\(\int \frac {(d+e x)^3}{(f+g x)^3 (d^2-e^2 x^2)^{7/2}} \, dx\) [587]

Optimal result

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

[Out]

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)+1/2*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)+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)

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1661, 1665, 821, 739, 210} \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt {e^2 f^2-d^2 g^2}}+\frac {3 e g^4 \sqrt {d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac {e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}+\frac {e^2 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt {d^2-e^2 x^2} (d g+e f)^5} \]

[In]

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

[Out]

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

Rule 210

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 739

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

Rule 821

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] + Dist[(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] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 1665

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

Rubi steps \begin{align*} \text {integral}& = \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 {\int \frac {\frac {d^3 e^2 \left (e^3 f^3+15 d e^2 f^2 g+15 d^2 e f g^2+5 d^3 g^3\right )}{(e f+d g)^3}-\frac {d^2 e^3 \left (5 e^3 f^3-33 d e^2 f^2 g-45 d^2 e f g^2-15 d^3 g^3\right ) x}{(e f+d g)^3}+\frac {4 d^3 e^4 g^2 (12 e f+5 d g) x^2}{(e f+d g)^3}+\frac {16 d^3 e^5 g^3 x^3}{(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 (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 {\int \frac {\frac {d^3 e^4 \left (2 e^4 f^4+17 d e^3 f^3 g+90 d^2 e^2 f^2 g^2+60 d^3 e f g^3+15 d^4 g^4\right )}{(e f+d g)^4}+\frac {3 d^3 e^5 g \left (2 e^2 f^2+45 d e f g+15 d^2 g^2\right ) x}{(e f+d g)^3}+\frac {3 d^3 e^6 g^2 \left (2 e^2 f^2+57 d e f g+25 d^2 g^2\right ) x^2}{(e f+d g)^4}+\frac {2 d^3 e^7 g^3 (e f+31 d g) x^3}{(e f+d g)^4}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4} \\ & = \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 {\int \frac {\frac {15 d^6 e^6 g^3 \left (10 e^2 f^2+5 d e f g+d^2 g^2\right )}{(e f+d g)^5}+\frac {45 d^6 e^7 g^4 (5 e f+d g) x}{(e f+d g)^5}+\frac {90 d^6 e^8 g^5 x^2}{(e f+d g)^5}}{(f+g x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6} \\ & = \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 {\int \frac {\frac {30 d^6 e^7 g^3 \left (10 e^2 f^2-5 d e f g-3 d^2 g^2\right )}{(e f+d g)^4}+\frac {15 d^6 e^8 g^4 (11 e f-13 d g) x}{(e f+d g)^4}}{(f+g x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^6 e^6 \left (e^2 f^2-d^2 g^2\right )} \\ & = \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 {\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{2 (e f-d g)^2 (e f+d g)^5} \\ & = \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 {\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\right ) \text {Subst}\left (\int \frac {1}{-e^2 f^2+d^2 g^2-x^2} \, dx,x,\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5} \\ & = \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 ) \tan ^{-1}\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.99 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 e^2 (e f+d g)^2}{d (d-e x)^3}+\frac {2 e^2 (e f+d g) (2 e f+17 d g)}{d^2 (d-e x)^2}+\frac {2 e^2 \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right )}{d^3 (d-e x)}+\frac {15 g^4 (e f+d g)}{(e f-d g) (f+g x)^2}+\frac {45 e g^4 (3 e f-2 d g)}{(e f-d g)^2 (f+g x)}\right )-\frac {15 i e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \log \left (\frac {4 (e f-d g)^2 (e f+d g)^5 \left (i d^2 g+i e^2 f x+\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}\right )}{e^2 g^2 \sqrt {e^2 f^2-d^2 g^2} \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) (f+g x)}\right )}{(e f-d g)^2 \sqrt {e^2 f^2-d^2 g^2}}}{30 (e f+d g)^5} \]

[In]

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

[Out]

(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*Sq
rt[e^2*f^2 - d^2*g^2]))/(30*(e*f + d*g)^5)

Maple [B] (verified)

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

Time = 0.48 (sec) , antiderivative size = 6396, normalized size of antiderivative = 16.07

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

[In]

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

[Out]

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 = 2.54 (sec) , antiderivative size = 5361, normalized size of antiderivative = 13.47 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)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 = 0.43 (sec) , antiderivative size = 1401, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(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 + sqrt(-e^2*x^2 + d^2)*abs(e))*f/(e*x))
/sqrt(e^2*f^2 - d^2*g^2))/((e^7*f^7*abs(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
*abs(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 + sqrt(-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 - 145*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e^2*f*g/x - 485*(d*e +
sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e*g^2/x + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e*f^2/x^2 + 245*(d*e + sqr
t(-e^2*x^2 + d^2)*abs(e))^2*d*f*g/x^2 + 745*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*g^2/(e*x^2) - 30*(d*e +
sqrt(-e^2*x^2 + d^2)*abs(e))^3*f^2/(e*x^3) - 195*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*f*g/(e^2*x^3) - 525*(
d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2*g^2/(e^3*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*f^2/(e^3*x^4
) + 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d*f*g/(e^4*x^4) + 150*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^2*g
^2/(e^5*x^4))/((d^3*e^5*f^5*abs(e) + 5*d^4*e^4*f^4*g*abs(e) + 10*d^5*e^3*f^3*g^2*abs(e) + 10*d^6*e^2*f^2*g^3*a
bs(e) + 5*d^7*e*f*g^4*abs(e) + d^8*g^5*abs(e))*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5)

Mupad [F(-1)]

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

[In]

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

[Out]

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