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

Optimal. Leaf size=311 \[ \frac{e \left (e x \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )+45 d^3 g^2\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^4}+\frac{e g^3 (4 e f-3 d g) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{(e f-d g) (d g+e f)^4 \sqrt{e^2 f^2-d^2 g^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(f+g x) (e f-d g) (d g+e f)^4}-\frac{e (5 d (e f-3 d g)-e x (21 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^3}+\frac{4 d e (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^2} \]

[Out]

(4*d*e*(d + e*x))/(5*(e*f + d*g)^2*(d^2 - e^2*x^2)^(5/2)) - (e*(5*d*(e*f - 3*d*g) - e*(e*f + 21*d*g)*x))/(15*d
*(e*f + d*g)^3*(d^2 - e^2*x^2)^(3/2)) + (e*(45*d^3*g^2 + e*(2*e^2*f^2 + 14*d*e*f*g + 57*d^2*g^2)*x))/(15*d^3*(
e*f + d*g)^4*Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/((e*f - d*g)*(e*f + d*g)^4*(f + g*x)) + (e*g^3*(
4*e*f - 3*d*g)*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)*(e*f + d*
g)^4*Sqrt[e^2*f^2 - d^2*g^2])

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Rubi [A]  time = 1.26448, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1647, 807, 725, 204} \[ \frac{e \left (e x \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )+45 d^3 g^2\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^4}+\frac{e g^3 (4 e f-3 d g) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{(e f-d g) (d g+e f)^4 \sqrt{e^2 f^2-d^2 g^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(f+g x) (e f-d g) (d g+e f)^4}-\frac{e (5 d (e f-3 d g)-e x (21 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^3}+\frac{4 d e (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*d*e*(d + e*x))/(5*(e*f + d*g)^2*(d^2 - e^2*x^2)^(5/2)) - (e*(5*d*(e*f - 3*d*g) - e*(e*f + 21*d*g)*x))/(15*d
*(e*f + d*g)^3*(d^2 - e^2*x^2)^(3/2)) + (e*(45*d^3*g^2 + e*(2*e^2*f^2 + 14*d*e*f*g + 57*d^2*g^2)*x))/(15*d^3*(
e*f + d*g)^4*Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/((e*f - d*g)*(e*f + d*g)^4*(f + g*x)) + (e*g^3*(
4*e*f - 3*d*g)*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)*(e*f + d*
g)^4*Sqrt[e^2*f^2 - d^2*g^2])

Rule 1647

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 807

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 725

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 204

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac{4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{\int \frac{\frac{d^3 e^2 \left (e^2 f^2+10 d e f g+5 d^2 g^2\right )}{(e f+d g)^2}-\frac{d^2 e^3 (e f-5 d g) (5 e f+3 d g) x}{(e f+d g)^2}+\frac{16 d^3 e^4 g^2 x^2}{(e f+d g)^2}}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac{4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d (e f-3 d g)-e (e f+21 d g) x)}{15 d (e f+d g)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{\int \frac{\frac{d^3 e^4 \left (2 e^3 f^3+12 d e^2 f^2 g+45 d^2 e f g^2+15 d^3 g^3\right )}{(e f+d g)^3}+\frac{d^3 e^5 g \left (4 e^2 f^2+69 d e f g+45 d^2 g^2\right ) x}{(e f+d g)^3}+\frac{2 d^3 e^6 g^2 (e f+21 d g) x^2}{(e f+d g)^3}}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac{4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d (e f-3 d g)-e (e f+21 d g) x)}{15 d (e f+d g)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e \left (45 d^3 g^2+e \left (2 e^2 f^2+14 d e f g+57 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^4 \sqrt{d^2-e^2 x^2}}+\frac{\int \frac{\frac{15 d^6 e^6 g^3 (4 e f+d g)}{(e f+d g)^4}+\frac{45 d^6 e^7 g^4 x}{(e f+d g)^4}}{(f+g x)^2 \sqrt{d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac{4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d (e f-3 d g)-e (e f+21 d g) x)}{15 d (e f+d g)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e \left (45 d^3 g^2+e \left (2 e^2 f^2+14 d e f g+57 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^4 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(e f-d g) (e f+d g)^4 (f+g x)}+\frac{\left (e g^3 (4 e f-3 d g)\right ) \int \frac{1}{(f+g x) \sqrt{d^2-e^2 x^2}} \, dx}{(e f-d g) (e f+d g)^4}\\ &=\frac{4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d (e f-3 d g)-e (e f+21 d g) x)}{15 d (e f+d g)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e \left (45 d^3 g^2+e \left (2 e^2 f^2+14 d e f g+57 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^4 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(e f-d g) (e f+d g)^4 (f+g x)}-\frac{\left (e g^3 (4 e f-3 d g)\right ) \operatorname{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 )}{(e f-d g) (e f+d g)^4}\\ &=\frac{4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (5 d (e f-3 d g)-e (e f+21 d g) x)}{15 d (e f+d g)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac{e \left (45 d^3 g^2+e \left (2 e^2 f^2+14 d e f g+57 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^4 \sqrt{d^2-e^2 x^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(e f-d g) (e f+d g)^4 (f+g x)}+\frac{e g^3 (4 e f-3 d g) \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 )}{(e f-d g) (e f+d g)^4 \sqrt{e^2 f^2-d^2 g^2}}\\ \end{align*}

Mathematica [A]  time = 0.648739, size = 341, normalized size = 1.1 \[ \frac{\frac{(d+e x) \left (e^2 f^2-d^2 g^2\right ) \left (d^4 e^2 g^2 \left (38 f^2+164 f g x+171 g^2 x^2\right )-3 d^3 e^3 g \left (19 f^2 g x-9 f^3+47 f g^2 x^2+24 g^3 x^3\right )+d^2 e^4 f \left (-29 f^2 g x+7 f^3+7 f g^2 x^2+43 g^3 x^3\right )-9 d^5 e g^3 (8 f+13 g x)+15 d^6 g^4+6 d e^5 f^2 x \left (-f^2+f g x+2 g^2 x^2\right )+2 e^6 f^3 x^2 (f+g x)\right )}{d^3 (d-e x)^2 \sqrt{d^2-e^2 x^2} (f+g x)}+15 e g^3 (4 e f-3 d g) \sqrt{e^2 f^2-d^2 g^2} \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{15 (e f-d g)^2 (d g+e f)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((e^2*f^2 - d^2*g^2)*(d + e*x)*(15*d^6*g^4 + 2*e^6*f^3*x^2*(f + g*x) - 9*d^5*e*g^3*(8*f + 13*g*x) + 6*d*e^5*f
^2*x*(-f^2 + f*g*x + 2*g^2*x^2) + d^4*e^2*g^2*(38*f^2 + 164*f*g*x + 171*g^2*x^2) - 3*d^3*e^3*g*(-9*f^3 + 19*f^
2*g*x + 47*f*g^2*x^2 + 24*g^3*x^3) + d^2*e^4*f*(7*f^3 - 29*f^2*g*x + 7*f*g^2*x^2 + 43*g^3*x^3)))/(d^3*(d - e*x
)^2*(f + g*x)*Sqrt[d^2 - e^2*x^2]) + 15*e*g^3*(4*e*f - 3*d*g)*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)^2*(e*f + d*g)^5)

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Maple [B]  time = 0.211, size = 6760, normalized size = 21.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.69988, size = 6650, normalized size = 21.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/15*(7*d^3*e^6*f^7 + 27*d^4*e^5*f^6*g + 31*d^5*e^4*f^5*g^2 - 99*d^6*e^3*f^4*g^3 - 23*d^7*e^2*f^3*g^4 + 72*d^
8*e*f^2*g^5 - 15*d^9*f*g^6 - (7*e^9*f^6*g + 27*d*e^8*f^5*g^2 + 31*d^2*e^7*f^4*g^3 - 99*d^3*e^6*f^3*g^4 - 23*d^
4*e^5*f^2*g^5 + 72*d^5*e^4*f*g^6 - 15*d^6*e^3*g^7)*x^4 - (7*e^9*f^7 + 6*d*e^8*f^6*g - 50*d^2*e^7*f^5*g^2 - 192
*d^3*e^6*f^4*g^3 + 274*d^4*e^5*f^3*g^4 + 141*d^5*e^4*f^2*g^5 - 231*d^6*e^3*f*g^6 + 45*d^7*e^2*g^7)*x^3 + 3*(7*
d*e^8*f^7 + 20*d^2*e^7*f^6*g + 4*d^3*e^6*f^5*g^2 - 130*d^4*e^5*f^4*g^3 + 76*d^5*e^4*f^3*g^4 + 95*d^6*e^3*f^2*g
^5 - 87*d^7*e^2*f*g^6 + 15*d^8*e*g^7)*x^2 - 15*(4*d^6*e^2*f^3*g^3 - 3*d^7*e*f^2*g^4 - (4*d^3*e^5*f^2*g^4 - 3*d
^4*e^4*f*g^5)*x^4 - (4*d^3*e^5*f^3*g^3 - 15*d^4*e^4*f^2*g^4 + 9*d^5*e^3*f*g^5)*x^3 + 3*(4*d^4*e^4*f^3*g^3 - 7*
d^5*e^3*f^2*g^4 + 3*d^6*e^2*f*g^5)*x^2 - (12*d^5*e^3*f^3*g^3 - 13*d^6*e^2*f^2*g^4 + 3*d^7*e*f*g^5)*x)*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)) - (21*d^2*e^7*f^7 + 74*d^3*e^6*f^6*g + 66*d^4*e^5
*f^5*g^2 - 328*d^5*e^4*f^4*g^3 + 30*d^6*e^3*f^3*g^4 + 239*d^7*e^2*f^2*g^5 - 117*d^8*e*f*g^6 + 15*d^9*g^7)*x +
(7*d^2*e^6*f^7 + 27*d^3*e^5*f^6*g + 31*d^4*e^4*f^5*g^2 - 99*d^5*e^3*f^4*g^3 - 23*d^6*e^2*f^3*g^4 + 72*d^7*e*f^
2*g^5 - 15*d^8*f*g^6 + (2*e^8*f^6*g + 12*d*e^7*f^5*g^2 + 41*d^2*e^6*f^4*g^3 - 84*d^3*e^5*f^3*g^4 - 43*d^4*e^4*
f^2*g^5 + 72*d^5*e^3*f*g^6)*x^3 + (2*e^8*f^7 + 6*d*e^7*f^6*g + 5*d^2*e^6*f^5*g^2 - 147*d^3*e^5*f^4*g^3 + 164*d
^4*e^4*f^3*g^4 + 141*d^5*e^3*f^2*g^5 - 171*d^6*e^2*f*g^6)*x^2 - (6*d*e^7*f^7 + 29*d^2*e^6*f^6*g + 51*d^3*e^5*f
^5*g^2 - 193*d^4*e^4*f^4*g^3 + 60*d^5*e^3*f^3*g^4 + 164*d^6*e^2*f^2*g^5 - 117*d^7*e*f*g^6)*x)*sqrt(-e^2*x^2 +
d^2))/(d^6*e^7*f^9 + 3*d^7*e^6*f^8*g + d^8*e^5*f^7*g^2 - 5*d^9*e^4*f^6*g^3 - 5*d^10*e^3*f^5*g^4 + d^11*e^2*f^4
*g^5 + 3*d^12*e*f^3*g^6 + d^13*f^2*g^7 - (d^3*e^10*f^8*g + 3*d^4*e^9*f^7*g^2 + d^5*e^8*f^6*g^3 - 5*d^6*e^7*f^5
*g^4 - 5*d^7*e^6*f^4*g^5 + d^8*e^5*f^3*g^6 + 3*d^9*e^4*f^2*g^7 + d^10*e^3*f*g^8)*x^4 - (d^3*e^10*f^9 - 8*d^5*e
^8*f^7*g^2 - 8*d^6*e^7*f^6*g^3 + 10*d^7*e^6*f^5*g^4 + 16*d^8*e^5*f^4*g^5 - 8*d^10*e^3*f^2*g^7 - 3*d^11*e^2*f*g
^8)*x^3 + 3*(d^4*e^9*f^9 + 2*d^5*e^8*f^8*g - 2*d^6*e^7*f^7*g^2 - 6*d^7*e^6*f^6*g^3 + 6*d^9*e^4*f^4*g^5 + 2*d^1
0*e^3*f^3*g^6 - 2*d^11*e^2*f^2*g^7 - d^12*e*f*g^8)*x^2 - (3*d^5*e^8*f^9 + 8*d^6*e^7*f^8*g - 16*d^8*e^5*f^6*g^3
- 10*d^9*e^4*f^5*g^4 + 8*d^10*e^3*f^4*g^5 + 8*d^11*e^2*f^3*g^6 - d^13*f*g^8)*x), 1/15*(7*d^3*e^6*f^7 + 27*d^4
*e^5*f^6*g + 31*d^5*e^4*f^5*g^2 - 99*d^6*e^3*f^4*g^3 - 23*d^7*e^2*f^3*g^4 + 72*d^8*e*f^2*g^5 - 15*d^9*f*g^6 -
(7*e^9*f^6*g + 27*d*e^8*f^5*g^2 + 31*d^2*e^7*f^4*g^3 - 99*d^3*e^6*f^3*g^4 - 23*d^4*e^5*f^2*g^5 + 72*d^5*e^4*f*
g^6 - 15*d^6*e^3*g^7)*x^4 - (7*e^9*f^7 + 6*d*e^8*f^6*g - 50*d^2*e^7*f^5*g^2 - 192*d^3*e^6*f^4*g^3 + 274*d^4*e^
5*f^3*g^4 + 141*d^5*e^4*f^2*g^5 - 231*d^6*e^3*f*g^6 + 45*d^7*e^2*g^7)*x^3 + 3*(7*d*e^8*f^7 + 20*d^2*e^7*f^6*g
+ 4*d^3*e^6*f^5*g^2 - 130*d^4*e^5*f^4*g^3 + 76*d^5*e^4*f^3*g^4 + 95*d^6*e^3*f^2*g^5 - 87*d^7*e^2*f*g^6 + 15*d^
8*e*g^7)*x^2 + 30*(4*d^6*e^2*f^3*g^3 - 3*d^7*e*f^2*g^4 - (4*d^3*e^5*f^2*g^4 - 3*d^4*e^4*f*g^5)*x^4 - (4*d^3*e^
5*f^3*g^3 - 15*d^4*e^4*f^2*g^4 + 9*d^5*e^3*f*g^5)*x^3 + 3*(4*d^4*e^4*f^3*g^3 - 7*d^5*e^3*f^2*g^4 + 3*d^6*e^2*f
*g^5)*x^2 - (12*d^5*e^3*f^3*g^3 - 13*d^6*e^2*f^2*g^4 + 3*d^7*e*f*g^5)*x)*sqrt(e^2*f^2 - d^2*g^2)*arctan((d*g*x
+ d*f - sqrt(-e^2*x^2 + d^2)*f)/(sqrt(e^2*f^2 - d^2*g^2)*x)) - (21*d^2*e^7*f^7 + 74*d^3*e^6*f^6*g + 66*d^4*e^
5*f^5*g^2 - 328*d^5*e^4*f^4*g^3 + 30*d^6*e^3*f^3*g^4 + 239*d^7*e^2*f^2*g^5 - 117*d^8*e*f*g^6 + 15*d^9*g^7)*x +
(7*d^2*e^6*f^7 + 27*d^3*e^5*f^6*g + 31*d^4*e^4*f^5*g^2 - 99*d^5*e^3*f^4*g^3 - 23*d^6*e^2*f^3*g^4 + 72*d^7*e*f
^2*g^5 - 15*d^8*f*g^6 + (2*e^8*f^6*g + 12*d*e^7*f^5*g^2 + 41*d^2*e^6*f^4*g^3 - 84*d^3*e^5*f^3*g^4 - 43*d^4*e^4
*f^2*g^5 + 72*d^5*e^3*f*g^6)*x^3 + (2*e^8*f^7 + 6*d*e^7*f^6*g + 5*d^2*e^6*f^5*g^2 - 147*d^3*e^5*f^4*g^3 + 164*
d^4*e^4*f^3*g^4 + 141*d^5*e^3*f^2*g^5 - 171*d^6*e^2*f*g^6)*x^2 - (6*d*e^7*f^7 + 29*d^2*e^6*f^6*g + 51*d^3*e^5*
f^5*g^2 - 193*d^4*e^4*f^4*g^3 + 60*d^5*e^3*f^3*g^4 + 164*d^6*e^2*f^2*g^5 - 117*d^7*e*f*g^6)*x)*sqrt(-e^2*x^2 +
d^2))/(d^6*e^7*f^9 + 3*d^7*e^6*f^8*g + d^8*e^5*f^7*g^2 - 5*d^9*e^4*f^6*g^3 - 5*d^10*e^3*f^5*g^4 + d^11*e^2*f^
4*g^5 + 3*d^12*e*f^3*g^6 + d^13*f^2*g^7 - (d^3*e^10*f^8*g + 3*d^4*e^9*f^7*g^2 + d^5*e^8*f^6*g^3 - 5*d^6*e^7*f^
5*g^4 - 5*d^7*e^6*f^4*g^5 + d^8*e^5*f^3*g^6 + 3*d^9*e^4*f^2*g^7 + d^10*e^3*f*g^8)*x^4 - (d^3*e^10*f^9 - 8*d^5*
e^8*f^7*g^2 - 8*d^6*e^7*f^6*g^3 + 10*d^7*e^6*f^5*g^4 + 16*d^8*e^5*f^4*g^5 - 8*d^10*e^3*f^2*g^7 - 3*d^11*e^2*f*
g^8)*x^3 + 3*(d^4*e^9*f^9 + 2*d^5*e^8*f^8*g - 2*d^6*e^7*f^7*g^2 - 6*d^7*e^6*f^6*g^3 + 6*d^9*e^4*f^4*g^5 + 2*d^
10*e^3*f^3*g^6 - 2*d^11*e^2*f^2*g^7 - d^12*e*f*g^8)*x^2 - (3*d^5*e^8*f^9 + 8*d^6*e^7*f^8*g - 16*d^8*e^5*f^6*g^
3 - 10*d^9*e^4*f^5*g^4 + 8*d^10*e^3*f^4*g^5 + 8*d^11*e^2*f^3*g^6 - d^13*f*g^8)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (g x + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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