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

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

[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)

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Rubi [A]
time = 0.40, antiderivative size = 242, normalized size of antiderivative = 1.00, 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 = {1661, 837, 12, 739, 210} \begin {gather*} \frac {g^3 \text {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} \end {gather*}

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 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*} \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 \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*}

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Mathematica [A]
time = 10.29, size = 225, normalized size = 0.93 \begin {gather*} \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} \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 )}{15 (-e f+d g) (e f+d g)^4} \end {gather*}

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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1666\) vs. \(2(223)=446\).
time = 0.10, size = 1667, normalized size = 6.89

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

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d*g-%e*f>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (225) = 450\).
time = 3.75, size = 1742, normalized size = 7.20 \begin {gather*} \left [\frac {32 \, d^{7} g^{4} + 7 \, f^{4} x^{3} e^{7} - 15 \, {\left (d^{3} g^{3} x^{3} e^{3} - 3 \, d^{4} g^{3} x^{2} e^{2} + 3 \, d^{5} g^{3} x e - d^{6} g^{3}\right )} \sqrt {d^{2} g^{2} - f^{2} e^{2}} \log \left (\frac {d^{3} g^{2} + d f g x e^{2} - \sqrt {d^{2} g^{2} - f^{2} e^{2}} {\left (d^{2} g + f x e^{2} + \sqrt {-x^{2} e^{2} + d^{2}} d g\right )} + {\left (d^{2} g^{2} - f^{2} e^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{g x + f}\right ) + 3 \, {\left (8 \, d f^{3} g x^{3} - 7 \, d f^{4} x^{2}\right )} e^{6} + {\left (25 \, d^{2} f^{2} g^{2} x^{3} - 72 \, d^{2} f^{3} g x^{2} + 21 \, d^{2} f^{4} x\right )} e^{5} - {\left (24 \, d^{3} f g^{3} x^{3} + 75 \, d^{3} f^{2} g^{2} x^{2} - 72 \, d^{3} f^{3} g x + 7 \, d^{3} f^{4}\right )} e^{4} - {\left (32 \, d^{4} g^{4} x^{3} - 72 \, d^{4} f g^{3} x^{2} - 75 \, d^{4} f^{2} g^{2} x + 24 \, d^{4} f^{3} g\right )} e^{3} + {\left (96 \, d^{5} g^{4} x^{2} - 72 \, d^{5} f g^{3} x - 25 \, d^{5} f^{2} g^{2}\right )} e^{2} - 24 \, {\left (4 \, d^{6} g^{4} x - d^{6} f g^{3}\right )} e + {\left (32 \, d^{6} g^{4} - 2 \, f^{4} x^{2} e^{6} - 3 \, {\left (3 \, d f^{3} g x^{2} - 2 \, d f^{4} x\right )} e^{5} - {\left (20 \, d^{2} f^{2} g^{2} x^{2} - 27 \, d^{2} f^{3} g x + 7 \, d^{2} f^{4}\right )} e^{4} + 3 \, {\left (3 \, d^{3} f g^{3} x^{2} + 15 \, d^{3} f^{2} g^{2} x - 8 \, d^{3} f^{3} g\right )} e^{3} + {\left (22 \, d^{4} g^{4} x^{2} - 27 \, d^{4} f g^{3} x - 25 \, d^{4} f^{2} g^{2}\right )} e^{2} - 3 \, {\left (17 \, d^{5} g^{4} x - 8 \, d^{5} f g^{3}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{11} g^{5} + d^{3} f^{5} x^{3} e^{8} + 3 \, {\left (d^{4} f^{4} g x^{3} - d^{4} f^{5} x^{2}\right )} e^{7} + {\left (2 \, d^{5} f^{3} g^{2} x^{3} - 9 \, d^{5} f^{4} g x^{2} + 3 \, d^{5} f^{5} x\right )} e^{6} - {\left (2 \, d^{6} f^{2} g^{3} x^{3} + 6 \, d^{6} f^{3} g^{2} x^{2} - 9 \, d^{6} f^{4} g x + d^{6} f^{5}\right )} e^{5} - 3 \, {\left (d^{7} f g^{4} x^{3} - 2 \, d^{7} f^{2} g^{3} x^{2} - 2 \, d^{7} f^{3} g^{2} x + d^{7} f^{4} g\right )} e^{4} - {\left (d^{8} g^{5} x^{3} - 9 \, d^{8} f g^{4} x^{2} + 6 \, d^{8} f^{2} g^{3} x + 2 \, d^{8} f^{3} g^{2}\right )} e^{3} + {\left (3 \, d^{9} g^{5} x^{2} - 9 \, d^{9} f g^{4} x + 2 \, d^{9} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (d^{10} g^{5} x - d^{10} f g^{4}\right )} e\right )}}, \frac {32 \, d^{7} g^{4} + 7 \, f^{4} x^{3} e^{7} - 30 \, {\left (d^{3} g^{3} x^{3} e^{3} - 3 \, d^{4} g^{3} x^{2} e^{2} + 3 \, d^{5} g^{3} x e - d^{6} g^{3}\right )} \sqrt {-d^{2} g^{2} + f^{2} e^{2}} \arctan \left (\frac {\sqrt {-d^{2} g^{2} + f^{2} e^{2}} {\left (d g x + d f - \sqrt {-x^{2} e^{2} + d^{2}} f\right )}}{d^{2} g^{2} x - f^{2} x e^{2}}\right ) + 3 \, {\left (8 \, d f^{3} g x^{3} - 7 \, d f^{4} x^{2}\right )} e^{6} + {\left (25 \, d^{2} f^{2} g^{2} x^{3} - 72 \, d^{2} f^{3} g x^{2} + 21 \, d^{2} f^{4} x\right )} e^{5} - {\left (24 \, d^{3} f g^{3} x^{3} + 75 \, d^{3} f^{2} g^{2} x^{2} - 72 \, d^{3} f^{3} g x + 7 \, d^{3} f^{4}\right )} e^{4} - {\left (32 \, d^{4} g^{4} x^{3} - 72 \, d^{4} f g^{3} x^{2} - 75 \, d^{4} f^{2} g^{2} x + 24 \, d^{4} f^{3} g\right )} e^{3} + {\left (96 \, d^{5} g^{4} x^{2} - 72 \, d^{5} f g^{3} x - 25 \, d^{5} f^{2} g^{2}\right )} e^{2} - 24 \, {\left (4 \, d^{6} g^{4} x - d^{6} f g^{3}\right )} e + {\left (32 \, d^{6} g^{4} - 2 \, f^{4} x^{2} e^{6} - 3 \, {\left (3 \, d f^{3} g x^{2} - 2 \, d f^{4} x\right )} e^{5} - {\left (20 \, d^{2} f^{2} g^{2} x^{2} - 27 \, d^{2} f^{3} g x + 7 \, d^{2} f^{4}\right )} e^{4} + 3 \, {\left (3 \, d^{3} f g^{3} x^{2} + 15 \, d^{3} f^{2} g^{2} x - 8 \, d^{3} f^{3} g\right )} e^{3} + {\left (22 \, d^{4} g^{4} x^{2} - 27 \, d^{4} f g^{3} x - 25 \, d^{4} f^{2} g^{2}\right )} e^{2} - 3 \, {\left (17 \, d^{5} g^{4} x - 8 \, d^{5} f g^{3}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{11} g^{5} + d^{3} f^{5} x^{3} e^{8} + 3 \, {\left (d^{4} f^{4} g x^{3} - d^{4} f^{5} x^{2}\right )} e^{7} + {\left (2 \, d^{5} f^{3} g^{2} x^{3} - 9 \, d^{5} f^{4} g x^{2} + 3 \, d^{5} f^{5} x\right )} e^{6} - {\left (2 \, d^{6} f^{2} g^{3} x^{3} + 6 \, d^{6} f^{3} g^{2} x^{2} - 9 \, d^{6} f^{4} g x + d^{6} f^{5}\right )} e^{5} - 3 \, {\left (d^{7} f g^{4} x^{3} - 2 \, d^{7} f^{2} g^{3} x^{2} - 2 \, d^{7} f^{3} g^{2} x + d^{7} f^{4} g\right )} e^{4} - {\left (d^{8} g^{5} x^{3} - 9 \, d^{8} f g^{4} x^{2} + 6 \, d^{8} f^{2} g^{3} x + 2 \, d^{8} f^{3} g^{2}\right )} e^{3} + {\left (3 \, d^{9} g^{5} x^{2} - 9 \, d^{9} f g^{4} x + 2 \, d^{9} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (d^{10} g^{5} x - d^{10} f g^{4}\right )} e\right )}}\right ] \end {gather*}

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*(32*d^7*g^4 + 7*f^4*x^3*e^7 - 15*(d^3*g^3*x^3*e^3 - 3*d^4*g^3*x^2*e^2 + 3*d^5*g^3*x*e - d^6*g^3)*sqrt(d^
2*g^2 - f^2*e^2)*log((d^3*g^2 + d*f*g*x*e^2 - sqrt(d^2*g^2 - f^2*e^2)*(d^2*g + f*x*e^2 + sqrt(-x^2*e^2 + d^2)*
d*g) + (d^2*g^2 - f^2*e^2)*sqrt(-x^2*e^2 + d^2))/(g*x + f)) + 3*(8*d*f^3*g*x^3 - 7*d*f^4*x^2)*e^6 + (25*d^2*f^
2*g^2*x^3 - 72*d^2*f^3*g*x^2 + 21*d^2*f^4*x)*e^5 - (24*d^3*f*g^3*x^3 + 75*d^3*f^2*g^2*x^2 - 72*d^3*f^3*g*x + 7
*d^3*f^4)*e^4 - (32*d^4*g^4*x^3 - 72*d^4*f*g^3*x^2 - 75*d^4*f^2*g^2*x + 24*d^4*f^3*g)*e^3 + (96*d^5*g^4*x^2 -
72*d^5*f*g^3*x - 25*d^5*f^2*g^2)*e^2 - 24*(4*d^6*g^4*x - d^6*f*g^3)*e + (32*d^6*g^4 - 2*f^4*x^2*e^6 - 3*(3*d*f
^3*g*x^2 - 2*d*f^4*x)*e^5 - (20*d^2*f^2*g^2*x^2 - 27*d^2*f^3*g*x + 7*d^2*f^4)*e^4 + 3*(3*d^3*f*g^3*x^2 + 15*d^
3*f^2*g^2*x - 8*d^3*f^3*g)*e^3 + (22*d^4*g^4*x^2 - 27*d^4*f*g^3*x - 25*d^4*f^2*g^2)*e^2 - 3*(17*d^5*g^4*x - 8*
d^5*f*g^3)*e)*sqrt(-x^2*e^2 + d^2))/(d^11*g^5 + d^3*f^5*x^3*e^8 + 3*(d^4*f^4*g*x^3 - d^4*f^5*x^2)*e^7 + (2*d^5
*f^3*g^2*x^3 - 9*d^5*f^4*g*x^2 + 3*d^5*f^5*x)*e^6 - (2*d^6*f^2*g^3*x^3 + 6*d^6*f^3*g^2*x^2 - 9*d^6*f^4*g*x + d
^6*f^5)*e^5 - 3*(d^7*f*g^4*x^3 - 2*d^7*f^2*g^3*x^2 - 2*d^7*f^3*g^2*x + d^7*f^4*g)*e^4 - (d^8*g^5*x^3 - 9*d^8*f
*g^4*x^2 + 6*d^8*f^2*g^3*x + 2*d^8*f^3*g^2)*e^3 + (3*d^9*g^5*x^2 - 9*d^9*f*g^4*x + 2*d^9*f^2*g^3)*e^2 - 3*(d^1
0*g^5*x - d^10*f*g^4)*e), 1/15*(32*d^7*g^4 + 7*f^4*x^3*e^7 - 30*(d^3*g^3*x^3*e^3 - 3*d^4*g^3*x^2*e^2 + 3*d^5*g
^3*x*e - d^6*g^3)*sqrt(-d^2*g^2 + f^2*e^2)*arctan(sqrt(-d^2*g^2 + f^2*e^2)*(d*g*x + d*f - sqrt(-x^2*e^2 + d^2)
*f)/(d^2*g^2*x - f^2*x*e^2)) + 3*(8*d*f^3*g*x^3 - 7*d*f^4*x^2)*e^6 + (25*d^2*f^2*g^2*x^3 - 72*d^2*f^3*g*x^2 +
21*d^2*f^4*x)*e^5 - (24*d^3*f*g^3*x^3 + 75*d^3*f^2*g^2*x^2 - 72*d^3*f^3*g*x + 7*d^3*f^4)*e^4 - (32*d^4*g^4*x^3
- 72*d^4*f*g^3*x^2 - 75*d^4*f^2*g^2*x + 24*d^4*f^3*g)*e^3 + (96*d^5*g^4*x^2 - 72*d^5*f*g^3*x - 25*d^5*f^2*g^2
)*e^2 - 24*(4*d^6*g^4*x - d^6*f*g^3)*e + (32*d^6*g^4 - 2*f^4*x^2*e^6 - 3*(3*d*f^3*g*x^2 - 2*d*f^4*x)*e^5 - (20
*d^2*f^2*g^2*x^2 - 27*d^2*f^3*g*x + 7*d^2*f^4)*e^4 + 3*(3*d^3*f*g^3*x^2 + 15*d^3*f^2*g^2*x - 8*d^3*f^3*g)*e^3
+ (22*d^4*g^4*x^2 - 27*d^4*f*g^3*x - 25*d^4*f^2*g^2)*e^2 - 3*(17*d^5*g^4*x - 8*d^5*f*g^3)*e)*sqrt(-x^2*e^2 + d
^2))/(d^11*g^5 + d^3*f^5*x^3*e^8 + 3*(d^4*f^4*g*x^3 - d^4*f^5*x^2)*e^7 + (2*d^5*f^3*g^2*x^3 - 9*d^5*f^4*g*x^2
+ 3*d^5*f^5*x)*e^6 - (2*d^6*f^2*g^3*x^3 + 6*d^6*f^3*g^2*x^2 - 9*d^6*f^4*g*x + d^6*f^5)*e^5 - 3*(d^7*f*g^4*x^3
- 2*d^7*f^2*g^3*x^2 - 2*d^7*f^3*g^2*x + d^7*f^4*g)*e^4 - (d^8*g^5*x^3 - 9*d^8*f*g^4*x^2 + 6*d^8*f^2*g^3*x + 2*
d^8*f^3*g^2)*e^3 + (3*d^9*g^5*x^2 - 9*d^9*f*g^4*x + 2*d^9*f^2*g^3)*e^2 - 3*(d^10*g^5*x - d^10*f*g^4)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 607 vs. \(2 (225) = 450\).
time = 2.00, size = 607, normalized size = 2.51 \begin {gather*} -\frac {2 \, g^{3} \arctan \left (\frac {d g + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f e^{\left (-1\right )}}{x}}{\sqrt {-d^{2} g^{2} + f^{2} e^{2}}}\right )}{{\left (d^{3} g^{3} + 3 \, d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} + f^{3} e^{3}\right )} \sqrt {-d^{2} g^{2} + f^{2} e^{2}}} - \frac {2 \, {\left (\frac {115 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} g^{2} e^{\left (-2\right )}}{x} - \frac {185 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} g^{2} e^{\left (-4\right )}}{x^{2}} + \frac {135 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} g^{2} e^{\left (-6\right )}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} g^{2} e^{\left (-8\right )}}{x^{4}} - 32 \, d^{2} g^{2} - 24 \, d f g e + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f g e^{\left (-1\right )}}{x} - \frac {135 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f g e^{\left (-3\right )}}{x^{2}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f g e^{\left (-5\right )}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d f g e^{\left (-7\right )}}{x^{4}} - 7 \, f^{2} e^{2} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{2} e^{\left (-2\right )}}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{2} e^{\left (-4\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{2} e^{\left (-6\right )}}{x^{4}} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{2}}{x}\right )}}{15 \, {\left (d^{6} g^{3} + 3 \, d^{5} f g^{2} e + 3 \, d^{4} f^{2} g e^{2} + d^{3} f^{3} e^{3}\right )} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{5}} \end {gather*}

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*g^3*arctan((d*g + (d*e + sqrt(-x^2*e^2 + d^2)*e)*f*e^(-1)/x)/sqrt(-d^2*g^2 + f^2*e^2))/((d^3*g^3 + 3*d^2*f*
g^2*e + 3*d*f^2*g*e^2 + f^3*e^3)*sqrt(-d^2*g^2 + f^2*e^2)) - 2/15*(115*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*g^2*
e^(-2)/x - 185*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*g^2*e^(-4)/x^2 + 135*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2*
g^2*e^(-6)/x^3 - 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*g^2*e^(-8)/x^4 - 32*d^2*g^2 - 24*d*f*g*e + 75*(d*e +
sqrt(-x^2*e^2 + d^2)*e)*d*f*g*e^(-1)/x - 135*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d*f*g*e^(-3)/x^2 + 105*(d*e + sq
rt(-x^2*e^2 + d^2)*e)^3*d*f*g*e^(-5)/x^3 - 45*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d*f*g*e^(-7)/x^4 - 7*f^2*e^2 -
40*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*f^2*e^(-2)/x^2 + 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*f^2*e^(-4)/x^3 - 15*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^4*f^2*e^(-6)/x^4 + 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*f^2/x)/((d^6*g^3 + 3*d^5*f*
g^2*e + 3*d^4*f^2*g*e^2 + d^3*f^3*e^3)*((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)^5)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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