^4)*x^2 - 3*(2*d*e^5 
*f^4 + 9*d^2*e^4*f^3*g + 15*d^3*e^3*f^2*g^2 - 9*d^4*e^2*f*g^3 - 17*d^5*e*g 
^4)*x)*sqrt(-e^2*x^2 + d^2))/(d^6*e^5*f^5 + 3*d^7*e^4*f^4*g + 2*d^8*e^3*f^ 
3*g^2 - 2*d^9*e^2*f^2*g^3 - 3*d^10*e*f*g^4 - d^11*g^5 - (d^3*e^8*f^5 + 3*d 
^4*e^7*f^4*g + 2*d^5*e^6*f^3*g^2 - 2*d^6*e^5*f^2*g^3 - 3*d^7*e^4*f*g^4 - d 
^8*e^3*g^5)*x^3 + 3*(d^4*e^7*f^5 + 3*d^5*e^6*f^4*g + 2*d^6*e^5*f^3*g^2 - 2 
*d^7*e^4*f^2*g^3 - 3*d^8*e^3*f*g^4 - d^9*e^2*g^5)*x^2 - 3*(d^5*e^6*f^5 + 3 
*d^6*e^5*f^4*g + 2*d^7*e^4*f^3*g^2 - 2*d^8*e^3*f^2*g^3 - 3*d^9*e^2*f*g^4 - 
 d^10*e*g^5)*x), 1/15*(7*d^3*e^4*f^4 + 24*d^4*e^3*f^3*g + 25*d^5*e^2*f^2*g 
^2 - 24*d^6*e*f*g^3 - 32*d^7*g^4 - (7*e^7*f^4 + 24*d*e^6*f^3*g + 25*d^2...
 

Sympy [F]

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

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

Maxima [F(-2)]

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

integrate((e*x+d)^3/(g*x+f)/(-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. 642 vs. \(2 (223) = 446\).

Time = 0.18 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.65 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {2 \, e g^{3} \arctan \left (\frac {d g + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} f}{e x}}{\sqrt {e^{2} f^{2} - d^{2} g^{2}}}\right )}{{\left (e^{3} f^{3} {\left | e \right |} + 3 \, d e^{2} f^{2} g {\left | e \right |} + 3 \, d^{2} e f g^{2} {\left | e \right |} + d^{3} g^{3} {\left | e \right |}\right )} \sqrt {e^{2} f^{2} - d^{2} g^{2}}} + \frac {2 \, {\left (7 \, e^{3} f^{2} + 24 \, d e^{2} f g + 32 \, d^{2} e g^{2} - \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e f^{2}}{x} - \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d f g}{x} - \frac {115 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} g^{2}}{e x} + \frac {40 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} f^{2}}{e x^{2}} + \frac {135 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d f g}{e^{2} x^{2}} + \frac {185 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} g^{2}}{e^{3} x^{2}} - \frac {30 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} f^{2}}{e^{3} x^{3}} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d f g}{e^{4} x^{3}} - \frac {135 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2} g^{2}}{e^{5} x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} f^{2}}{e^{5} x^{4}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d f g}{e^{6} x^{4}} + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2} g^{2}}{e^{7} x^{4}}\right )}}{15 \, {\left (d^{3} e^{3} f^{3} {\left | e \right |} + 3 \, d^{4} e^{2} f^{2} g {\left | e \right |} + 3 \, d^{5} e f g^{2} {\left | e \right |} + d^{6} g^{3} {\left | e \right |}\right )} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5}} \] Input:

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

Output:

-2*e*g^3*arctan((d*g + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*f/(e*x))/sqrt(e 
^2*f^2 - d^2*g^2))/((e^3*f^3*abs(e) + 3*d*e^2*f^2*g*abs(e) + 3*d^2*e*f*g^2 
*abs(e) + d^3*g^3*abs(e))*sqrt(e^2*f^2 - d^2*g^2)) + 2/15*(7*e^3*f^2 + 24* 
d*e^2*f*g + 32*d^2*e*g^2 - 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e*f^2/x 
- 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*f*g/x - 115*(d*e + sqrt(-e^2*x^ 
2 + d^2)*abs(e))*d^2*g^2/(e*x) + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2* 
f^2/(e*x^2) + 135*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*f*g/(e^2*x^2) + 
185*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*g^2/(e^3*x^2) - 30*(d*e + sq 
rt(-e^2*x^2 + d^2)*abs(e))^3*f^2/(e^3*x^3) - 105*(d*e + sqrt(-e^2*x^2 + d^ 
2)*abs(e))^3*d*f*g/(e^4*x^3) - 135*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d 
^2*g^2/(e^5*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*f^2/(e^5*x^4) 
+ 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d*f*g/(e^6*x^4) + 45*(d*e + sqr 
t(-e^2*x^2 + d^2)*abs(e))^4*d^2*g^2/(e^7*x^4))/((d^3*e^3*f^3*abs(e) + 3*d^ 
4*e^2*f^2*g*abs(e) + 3*d^5*e*f*g^2*abs(e) + d^6*g^3*abs(e))*((d*e + sqrt(- 
e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1144, normalized size of antiderivative = 4.73 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(32*sqrt(d**2 - e**2*x**2)*d**6*g**4 + 24*sqrt(d**2 - e**2*x**2)*d**5*e*f* 
g**3 - 51*sqrt(d**2 - e**2*x**2)*d**5*e*g**4*x - 25*sqrt(d**2 - e**2*x**2) 
*d**4*e**2*f**2*g**2 - 27*sqrt(d**2 - e**2*x**2)*d**4*e**2*f*g**3*x + 22*s 
qrt(d**2 - e**2*x**2)*d**4*e**2*g**4*x**2 - 24*sqrt(d**2 - e**2*x**2)*d**3 
*e**3*f**3*g + 45*sqrt(d**2 - e**2*x**2)*d**3*e**3*f**2*g**2*x + 9*sqrt(d* 
*2 - e**2*x**2)*d**3*e**3*f*g**3*x**2 - 7*sqrt(d**2 - e**2*x**2)*d**2*e**4 
*f**4 + 27*sqrt(d**2 - e**2*x**2)*d**2*e**4*f**3*g*x - 20*sqrt(d**2 - e**2 
*x**2)*d**2*e**4*f**2*g**2*x**2 + 6*sqrt(d**2 - e**2*x**2)*d*e**5*f**4*x - 
 9*sqrt(d**2 - e**2*x**2)*d*e**5*f**3*g*x**2 - 2*sqrt(d**2 - e**2*x**2)*e* 
*6*f**4*x**2 + 15*sqrt(d**2*g**2 - e**2*f**2)*log( - sqrt(d**2*g**2 - e**2 
*f**2)*sqrt(d**2 - e**2*x**2) + d**2*g + e**2*f*x)*d**6*g**3 - 45*sqrt(d** 
2*g**2 - e**2*f**2)*log( - sqrt(d**2*g**2 - e**2*f**2)*sqrt(d**2 - e**2*x* 
*2) + d**2*g + e**2*f*x)*d**5*e*g**3*x + 45*sqrt(d**2*g**2 - e**2*f**2)*lo 
g( - sqrt(d**2*g**2 - e**2*f**2)*sqrt(d**2 - e**2*x**2) + d**2*g + e**2*f* 
x)*d**4*e**2*g**3*x**2 - 15*sqrt(d**2*g**2 - e**2*f**2)*log( - sqrt(d**2*g 
**2 - e**2*f**2)*sqrt(d**2 - e**2*x**2) + d**2*g + e**2*f*x)*d**3*e**3*g** 
3*x**3 - 15*sqrt(d**2*g**2 - e**2*f**2)*log(f + g*x)*d**6*g**3 + 45*sqrt(d 
**2*g**2 - e**2*f**2)*log(f + g*x)*d**5*e*g**3*x - 45*sqrt(d**2*g**2 - e** 
2*f**2)*log(f + g*x)*d**4*e**2*g**3*x**2 + 15*sqrt(d**2*g**2 - e**2*f**2)* 
log(f + g*x)*d**3*e**3*g**3*x**3)/(15*d**3*(d**8*g**5 + 3*d**7*e*f*g**4...