\(\int \frac {(d+e x)^3}{(f+g x) (d^2-e^2 x^2)^{7/2}} \, dx\) [585]

Optimal result

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

[Out]

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)+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)+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)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1661, 837, 12, 739, 210} \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {g^3 \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )}{(d g+e f)^3 \sqrt {e^2 f^2-d^2 g^2}}-\frac {5 d (e f-d g)-e x (11 d g+e f)}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^2}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}+\frac {15 d^3 g^2+e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )}{15 d^3 \sqrt {d^2-e^2 x^2} (d g+e f)^3} \]

[In]

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

[Out]

(4*d*(d + e*x))/(5*(e*f + d*g)*(d^2 - e^2*x^2)^(5/2)) - (5*d*(e*f - d*g) - e*(e*f + 11*d*g)*x)/(15*d*(e*f + d*
g)^2*(d^2 - e^2*x^2)^(3/2)) + (15*d^3*g^2 + e*(2*e^2*f^2 + 9*d*e*f*g + 22*d^2*g^2)*x)/(15*d^3*(e*f + d*g)^3*Sq
rt[d^2 - e^2*x^2]) + (g^3*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
)^3*Sqrt[e^2*f^2 - d^2*g^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 837

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] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[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] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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]

Rubi steps \begin{align*} \text {integral}& = \frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {\frac {d^3 e^2 (e f+5 d g)}{e f+d g}-\frac {d^2 e^3 (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 (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 {\int \frac {-\frac {d^3 e^4 (e f-d g) \left (2 e^2 f^2+7 d e f g+15 d^2 g^2\right )}{e f+d g}-\frac {2 d^3 e^5 g (e f-d g) (e f+11 d g) x}{e f+d g}}{(f+g x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4 \left (e^2 f^2-d^2 g^2\right )} \\ & = \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 {\int \frac {15 d^6 e^6 g^3 (e f-d g)^2}{(e f+d g) (f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6 \left (e^2 f^2-d^2 g^2\right )^2} \\ & = \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 \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{(e f+d g)^3} \\ & = \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 \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 )}{(e f+d g)^3} \\ & = \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 \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)^3 \sqrt {e^2 f^2-d^2 g^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.30 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\left (-e^2 f^2+d^2 g^2\right ) (d+e x) \left (32 d^4 g^2+2 e^4 f^2 x^2+3 d^3 e g (8 f-17 g x)+3 d e^3 f x (-2 f+3 g x)+d^2 e^2 \left (7 f^2-27 f g x+22 g^2 x^2\right )\right )}{d^3 (d-e x)^2 \sqrt {d^2-e^2 x^2}}-15 g^3 \sqrt {e^2 f^2-d^2 g^2} \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{15 (-e f+d g) (e f+d g)^4} \]

[In]

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

[Out]

(((-(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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1666\) vs. \(2(223)=446\).

Time = 0.52 (sec) , antiderivative size = 1667, normalized size of antiderivative = 6.89

method	result	size
default	\(\text {Expression too large to display}\)	\(1667\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

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

Time = 0.37 (sec) , antiderivative size = 1767, normalized size of antiderivative = 7.30 \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[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 +
25*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*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 + 25*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 - 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)*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)) - 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)]

Sympy [F]

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

[In]

integrate((e*x+d)**3/(g*x+f)/(-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)), x)

Maxima [F(-2)]

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

[In]

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

Time = 0.31 (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}} \]

[In]

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

[Out]

-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 + 2
4*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)*a
bs(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 + sqrt(-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 + s
qrt(-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 + sqrt(-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. \[ \int \frac {(d+e x)^3}{(f+g x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{\left (f+g\,x\right )\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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