\(\int \frac {(d+e x)^3}{(f+g x)^2 (d^2-e^2 x^2)^{7/2}} \, dx\) [586]

Optimal result

Integrand size = 31, antiderivative size = 311 \[ \int \frac {(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 d e (d+e x)}{5 (e f+d g)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e (5 d (e f-3 d g)-e (e f+21 d g) x)}{15 d (e f+d g)^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e \left (45 d^3 g^2+e \left (2 e^2 f^2+14 d e f g+57 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^4 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{(e f-d g) (e f+d g)^4 (f+g x)}+\frac {e g^3 (4 e f-3 d g) \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{(e f-d g) (e f+d g)^4 \sqrt {e^2 f^2-d^2 g^2}} \]

[Out]

4/5*d*e*(e*x+d)/(d*g+e*f)^2/(-e^2*x^2+d^2)^(5/2)-1/15*e*(5*d*(-3*d*g+e*f)-e*(21*d*g+e*f)*x)/d/(d*g+e*f)^3/(-e^
2*x^2+d^2)^(3/2)+e*g^3*(-3*d*g+4*e*f)*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)/(d*g+e*f)^4/(-d^2*g^2+e^2*f^2)^(1/2)+1/15*e*(45*d^3*g^2+e*(57*d^2*g^2+14*d*e*f*g+2*e^2*f^2)*x)/d^3/(d
*g+e*f)^4/(-e^2*x^2+d^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)

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1661, 821, 739, 210} \[ \int \frac {(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e g^3 (4 e f-3 d g) \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )}{(e f-d g) (d g+e f)^4 \sqrt {e^2 f^2-d^2 g^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{(f+g x) (e f-d g) (d g+e f)^4}-\frac {e (5 d (e f-3 d g)-e x (21 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^3}+\frac {4 d e (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^2}+\frac {e \left (45 d^3 g^2+e x \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt {d^2-e^2 x^2} (d g+e f)^4} \]

[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 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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \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) (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] (verified)

Time = 10.43 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\left (e^2 f^2-d^2 g^2\right ) (d+e x) \left (15 d^6 g^4+2 e^6 f^3 x^2 (f+g x)-9 d^5 e g^3 (8 f+13 g x)+6 d e^5 f^2 x \left (-f^2+f g x+2 g^2 x^2\right )+d^4 e^2 g^2 \left (38 f^2+164 f g x+171 g^2 x^2\right )-3 d^3 e^3 g \left (-9 f^3+19 f^2 g x+47 f g^2 x^2+24 g^3 x^3\right )+d^2 e^4 f \left (7 f^3-29 f^2 g x+7 f g^2 x^2+43 g^3 x^3\right )\right )}{d^3 (d-e x)^2 (f+g x) \sqrt {d^2-e^2 x^2}}+15 e g^3 (4 e f-3 d g) \sqrt {e^2 f^2-d^2 g^2} \arctan \left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{15 (e f-d g)^2 (e f+d g)^5} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3288\) vs. \(2(291)=582\).

Time = 0.52 (sec) , antiderivative size = 3289, normalized size of antiderivative = 10.58

method	result	size
default	\(\text {Expression too large to display}\)	\(3289\)

[In]

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

[Out]

e^2/g^3*(1/5/e*g/(-e^2*x^2+d^2)^(5/2)+3*d*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)))-2*e*f*(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*e/g^4*(d^2*g^2-2*d*e*f*g+e^2*f^2)*(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)))))+1/g^5*(d^3*g^3-3*d^2*e*f*g^2+3*d*e^2*f^2*g-e^3*f^3)*(-1/(d^2*g^2-
e^2*f^2)*g^2/(x+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)^(5/2)-7*e^2*f*g/(d^2*g^2-e^2*f^2
)*(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)))))+6*e^2/(d^2*g^2-e^2*f^2)*g^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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1623 vs. \(2 (290) = 580\).

Time = 0.75 (sec) , antiderivative size = 3305, normalized size of antiderivative = 10.63 \[ \int \frac {(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^3/(g*x+f)^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)]

Sympy [F]

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

[In]

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

Maxima [F(-2)]

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

[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 >> 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 [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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