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

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

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Rubi [A]  time = 1.26, 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 = {1647, 807, 725, 204} \begin {gather*} \frac {e \left (e x \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )+45 d^3 g^2\right )}{15 d^3 \sqrt {d^2-e^2 x^2} (d g+e f)^4}+\frac {e g^3 (4 e f-3 d g) \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 )}{(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} \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 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])

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 807

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 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]

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 ) \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) (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 = 0.61, size = 341, normalized size = 1.10 \begin {gather*} \frac {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 {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )+\frac {(d+e x) \left (e^2 f^2-d^2 g^2\right ) \left (15 d^6 g^4-9 d^5 e g^3 (8 f+13 g x)+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 )+6 d e^5 f^2 x \left (-f^2+f g x+2 g^2 x^2\right )+2 e^6 f^3 x^2 (f+g x)\right )}{d^3 (d-e x)^2 \sqrt {d^2-e^2 x^2} (f+g x)}}{15 (e f-d g)^2 (d g+e f)^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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IntegrateAlgebraic [F]  time = 180.05, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [B]  time = 1.02, size = 3305, normalized size = 10.63

result too large to display

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*(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)]

________________________________________________________________________________________

giac [C]  time = 2.97, size = 4343, normalized size = 13.96

result too large to display

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]

-1/15*(15*(-45*I*d^9*g^12*e^6*log(d^2*g^4*e^2) - 75*I*d^8*f*g^11*e^7*log(d^2*g^4*e^2) + 90*I*d^7*f^2*g^10*e^8*
log(d^2*g^4*e^2) + 144*sqrt(d^2*g^2 - f^2*e^2)*d^8*g^10*abs(g)*e^6 + 210*I*d^6*f^3*g^9*e^9*log(d^2*g^4*e^2) +
346*sqrt(d^2*g^2 - f^2*e^2)*d^7*f*g^9*abs(g)*e^7 + 15*I*d^5*f^4*g^8*e^10*log(d^2*g^4*e^2) + 6*sqrt(d^2*g^2 - f
^2*e^2)*d^6*f^2*g^8*abs(g)*e^8 - 135*I*d^4*f^5*g^7*e^11*log(d^2*g^4*e^2) - 536*sqrt(d^2*g^2 - f^2*e^2)*d^5*f^3
*g^7*abs(g)*e^9 - 60*I*d^3*f^6*g^6*e^12*log(d^2*g^4*e^2) - 320*sqrt(d^2*g^2 - f^2*e^2)*d^4*f^4*g^6*abs(g)*e^10
+ 154*sqrt(d^2*g^2 - f^2*e^2)*d^3*f^5*g^5*abs(g)*e^11 + 166*sqrt(d^2*g^2 - f^2*e^2)*d^2*f^6*g^4*abs(g)*e^12 +
36*sqrt(d^2*g^2 - f^2*e^2)*d*f^7*g^3*abs(g)*e^13 + 4*sqrt(d^2*g^2 - f^2*e^2)*f^8*g^2*abs(g)*e^14)*sgn(1/(g*x
+ f))*sgn(g)/(30*I*sqrt(d^2*g^2 - f^2*e^2)*d^13*g^10*abs(g)*e^5 + 180*I*sqrt(d^2*g^2 - f^2*e^2)*d^12*f*g^9*abs
(g)*e^6 + 390*I*sqrt(d^2*g^2 - f^2*e^2)*d^11*f^2*g^8*abs(g)*e^7 + 240*I*sqrt(d^2*g^2 - f^2*e^2)*d^10*f^3*g^7*a
bs(g)*e^8 - 420*I*sqrt(d^2*g^2 - f^2*e^2)*d^9*f^4*g^6*abs(g)*e^9 - 840*I*sqrt(d^2*g^2 - f^2*e^2)*d^8*f^5*g^5*a
bs(g)*e^10 - 420*I*sqrt(d^2*g^2 - f^2*e^2)*d^7*f^6*g^4*abs(g)*e^11 + 240*I*sqrt(d^2*g^2 - f^2*e^2)*d^6*f^7*g^3
*abs(g)*e^12 + 390*I*sqrt(d^2*g^2 - f^2*e^2)*d^5*f^8*g^2*abs(g)*e^13 + 180*I*sqrt(d^2*g^2 - f^2*e^2)*d^4*f^9*g
*abs(g)*e^14 + 30*I*sqrt(d^2*g^2 - f^2*e^2)*d^3*f^10*abs(g)*e^15) + 15*(3*d*g^7*e - 4*f*g^6*e^2)*log(abs(f*g*e
^2 + sqrt(d^2*g^2 - f^2*e^2)*(sqrt(d^2*g^2/(g*x + f)^2 + 2*f*e^2/(g*x + f) - f^2*e^2/(g*x + f)^2 - e^2) + sqrt
(d^2*g^4 - f^2*g^2*e^2)/((g*x + f)*g))*abs(g)))/(sqrt(d^2*g^2 - f^2*e^2)*d^5*g^5*abs(g)*sgn(1/(g*x + f))*sgn(g
) + 3*sqrt(d^2*g^2 - f^2*e^2)*d^4*f*g^4*abs(g)*e*sgn(1/(g*x + f))*sgn(g) + 2*sqrt(d^2*g^2 - f^2*e^2)*d^3*f^2*g
^3*abs(g)*e^2*sgn(1/(g*x + f))*sgn(g) - 2*sqrt(d^2*g^2 - f^2*e^2)*d^2*f^3*g^2*abs(g)*e^3*sgn(1/(g*x + f))*sgn(
g) - 3*sqrt(d^2*g^2 - f^2*e^2)*d*f^4*g*abs(g)*e^4*sgn(1/(g*x + f))*sgn(g) - sqrt(d^2*g^2 - f^2*e^2)*f^5*abs(g)
*e^5*sgn(1/(g*x + f))*sgn(g)) - ((72*d^8*g^24*e^10*sgn(1/(g*x + f))^3*sgn(g)^3 - 187*d^7*f*g^23*e^11*sgn(1/(g*
x + f))^3*sgn(g)^3 + 146*d^6*f^2*g^22*e^12*sgn(1/(g*x + f))^3*sgn(g)^3 - 21*d^5*f^3*g^21*e^13*sgn(1/(g*x + f))
^3*sgn(g)^3 - 8*d^4*f^4*g^20*e^14*sgn(1/(g*x + f))^3*sgn(g)^3 - 2*d^3*f^5*g^19*e^15*sgn(1/(g*x + f))^3*sgn(g)^
3)/(d^13*g^24*e^4*sgn(1/(g*x + f))^4*sgn(g)^4 + d^12*f*g^23*e^5*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*
e^6*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^10*f^3*g^21*e^7*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(
g*x + f))^4*sgn(g)^4 + 3*d^8*f^5*g^19*e^9*sgn(1/(g*x + f))^4*sgn(g)^4 - d^7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*s
gn(g)^4 - d^6*f^7*g^17*e^11*sgn(1/(g*x + f))^4*sgn(g)^4) + (5*(9*d^9*g^26*e^9*sgn(1/(g*x + f))^3*sgn(g)^3 - 10
2*d^8*f*g^25*e^10*sgn(1/(g*x + f))^3*sgn(g)^3 + 220*d^7*f^2*g^24*e^11*sgn(1/(g*x + f))^3*sgn(g)^3 - 158*d^6*f^
3*g^23*e^12*sgn(1/(g*x + f))^3*sgn(g)^3 + 21*d^5*f^4*g^22*e^13*sgn(1/(g*x + f))^3*sgn(g)^3 + 8*d^4*f^5*g^21*e^
14*sgn(1/(g*x + f))^3*sgn(g)^3 + 2*d^3*f^6*g^20*e^15*sgn(1/(g*x + f))^3*sgn(g)^3)/(d^13*g^24*e^4*sgn(1/(g*x +
f))^4*sgn(g)^4 + d^12*f*g^23*e^5*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*e^6*sgn(1/(g*x + f))^4*sgn(g)^4
- 3*d^10*f^3*g^21*e^7*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^8*f^
5*g^19*e^9*sgn(1/(g*x + f))^4*sgn(g)^4 - d^7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*sgn(g)^4 - d^6*f^7*g^17*e^11*sgn
(1/(g*x + f))^4*sgn(g)^4) - (5*(36*d^10*g^28*e^8*sgn(1/(g*x + f))^3*sgn(g)^3 - 53*d^9*f*g^27*e^9*sgn(1/(g*x +
f))^3*sgn(g)^3 - 206*d^8*f^2*g^26*e^10*sgn(1/(g*x + f))^3*sgn(g)^3 + 512*d^7*f^3*g^25*e^11*sgn(1/(g*x + f))^3*
sgn(g)^3 - 350*d^6*f^4*g^24*e^12*sgn(1/(g*x + f))^3*sgn(g)^3 + 41*d^5*f^5*g^23*e^13*sgn(1/(g*x + f))^3*sgn(g)^
3 + 16*d^4*f^6*g^22*e^14*sgn(1/(g*x + f))^3*sgn(g)^3 + 4*d^3*f^7*g^21*e^15*sgn(1/(g*x + f))^3*sgn(g)^3)/(d^13*
g^24*e^4*sgn(1/(g*x + f))^4*sgn(g)^4 + d^12*f*g^23*e^5*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*e^6*sgn(1
/(g*x + f))^4*sgn(g)^4 - 3*d^10*f^3*g^21*e^7*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(g*x + f))
^4*sgn(g)^4 + 3*d^8*f^5*g^19*e^9*sgn(1/(g*x + f))^4*sgn(g)^4 - d^7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*sgn(g)^4 -
d^6*f^7*g^17*e^11*sgn(1/(g*x + f))^4*sgn(g)^4) + (5*(21*d^11*g^30*e^7*sgn(1/(g*x + f))^3*sgn(g)^3 - 178*d^10*
f*g^29*e^8*sgn(1/(g*x + f))^3*sgn(g)^3 + 287*d^9*f^2*g^28*e^9*sgn(1/(g*x + f))^3*sgn(g)^3 + 132*d^8*f^3*g^27*e
^10*sgn(1/(g*x + f))^3*sgn(g)^3 - 601*d^7*f^4*g^26*e^11*sgn(1/(g*x + f))^3*sgn(g)^3 + 398*d^6*f^5*g^25*e^12*sg
n(1/(g*x + f))^3*sgn(g)^3 - 39*d^5*f^6*g^24*e^13*sgn(1/(g*x + f))^3*sgn(g)^3 - 16*d^4*f^7*g^23*e^14*sgn(1/(g*x
+ f))^3*sgn(g)^3 - 4*d^3*f^8*g^22*e^15*sgn(1/(g*x + f))^3*sgn(g)^3)/(d^13*g^24*e^4*sgn(1/(g*x + f))^4*sgn(g)^
4 + d^12*f*g^23*e^5*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*e^6*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^10*f^3
*g^21*e^7*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^8*f^5*g^19*e^9*sg
n(1/(g*x + f))^4*sgn(g)^4 - d^7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*sgn(g)^4 - d^6*f^7*g^17*e^11*sgn(1/(g*x + f))
^4*sgn(g)^4) - (5*(27*d^12*g^32*e^6*sgn(1/(g*x + f))^3*sgn(g)^3 - 18*d^11*f*g^31*e^7*sgn(1/(g*x + f))^3*sgn(g)
^3 - 227*d^10*f^2*g^30*e^8*sgn(1/(g*x + f))^3*sgn(g)^3 + 406*d^9*f^3*g^29*e^9*sgn(1/(g*x + f))^3*sgn(g)^3 - 27
*d^8*f^4*g^28*e^10*sgn(1/(g*x + f))^3*sgn(g)^3 - 368*d^7*f^5*g^27*e^11*sgn(1/(g*x + f))^3*sgn(g)^3 + 235*d^6*f
^6*g^26*e^12*sgn(1/(g*x + f))^3*sgn(g)^3 - 18*d^5*f^7*g^25*e^13*sgn(1/(g*x + f))^3*sgn(g)^3 - 8*d^4*f^8*g^24*e
^14*sgn(1/(g*x + f))^3*sgn(g)^3 - 2*d^3*f^9*g^23*e^15*sgn(1/(g*x + f))^3*sgn(g)^3)/(d^13*g^24*e^4*sgn(1/(g*x +
f))^4*sgn(g)^4 + d^12*f*g^23*e^5*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*e^6*sgn(1/(g*x + f))^4*sgn(g)^
4 - 3*d^10*f^3*g^21*e^7*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^8*f
^5*g^19*e^9*sgn(1/(g*x + f))^4*sgn(g)^4 - d^7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*sgn(g)^4 - d^6*f^7*g^17*e^11*sg
n(1/(g*x + f))^4*sgn(g)^4) + (2*(36*d^13*g^34*e^5*sgn(1/(g*x + f))^3*sgn(g)^3 - 181*d^12*f*g^33*e^6*sgn(1/(g*x
+ f))^3*sgn(g)^3 + 203*d^11*f^2*g^32*e^7*sgn(1/(g*x + f))^3*sgn(g)^3 + 217*d^10*f^3*g^31*e^8*sgn(1/(g*x + f))
^3*sgn(g)^3 - 504*d^9*f^4*g^30*e^9*sgn(1/(g*x + f))^3*sgn(g)^3 + 113*d^8*f^5*g^29*e^10*sgn(1/(g*x + f))^3*sgn(
g)^3 + 256*d^7*f^6*g^28*e^11*sgn(1/(g*x + f))^3*sgn(g)^3 - 153*d^6*f^7*g^27*e^12*sgn(1/(g*x + f))^3*sgn(g)^3 +
8*d^5*f^8*g^26*e^13*sgn(1/(g*x + f))^3*sgn(g)^3 + 4*d^4*f^9*g^25*e^14*sgn(1/(g*x + f))^3*sgn(g)^3 + d^3*f^10*
g^24*e^15*sgn(1/(g*x + f))^3*sgn(g)^3)/(d^13*g^24*e^4*sgn(1/(g*x + f))^4*sgn(g)^4 + d^12*f*g^23*e^5*sgn(1/(g*x
+ f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*e^6*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^10*f^3*g^21*e^7*sgn(1/(g*x + f))^4*s
gn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^8*f^5*g^19*e^9*sgn(1/(g*x + f))^4*sgn(g)^4 - d^
7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*sgn(g)^4 - d^6*f^7*g^17*e^11*sgn(1/(g*x + f))^4*sgn(g)^4) - 15*(d^14*g^36*e
^4*sgn(1/(g*x + f))^3*sgn(g)^3 - 2*d^13*f*g^35*e^5*sgn(1/(g*x + f))^3*sgn(g)^3 - 2*d^12*f^2*g^34*e^6*sgn(1/(g*
x + f))^3*sgn(g)^3 + 6*d^11*f^3*g^33*e^7*sgn(1/(g*x + f))^3*sgn(g)^3 - 6*d^9*f^5*g^31*e^9*sgn(1/(g*x + f))^3*s
gn(g)^3 + 2*d^8*f^6*g^30*e^10*sgn(1/(g*x + f))^3*sgn(g)^3 + 2*d^7*f^7*g^29*e^11*sgn(1/(g*x + f))^3*sgn(g)^3 -
d^6*f^8*g^28*e^12*sgn(1/(g*x + f))^3*sgn(g)^3)/((d^13*g^24*e^4*sgn(1/(g*x + f))^4*sgn(g)^4 + d^12*f*g^23*e^5*s
gn(1/(g*x + f))^4*sgn(g)^4 - 3*d^11*f^2*g^22*e^6*sgn(1/(g*x + f))^4*sgn(g)^4 - 3*d^10*f^3*g^21*e^7*sgn(1/(g*x
+ f))^4*sgn(g)^4 + 3*d^9*f^4*g^20*e^8*sgn(1/(g*x + f))^4*sgn(g)^4 + 3*d^8*f^5*g^19*e^9*sgn(1/(g*x + f))^4*sgn(
g)^4 - d^7*f^6*g^18*e^10*sgn(1/(g*x + f))^4*sgn(g)^4 - d^6*f^7*g^17*e^11*sgn(1/(g*x + f))^4*sgn(g)^4)*(g*x + f
)*g))/((g*x + f)*g))/((g*x + f)*g))/((g*x + f)*g))/((g*x + f)*g))/((g*x + f)*g))/(d^2*g^2/(g*x + f)^2 + 2*f*e^
2/(g*x + f) - f^2*e^2/(g*x + f)^2 - e^2)^(5/2))/g^2

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maple [B]  time = 0.03, size = 6760, normalized size = 21.74 \begin {gather*} \text {output too large to display} \end {gather*}

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)

[Out]

result too large to display

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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?` f
or more details)Is (d*g-e*f)    *(d*g+e*f)    positive, negative or zero?

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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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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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