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

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

[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)

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Rubi [A]
time = 1.01, antiderivative size = 311, normalized size of antiderivative = 1.00, 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 = {1661, 821, 739, 210} \begin {gather*} \frac {e g^3 (4 e f-3 d g) \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 )}{(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} \end {gather*}

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 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*} \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 ) \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*}

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Mathematica [A]
time = 10.44, size = 341, normalized size = 1.10 \begin {gather*} \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} \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)^2 (e f+d g)^5} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(3288\) vs. \(2(291)=582\).
time = 0.10, size = 3289, normalized size = 10.58

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

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,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/(-(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)))))+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)/(-(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)-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/(-(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)))))+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)/(-(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))))

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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)^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(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. 1607 vs. \(2 (301) = 602\).
time = 5.30, size = 3252, normalized size = 10.46 \begin {gather*} \text {Too large to display} \end {gather*}

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*(15*d^9*g^7*x + 15*d^9*f*g^6 - 15*sqrt(d^2*g^2 - f^2*e^2)*(4*(d^3*f^2*g^4*x^4 + d^3*f^3*g^3*x^3)*e^5 -
3*(d^4*f*g^5*x^4 + 5*d^4*f^2*g^4*x^3 + 4*d^4*f^3*g^3*x^2)*e^4 + 3*(3*d^5*f*g^5*x^3 + 7*d^5*f^2*g^4*x^2 + 4*d^5
*f^3*g^3*x)*e^3 - (9*d^6*f*g^5*x^2 + 13*d^6*f^2*g^4*x + 4*d^6*f^3*g^3)*e^2 + 3*(d^7*f*g^5*x + d^7*f^2*g^4)*e)*
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)) + 7*(f^6*g*x^4 + f^7*x^3)*e^9 + 3*(9*d*f^5*g^2*x^4 + 2*d*f^6*g*x^3
- 7*d*f^7*x^2)*e^8 + (31*d^2*f^4*g^3*x^4 - 50*d^2*f^5*g^2*x^3 - 60*d^2*f^6*g*x^2 + 21*d^2*f^7*x)*e^7 - (99*d^3
*f^3*g^4*x^4 + 192*d^3*f^4*g^3*x^3 + 12*d^3*f^5*g^2*x^2 - 74*d^3*f^6*g*x + 7*d^3*f^7)*e^6 - (23*d^4*f^2*g^5*x^
4 - 274*d^4*f^3*g^4*x^3 - 390*d^4*f^4*g^3*x^2 - 66*d^4*f^5*g^2*x + 27*d^4*f^6*g)*e^5 + (72*d^5*f*g^6*x^4 + 141
*d^5*f^2*g^5*x^3 - 228*d^5*f^3*g^4*x^2 - 328*d^5*f^4*g^3*x - 31*d^5*f^5*g^2)*e^4 - 3*(5*d^6*g^7*x^4 + 77*d^6*f
*g^6*x^3 + 95*d^6*f^2*g^5*x^2 - 10*d^6*f^3*g^4*x - 33*d^6*f^4*g^3)*e^3 + (45*d^7*g^7*x^3 + 261*d^7*f*g^6*x^2 +
239*d^7*f^2*g^5*x + 23*d^7*f^3*g^4)*e^2 - 9*(5*d^8*g^7*x^2 + 13*d^8*f*g^6*x + 8*d^8*f^2*g^5)*e + (15*d^8*f*g^
6 - 2*(f^6*g*x^3 + f^7*x^2)*e^8 - 6*(2*d*f^5*g^2*x^3 + d*f^6*g*x^2 - d*f^7*x)*e^7 - (41*d^2*f^4*g^3*x^3 + 5*d^
2*f^5*g^2*x^2 - 29*d^2*f^6*g*x + 7*d^2*f^7)*e^6 + 3*(28*d^3*f^3*g^4*x^3 + 49*d^3*f^4*g^3*x^2 + 17*d^3*f^5*g^2*
x - 9*d^3*f^6*g)*e^5 + (43*d^4*f^2*g^5*x^3 - 164*d^4*f^3*g^4*x^2 - 193*d^4*f^4*g^3*x - 31*d^4*f^5*g^2)*e^4 - 3
*(24*d^5*f*g^6*x^3 + 47*d^5*f^2*g^5*x^2 - 20*d^5*f^3*g^4*x - 33*d^5*f^4*g^3)*e^3 + (171*d^6*f*g^6*x^2 + 164*d^
6*f^2*g^5*x + 23*d^6*f^3*g^4)*e^2 - 9*(13*d^7*f*g^6*x + 8*d^7*f^2*g^5)*e)*sqrt(-x^2*e^2 + d^2))/(d^13*f*g^8*x
+ d^13*f^2*g^7 - (d^3*f^8*g*x^4 + d^3*f^9*x^3)*e^10 - 3*(d^4*f^7*g^2*x^4 - d^4*f^9*x^2)*e^9 - (d^5*f^6*g^3*x^4
- 8*d^5*f^7*g^2*x^3 - 6*d^5*f^8*g*x^2 + 3*d^5*f^9*x)*e^8 + (5*d^6*f^5*g^4*x^4 + 8*d^6*f^6*g^3*x^3 - 6*d^6*f^7
*g^2*x^2 - 8*d^6*f^8*g*x + d^6*f^9)*e^7 + (5*d^7*f^4*g^5*x^4 - 10*d^7*f^5*g^4*x^3 - 18*d^7*f^6*g^3*x^2 + 3*d^7
*f^8*g)*e^6 - (d^8*f^3*g^6*x^4 + 16*d^8*f^4*g^5*x^3 - 16*d^8*f^6*g^3*x - d^8*f^7*g^2)*e^5 - (3*d^9*f^2*g^7*x^4
- 18*d^9*f^4*g^5*x^2 - 10*d^9*f^5*g^4*x + 5*d^9*f^6*g^3)*e^4 - (d^10*f*g^8*x^4 - 8*d^10*f^2*g^7*x^3 - 6*d^10*
f^3*g^6*x^2 + 8*d^10*f^4*g^5*x + 5*d^10*f^5*g^4)*e^3 + (3*d^11*f*g^8*x^3 - 6*d^11*f^2*g^7*x^2 - 8*d^11*f^3*g^6
*x + d^11*f^4*g^5)*e^2 - 3*(d^12*f*g^8*x^2 - d^12*f^3*g^6)*e), -1/15*(15*d^9*g^7*x + 15*d^9*f*g^6 - 30*sqrt(-d
^2*g^2 + f^2*e^2)*(4*(d^3*f^2*g^4*x^4 + d^3*f^3*g^3*x^3)*e^5 - 3*(d^4*f*g^5*x^4 + 5*d^4*f^2*g^4*x^3 + 4*d^4*f^
3*g^3*x^2)*e^4 + 3*(3*d^5*f*g^5*x^3 + 7*d^5*f^2*g^4*x^2 + 4*d^5*f^3*g^3*x)*e^3 - (9*d^6*f*g^5*x^2 + 13*d^6*f^2
*g^4*x + 4*d^6*f^3*g^3)*e^2 + 3*(d^7*f*g^5*x + d^7*f^2*g^4)*e)*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)) + 7*(f^6*g*x^4 + f^7*x^3)*e^9 + 3*(9*d*f^5*g^2*x^4 + 2*d*f^6*
g*x^3 - 7*d*f^7*x^2)*e^8 + (31*d^2*f^4*g^3*x^4 - 50*d^2*f^5*g^2*x^3 - 60*d^2*f^6*g*x^2 + 21*d^2*f^7*x)*e^7 - (
99*d^3*f^3*g^4*x^4 + 192*d^3*f^4*g^3*x^3 + 12*d^3*f^5*g^2*x^2 - 74*d^3*f^6*g*x + 7*d^3*f^7)*e^6 - (23*d^4*f^2*
g^5*x^4 - 274*d^4*f^3*g^4*x^3 - 390*d^4*f^4*g^3*x^2 - 66*d^4*f^5*g^2*x + 27*d^4*f^6*g)*e^5 + (72*d^5*f*g^6*x^4
+ 141*d^5*f^2*g^5*x^3 - 228*d^5*f^3*g^4*x^2 - 328*d^5*f^4*g^3*x - 31*d^5*f^5*g^2)*e^4 - 3*(5*d^6*g^7*x^4 + 77
*d^6*f*g^6*x^3 + 95*d^6*f^2*g^5*x^2 - 10*d^6*f^3*g^4*x - 33*d^6*f^4*g^3)*e^3 + (45*d^7*g^7*x^3 + 261*d^7*f*g^6
*x^2 + 239*d^7*f^2*g^5*x + 23*d^7*f^3*g^4)*e^2 - 9*(5*d^8*g^7*x^2 + 13*d^8*f*g^6*x + 8*d^8*f^2*g^5)*e + (15*d^
8*f*g^6 - 2*(f^6*g*x^3 + f^7*x^2)*e^8 - 6*(2*d*f^5*g^2*x^3 + d*f^6*g*x^2 - d*f^7*x)*e^7 - (41*d^2*f^4*g^3*x^3
+ 5*d^2*f^5*g^2*x^2 - 29*d^2*f^6*g*x + 7*d^2*f^7)*e^6 + 3*(28*d^3*f^3*g^4*x^3 + 49*d^3*f^4*g^3*x^2 + 17*d^3*f^
5*g^2*x - 9*d^3*f^6*g)*e^5 + (43*d^4*f^2*g^5*x^3 - 164*d^4*f^3*g^4*x^2 - 193*d^4*f^4*g^3*x - 31*d^4*f^5*g^2)*e
^4 - 3*(24*d^5*f*g^6*x^3 + 47*d^5*f^2*g^5*x^2 - 20*d^5*f^3*g^4*x - 33*d^5*f^4*g^3)*e^3 + (171*d^6*f*g^6*x^2 +
164*d^6*f^2*g^5*x + 23*d^6*f^3*g^4)*e^2 - 9*(13*d^7*f*g^6*x + 8*d^7*f^2*g^5)*e)*sqrt(-x^2*e^2 + d^2))/(d^13*f*
g^8*x + d^13*f^2*g^7 - (d^3*f^8*g*x^4 + d^3*f^9*x^3)*e^10 - 3*(d^4*f^7*g^2*x^4 - d^4*f^9*x^2)*e^9 - (d^5*f^6*g
^3*x^4 - 8*d^5*f^7*g^2*x^3 - 6*d^5*f^8*g*x^2 + 3*d^5*f^9*x)*e^8 + (5*d^6*f^5*g^4*x^4 + 8*d^6*f^6*g^3*x^3 - 6*d
^6*f^7*g^2*x^2 - 8*d^6*f^8*g*x + d^6*f^9)*e^7 + (5*d^7*f^4*g^5*x^4 - 10*d^7*f^5*g^4*x^3 - 18*d^7*f^6*g^3*x^2 +
3*d^7*f^8*g)*e^6 - (d^8*f^3*g^6*x^4 + 16*d^8*f^4*g^5*x^3 - 16*d^8*f^6*g^3*x - d^8*f^7*g^2)*e^5 - (3*d^9*f^2*g
^7*x^4 - 18*d^9*f^4*g^5*x^2 - 10*d^9*f^5*g^4*x + 5*d^9*f^6*g^3)*e^4 - (d^10*f*g^8*x^4 - 8*d^10*f^2*g^7*x^3 - 6
*d^10*f^3*g^6*x^2 + 8*d^10*f^4*g^5*x + 5*d^10*f^5*g^4)*e^3 + (3*d^11*f*g^8*x^3 - 6*d^11*f^2*g^7*x^2 - 8*d^11*f
^3*g^6*x + d^11*f^4*g^5)*e^2 - 3*(d^12*f*g^8*x^2 - d^12*f^3*g^6)*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 )^{2}}\, dx \end {gather*}

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]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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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 )}^2\,{\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)^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)

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