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

Optimal. Leaf size=242 \[ \frac{e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )+15 d^3 g^2}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^3}+\frac{g^3 \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{(d g+e f)^3 \sqrt{e^2 f^2-d^2 g^2}}-\frac{5 d (e f-d g)-e x (11 d g+e f)}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^2}+\frac{4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.617526, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {1647, 823, 12, 725, 204} \[ \frac{e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )+15 d^3 g^2}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^3}+\frac{g^3 \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{(d g+e f)^3 \sqrt{e^2 f^2-d^2 g^2}}-\frac{5 d (e f-d g)-e x (11 d g+e f)}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^2}+\frac{4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)} \]

Antiderivative was successfully verified.

[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 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 823

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] + Di
st[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 12

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

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) \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{\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 \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)^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]  time = 0.410837, size = 225, normalized size = 0.93 \[ \frac{\frac{(d+e x) \left (d^2 g^2-e^2 f^2\right ) \left (d^2 e^2 \left (7 f^2-27 f g x+22 g^2 x^2\right )+3 d^3 e g (8 f-17 g x)+32 d^4 g^2+3 d e^3 f x (3 g x-2 f)+2 e^4 f^2 x^2\right )}{d^3 (d-e x)^2 \sqrt{d^2-e^2 x^2}}-15 g^3 \sqrt{e^2 f^2-d^2 g^2} \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{15 (d g-e f) (d g+e f)^4} \]

Antiderivative was successfully verified.

[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]  time = 0.259, size = 3961, normalized size = 16.4 \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)/(-e^2*x^2+d^2)^(7/2),x)

[Out]

22/15*e/g/d^4*x/(-e^2*x^2+d^2)^(1/2)-g^4/(d^2*g^2-e^2*f^2)^3*f*d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^
2*f^2)/g^2)^(1/2)*e^2*x-3*g^2/(d^2*g^2-e^2*f^2)^3*f^3/d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^
2)^(1/2)*e^4*x+g/(d^2*g^2-e^2*f^2)^3*f^4/d^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*e^
5*x-3/5/g^2*e^4*f^3/(d^2*g^2-e^2*f^2)/d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*x+8/15/
g^3*e^5*f^4/(d^2*g^2-e^2*f^2)/d^6/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x-1/3*g^2/(d^
2*g^2-e^2*f^2)^2*e^2*f*d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x+4/15/g^3*e^5*f^4/(d^
2*g^2-e^2*f^2)/d^4/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x-8/5/g^2*e^4*f^3/(d^2*g^2-e
^2*f^2)/d^5/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x-1/5/g^2/(d^2*g^2-e^2*f^2)/(-(x+f/
g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*e^3*f^3-g^2/(d^2*g^2-e^2*f^2)^3/(-(x+f/g)^2*e^2+2*e^2*
f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*e^3*f^3-g^5/(d^2*g^2-e^2*f^2)^3/((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)*(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*
g^2-e^2*f^2)/g^2)^(1/2))/(x+f/g))*d^3-3/5/(d^2*g^2-e^2*f^2)/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2
)/g^2)^(5/2)*d^2*e*f-2/(d^2*g^2-e^2*f^2)^2*e^4*f^3/d^3/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2
)^(1/2)*x+3/5/g/(d^2*g^2-e^2*f^2)/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*d*e^2*f^2-g^2
/(d^2*g^2-e^2*f^2)^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*d^2*e*f-2/3*g^2/(d^2*g^2-e
^2*f^2)^2*e^2*f/d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x+3/5/g/(-e^2*x^2+d^2)^(5/2)*
d+4/5/g*e^3*f^2/(d^2*g^2-e^2*f^2)/d^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x-4/5/g^2
*e^4*f^3/(d^2*g^2-e^2*f^2)/d^3/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x+3*g^4/(d^2*g^2
-e^2*f^2)^3/((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)*(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2))/(x+f/g))*d^2*e*f-3*g^3/(d^2*g^2-e^2*
f^2)^3/((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)*(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2))/(x+f/g))*d*e^2*f^2+1/5/g^3*e^5*f^4/(d^2*g
^2-e^2*f^2)/d^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*x+8/5/g*e^3*f^2/(d^2*g^2-e^2*f^
2)/d^4/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x+2*g/(d^2*g^2-e^2*f^2)^2*e^3*f^2/d^2/(-
(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x+2/3/g/(d^2*g^2-e^2*f^2)^2*e^5*f^4/d^4/(-(x+f/g)
^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x+11/15*e/g/d^2*x/(-e^2*x^2+d^2)^(3/2)+g^2/(d^2*g^2-e^2*
f^2)^3/((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)*(-(x+f/g)^2*e^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^3*f^3+3/5/g*e^3*f^2/(d^2*g^2-e
^2*f^2)/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*x+g/(d^2*g^2-e^2*f^2)^2*e^3*f^2/(-(x+f/
g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x+3*g^3/(d^2*g^2-e^2*f^2)^3*f^2/(-(x+f/g)^2*e^2+2*e^2*
f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*e^3*x-1/(d^2*g^2-e^2*f^2)^2*e^4*f^3/d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/
g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x+4/15*e^3/g^3*f^2/d^4*x/(-e^2*x^2+d^2)^(3/2)+8/15*e^3/g^3*f^2/d^6*x/(-e^2*x^2
+d^2)^(1/2)-1/5*e^2*f/(d^2*g^2-e^2*f^2)*d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*x-4/1
5*e^2*f/(d^2*g^2-e^2*f^2)/d/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x-8/15*e^2*f/(d^2*g
^2-e^2*f^2)/d^3/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*x-1/3/(d^2*g^2-e^2*f^2)^2/(-(x+
f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*e^3*f^3+1/5*g/(d^2*g^2-e^2*f^2)/(-(x+f/g)^2*e^2+2*e^
2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(5/2)*d^3+1/3*g^3/(d^2*g^2-e^2*f^2)^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(
d^2*g^2-e^2*f^2)/g^2)^(3/2)*d^3+g^5/(d^2*g^2-e^2*f^2)^3/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^
2)^(1/2)*d^3+4/5*e/g*x/(-e^2*x^2+d^2)^(5/2)-1/5*e/g^2/(-e^2*x^2+d^2)^(5/2)*f-3/5*e^2/g^2/d*f*x/(-e^2*x^2+d^2)^
(5/2)-4/5*e^2/g^2/d^3*f*x/(-e^2*x^2+d^2)^(3/2)-8/5*e^2/g^2/d^5*f*x/(-e^2*x^2+d^2)^(1/2)+1/5*e^3/g^3*f^2*x/d^2/
(-e^2*x^2+d^2)^(5/2)+g/(d^2*g^2-e^2*f^2)^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*d*e^
2*f^2-3*g^4/(d^2*g^2-e^2*f^2)^3/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*d^2*e*f+3*g^3/(
d^2*g^2-e^2*f^2)^3/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(1/2)*d*e^2*f^2+1/3/g/(d^2*g^2-e^2
*f^2)^2*e^5*f^4/d^2/(-(x+f/g)^2*e^2+2*e^2*f/g*(x+f/g)+(d^2*g^2-e^2*f^2)/g^2)^(3/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.26396, size = 3537, normalized size = 14.62 \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)/(-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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.28356, size = 4004, normalized size = 16.55 \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)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-2*(d^3*g^6*e^2 - 3*d^2*f*g^5*e^3 + 3*d*f^2*g^4*e^4 - f^3*g^3*e^5)*arctan((d*g*e + (d*e + sqrt(-x^2*e^2 + d^2)
*e)*f/x)/sqrt(-d^2*g^2*e^2 + f^2*e^4))/((d^6*g^6*e - 3*d^4*f^2*g^4*e^3 + 3*d^2*f^4*g^2*e^5 - f^6*e^7)*sqrt(-d^
2*g^2*e^2 + f^2*e^4)) - 1/15*sqrt(-x^2*e^2 + d^2)*((((((22*d^18*g^17*e^9 + 339*d^17*f*g^16*e^10 + 2447*d^16*f^
2*g^15*e^11 + 10985*d^15*f^3*g^14*e^12 + 34335*d^14*f^4*g^13*e^13 + 79261*d^13*f^5*g^12*e^14 + 139867*d^12*f^6
*g^11*e^15 + 192621*d^11*f^7*g^10*e^16 + 209495*d^10*f^8*g^9*e^17 + 180895*d^9*f^9*g^8*e^18 + 123981*d^8*f^10*
g^7*e^19 + 67067*d^7*f^11*g^6*e^20 + 28301*d^6*f^12*g^5*e^21 + 9135*d^5*f^13*g^4*e^22 + 2185*d^4*f^14*g^3*e^23
+ 367*d^3*f^15*g^2*e^24 + 39*d^2*f^16*g*e^25 + 2*d*f^17*e^26)*x/(d^22*g^18*e^4 + 18*d^21*f*g^17*e^5 + 153*d^2
0*f^2*g^16*e^6 + 816*d^19*f^3*g^15*e^7 + 3060*d^18*f^4*g^14*e^8 + 8568*d^17*f^5*g^13*e^9 + 18564*d^16*f^6*g^12
*e^10 + 31824*d^15*f^7*g^11*e^11 + 43758*d^14*f^8*g^10*e^12 + 48620*d^13*f^9*g^9*e^13 + 43758*d^12*f^10*g^8*e^
14 + 31824*d^11*f^11*g^7*e^15 + 18564*d^10*f^12*g^6*e^16 + 8568*d^9*f^13*g^5*e^17 + 3060*d^8*f^14*g^4*e^18 + 8
16*d^7*f^15*g^3*e^19 + 153*d^6*f^16*g^2*e^20 + 18*d^5*f^17*g*e^21 + d^4*f^18*e^22) + 15*(d^19*g^17*e^8 + 15*d^
18*f*g^16*e^9 + 105*d^17*f^2*g^15*e^10 + 455*d^16*f^3*g^14*e^11 + 1365*d^15*f^4*g^13*e^12 + 3003*d^14*f^5*g^12
*e^13 + 5005*d^13*f^6*g^11*e^14 + 6435*d^12*f^7*g^10*e^15 + 6435*d^11*f^8*g^9*e^16 + 5005*d^10*f^9*g^8*e^17 +
3003*d^9*f^10*g^7*e^18 + 1365*d^8*f^11*g^6*e^19 + 455*d^7*f^12*g^5*e^20 + 105*d^6*f^13*g^4*e^21 + 15*d^5*f^14*
g^3*e^22 + d^4*f^15*g^2*e^23)/(d^22*g^18*e^4 + 18*d^21*f*g^17*e^5 + 153*d^20*f^2*g^16*e^6 + 816*d^19*f^3*g^15*
e^7 + 3060*d^18*f^4*g^14*e^8 + 8568*d^17*f^5*g^13*e^9 + 18564*d^16*f^6*g^12*e^10 + 31824*d^15*f^7*g^11*e^11 +
43758*d^14*f^8*g^10*e^12 + 48620*d^13*f^9*g^9*e^13 + 43758*d^12*f^10*g^8*e^14 + 31824*d^11*f^11*g^7*e^15 + 185
64*d^10*f^12*g^6*e^16 + 8568*d^9*f^13*g^5*e^17 + 3060*d^8*f^14*g^4*e^18 + 816*d^7*f^15*g^3*e^19 + 153*d^6*f^16
*g^2*e^20 + 18*d^5*f^17*g*e^21 + d^4*f^18*e^22))*x - 5*(11*d^20*g^17*e^7 + 171*d^19*f*g^16*e^8 + 1246*d^18*f^2
*g^15*e^9 + 5650*d^17*f^3*g^14*e^10 + 17850*d^16*f^4*g^13*e^11 + 41678*d^15*f^5*g^12*e^12 + 74438*d^14*f^6*g^1
1*e^13 + 103818*d^13*f^7*g^10*e^14 + 114400*d^12*f^8*g^9*e^15 + 100100*d^11*f^9*g^8*e^16 + 69498*d^10*f^10*g^7
*e^17 + 38038*d^9*f^11*g^6*e^18 + 16198*d^8*f^12*g^5*e^19 + 5250*d^7*f^13*g^4*e^20 + 1250*d^6*f^14*g^3*e^21 +
206*d^5*f^15*g^2*e^22 + 21*d^4*f^16*g*e^23 + d^3*f^17*e^24)/(d^22*g^18*e^4 + 18*d^21*f*g^17*e^5 + 153*d^20*f^2
*g^16*e^6 + 816*d^19*f^3*g^15*e^7 + 3060*d^18*f^4*g^14*e^8 + 8568*d^17*f^5*g^13*e^9 + 18564*d^16*f^6*g^12*e^10
+ 31824*d^15*f^7*g^11*e^11 + 43758*d^14*f^8*g^10*e^12 + 48620*d^13*f^9*g^9*e^13 + 43758*d^12*f^10*g^8*e^14 +
31824*d^11*f^11*g^7*e^15 + 18564*d^10*f^12*g^6*e^16 + 8568*d^9*f^13*g^5*e^17 + 3060*d^8*f^14*g^4*e^18 + 816*d^
7*f^15*g^3*e^19 + 153*d^6*f^16*g^2*e^20 + 18*d^5*f^17*g*e^21 + d^4*f^18*e^22))*x - 5*(7*d^21*g^17*e^6 + 105*d^
20*f*g^16*e^7 + 734*d^19*f^2*g^15*e^8 + 3170*d^18*f^3*g^14*e^9 + 9450*d^17*f^4*g^13*e^10 + 20566*d^16*f^5*g^12
*e^11 + 33670*d^15*f^6*g^11*e^12 + 42042*d^14*f^7*g^10*e^13 + 40040*d^13*f^8*g^9*e^14 + 28600*d^12*f^9*g^8*e^1
5 + 14586*d^11*f^10*g^7*e^16 + 4550*d^10*f^11*g^6*e^17 + 182*d^9*f^12*g^5*e^18 - 630*d^8*f^13*g^4*e^19 - 350*d
^7*f^14*g^3*e^20 - 98*d^6*f^15*g^2*e^21 - 15*d^5*f^16*g*e^22 - d^4*f^17*e^23)/(d^22*g^18*e^4 + 18*d^21*f*g^17*
e^5 + 153*d^20*f^2*g^16*e^6 + 816*d^19*f^3*g^15*e^7 + 3060*d^18*f^4*g^14*e^8 + 8568*d^17*f^5*g^13*e^9 + 18564*
d^16*f^6*g^12*e^10 + 31824*d^15*f^7*g^11*e^11 + 43758*d^14*f^8*g^10*e^12 + 48620*d^13*f^9*g^9*e^13 + 43758*d^1
2*f^10*g^8*e^14 + 31824*d^11*f^11*g^7*e^15 + 18564*d^10*f^12*g^6*e^16 + 8568*d^9*f^13*g^5*e^17 + 3060*d^8*f^14
*g^4*e^18 + 816*d^7*f^15*g^3*e^19 + 153*d^6*f^16*g^2*e^20 + 18*d^5*f^17*g*e^21 + d^4*f^18*e^22))*x + 15*(3*d^2
2*g^17*e^5 + 48*d^21*f*g^16*e^6 + 361*d^20*f^2*g^15*e^7 + 1695*d^19*f^3*g^14*e^8 + 5565*d^18*f^4*g^13*e^9 + 13
559*d^17*f^5*g^12*e^10 + 25389*d^16*f^6*g^11*e^11 + 37323*d^15*f^7*g^10*e^12 + 43615*d^14*f^8*g^9*e^13 + 40755
*d^13*f^9*g^8*e^14 + 30459*d^12*f^10*g^7*e^15 + 18109*d^11*f^11*g^6*e^16 + 8463*d^10*f^12*g^5*e^17 + 3045*d^9*
f^13*g^4*e^18 + 815*d^8*f^14*g^3*e^19 + 153*d^7*f^15*g^2*e^20 + 18*d^6*f^16*g*e^21 + d^5*f^17*e^22)/(d^22*g^18
*e^4 + 18*d^21*f*g^17*e^5 + 153*d^20*f^2*g^16*e^6 + 816*d^19*f^3*g^15*e^7 + 3060*d^18*f^4*g^14*e^8 + 8568*d^17
*f^5*g^13*e^9 + 18564*d^16*f^6*g^12*e^10 + 31824*d^15*f^7*g^11*e^11 + 43758*d^14*f^8*g^10*e^12 + 48620*d^13*f^
9*g^9*e^13 + 43758*d^12*f^10*g^8*e^14 + 31824*d^11*f^11*g^7*e^15 + 18564*d^10*f^12*g^6*e^16 + 8568*d^9*f^13*g^
5*e^17 + 3060*d^8*f^14*g^4*e^18 + 816*d^7*f^15*g^3*e^19 + 153*d^6*f^16*g^2*e^20 + 18*d^5*f^17*g*e^21 + d^4*f^1
8*e^22))*x + (32*d^23*g^17*e^4 + 504*d^22*f*g^16*e^5 + 3727*d^21*f^2*g^15*e^6 + 17185*d^20*f^3*g^14*e^7 + 5533
5*d^19*f^4*g^13*e^8 + 132041*d^18*f^5*g^12*e^9 + 241787*d^17*f^6*g^11*e^10 + 347061*d^16*f^7*g^10*e^11 + 39539
5*d^15*f^8*g^9*e^12 + 359645*d^14*f^9*g^8*e^13 + 261261*d^13*f^10*g^7*e^14 + 150787*d^12*f^11*g^6*e^15 + 68341
*d^11*f^12*g^5*e^16 + 23835*d^10*f^13*g^4*e^17 + 6185*d^9*f^14*g^3*e^18 + 1127*d^8*f^15*g^2*e^19 + 129*d^7*f^1
6*g*e^20 + 7*d^6*f^17*e^21)/(d^22*g^18*e^4 + 18*d^21*f*g^17*e^5 + 153*d^20*f^2*g^16*e^6 + 816*d^19*f^3*g^15*e^
7 + 3060*d^18*f^4*g^14*e^8 + 8568*d^17*f^5*g^13*e^9 + 18564*d^16*f^6*g^12*e^10 + 31824*d^15*f^7*g^11*e^11 + 43
758*d^14*f^8*g^10*e^12 + 48620*d^13*f^9*g^9*e^13 + 43758*d^12*f^10*g^8*e^14 + 31824*d^11*f^11*g^7*e^15 + 18564
*d^10*f^12*g^6*e^16 + 8568*d^9*f^13*g^5*e^17 + 3060*d^8*f^14*g^4*e^18 + 816*d^7*f^15*g^3*e^19 + 153*d^6*f^16*g
^2*e^20 + 18*d^5*f^17*g*e^21 + d^4*f^18*e^22))/(x^2*e^2 - d^2)^3