3.6.87 \(\int \frac {(d+e x)^3}{(f+g x)^3 (d^2-e^2 x^2)^{7/2}} \, dx\) [587]

Optimal. Leaf size=398 \[ \frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5 \sqrt {e^2 f^2-d^2 g^2}} \]

[Out]

4/5*d*e^2*(e*x+d)/(d*g+e*f)^3/(-e^2*x^2+d^2)^(5/2)-1/15*e^2*(5*d*(-5*d*g+e*f)-e*(31*d*g+e*f)*x)/d/(d*g+e*f)^4/
(-e^2*x^2+d^2)^(3/2)+1/2*e^2*g^3*(13*d^2*g^2-30*d*e*f*g+20*e^2*f^2)*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)^2/(d*g+e*f)^5/(-d^2*g^2+e^2*f^2)^(1/2)+1/15*e^2*(90*d^3*g^2+e*(107*d^2*
g^2+19*d*e*f*g+2*e^2*f^2)*x)/d^3/(d*g+e*f)^5/(-e^2*x^2+d^2)^(1/2)+1/2*g^4*(-e^2*x^2+d^2)^(1/2)/(-d*g+e*f)/(d*g
+e*f)^4/(g*x+f)^2+3/2*e*g^4*(-2*d*g+3*e*f)*(-e^2*x^2+d^2)^(1/2)/(-d*g+e*f)^2/(d*g+e*f)^5/(g*x+f)

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Rubi [A]
time = 2.84, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1661, 1665, 821, 739, 210} \begin {gather*} \frac {e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \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 )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt {e^2 f^2-d^2 g^2}}+\frac {3 e g^4 \sqrt {d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac {e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}+\frac {e^2 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 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)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*d*e^2*(d + e*x))/(5*(e*f + d*g)^3*(d^2 - e^2*x^2)^(5/2)) - (e^2*(5*d*(e*f - 5*d*g) - e*(e*f + 31*d*g)*x))/(
15*d*(e*f + d*g)^4*(d^2 - e^2*x^2)^(3/2)) + (e^2*(90*d^3*g^2 + e*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2)*x))/(1
5*d^3*(e*f + d*g)^5*Sqrt[d^2 - e^2*x^2]) + (g^4*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)*(e*f + d*g)^4*(f + g*x)^2)
+ (3*e*g^4*(3*e*f - 2*d*g)*Sqrt[d^2 - e^2*x^2])/(2*(e*f - d*g)^2*(e*f + d*g)^5*(f + g*x)) + (e^2*g^3*(20*e^2*
f^2 - 30*d*e*f*g + 13*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])])/(2*(e*
f - d*g)^2*(e*f + d*g)^5*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]

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {\frac {d^3 e^2 \left (e^3 f^3+15 d e^2 f^2 g+15 d^2 e f g^2+5 d^3 g^3\right )}{(e f+d g)^3}-\frac {d^2 e^3 \left (5 e^3 f^3-33 d e^2 f^2 g-45 d^2 e f g^2-15 d^3 g^3\right ) x}{(e f+d g)^3}+\frac {4 d^3 e^4 g^2 (12 e f+5 d g) x^2}{(e f+d g)^3}+\frac {16 d^3 e^5 g^3 x^3}{(e f+d g)^3}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {\frac {d^3 e^4 \left (2 e^4 f^4+17 d e^3 f^3 g+90 d^2 e^2 f^2 g^2+60 d^3 e f g^3+15 d^4 g^4\right )}{(e f+d g)^4}+\frac {3 d^3 e^5 g \left (2 e^2 f^2+45 d e f g+15 d^2 g^2\right ) x}{(e f+d g)^3}+\frac {3 d^3 e^6 g^2 \left (2 e^2 f^2+57 d e f g+25 d^2 g^2\right ) x^2}{(e f+d g)^4}+\frac {2 d^3 e^7 g^3 (e f+31 d g) x^3}{(e f+d g)^4}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {\frac {15 d^6 e^6 g^3 \left (10 e^2 f^2+5 d e f g+d^2 g^2\right )}{(e f+d g)^5}+\frac {45 d^6 e^7 g^4 (5 e f+d g) x}{(e f+d g)^5}+\frac {90 d^6 e^8 g^5 x^2}{(e f+d g)^5}}{(f+g x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {\int \frac {\frac {30 d^6 e^7 g^3 \left (10 e^2 f^2-5 d e f g-3 d^2 g^2\right )}{(e f+d g)^4}+\frac {15 d^6 e^8 g^4 (11 e f-13 d g) x}{(e f+d g)^4}}{(f+g x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^6 e^6 \left (e^2 f^2-d^2 g^2\right )}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{2 (e f-d g)^2 (e f+d g)^5}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}-\frac {\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\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 )}{2 (e f-d g)^2 (e f+d g)^5}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \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 )}{2 (e f-d g)^2 (e f+d g)^5 \sqrt {e^2 f^2-d^2 g^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.96, size = 387, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 e^2 (e f+d g)^2}{d (d-e x)^3}+\frac {2 e^2 (e f+d g) (2 e f+17 d g)}{d^2 (d-e x)^2}+\frac {2 e^2 \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right )}{d^3 (d-e x)}+\frac {15 g^4 (e f+d g)}{(e f-d g) (f+g x)^2}+\frac {45 e g^4 (3 e f-2 d g)}{(e f-d g)^2 (f+g x)}\right )-\frac {15 i e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \log \left (\frac {4 (e f-d g)^2 (e f+d g)^5 \left (i d^2 g+i e^2 f x+\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}\right )}{e^2 g^2 \sqrt {e^2 f^2-d^2 g^2} \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) (f+g x)}\right )}{(e f-d g)^2 \sqrt {e^2 f^2-d^2 g^2}}}{30 (e f+d g)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*((6*e^2*(e*f + d*g)^2)/(d*(d - e*x)^3) + (2*e^2*(e*f + d*g)*(2*e*f + 17*d*g))/(d^2*(d - e
*x)^2) + (2*e^2*(2*e^2*f^2 + 19*d*e*f*g + 107*d^2*g^2))/(d^3*(d - e*x)) + (15*g^4*(e*f + d*g))/((e*f - d*g)*(f
+ g*x)^2) + (45*e*g^4*(3*e*f - 2*d*g))/((e*f - d*g)^2*(f + g*x))) - ((15*I)*e^2*g^3*(20*e^2*f^2 - 30*d*e*f*g
+ 13*d^2*g^2)*Log[(4*(e*f - d*g)^2*(e*f + d*g)^5*(I*d^2*g + I*e^2*f*x + Sqrt[e^2*f^2 - d^2*g^2]*Sqrt[d^2 - e^2
*x^2]))/(e^2*g^2*Sqrt[e^2*f^2 - d^2*g^2]*(20*e^2*f^2 - 30*d*e*f*g + 13*d^2*g^2)*(f + g*x))])/((e*f - d*g)^2*Sq
rt[e^2*f^2 - d^2*g^2]))/(30*(e*f + d*g)^5)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6395\) vs. \(2(370)=740\).
time = 0.09, size = 6396, normalized size = 16.07

method	result	size
default	\(\text {Expression too large to display}\)	\(6396\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)^3/(-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. 2615 vs. \(2 (373) = 746\).
time = 27.97, size = 5268, normalized size = 13.24 \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)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

[-1/30*(15*d^11*g^10*x^2 + 30*d^11*f*g^9*x + 15*d^11*f^2*g^8 + 15*sqrt(d^2*g^2 - f^2*e^2)*(20*(d^3*f^4*g^5*x^5
+ 2*d^3*f^5*g^4*x^4 + d^3*f^6*g^3*x^3)*e^7 - 30*(d^4*f^3*g^6*x^5 + 4*d^4*f^4*g^5*x^4 + 5*d^4*f^5*g^4*x^3 + 2*
d^4*f^6*g^3*x^2)*e^6 + (13*d^5*f^2*g^7*x^5 + 116*d^5*f^3*g^6*x^4 + 253*d^5*f^4*g^5*x^3 + 210*d^5*f^5*g^4*x^2 +
60*d^5*f^6*g^3*x)*e^5 - (39*d^6*f^2*g^7*x^4 + 168*d^6*f^3*g^6*x^3 + 239*d^6*f^4*g^5*x^2 + 130*d^6*f^5*g^4*x +
20*d^6*f^6*g^3)*e^4 + 3*(13*d^7*f^2*g^7*x^3 + 36*d^7*f^3*g^6*x^2 + 33*d^7*f^4*g^5*x + 10*d^7*f^5*g^4)*e^3 - 1
3*(d^8*f^2*g^7*x^2 + 2*d^8*f^3*g^6*x + d^8*f^4*g^5)*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)) - 14*(f^8*
g^2*x^5 + 2*f^9*g*x^4 + f^10*x^3)*e^11 - 6*(10*d*f^7*g^3*x^5 + 13*d*f^8*g^2*x^4 - 4*d*f^9*g*x^3 - 7*d*f^10*x^2
)*e^10 - 6*(13*d^2*f^6*g^4*x^5 - 4*d^2*f^7*g^3*x^4 - 40*d^2*f^8*g^2*x^3 - 16*d^2*f^9*g*x^2 + 7*d^2*f^10*x)*e^9
+ 2*(240*d^3*f^5*g^5*x^5 + 597*d^3*f^6*g^4*x^4 + 384*d^3*f^7*g^3*x^3 - 56*d^3*f^8*g^2*x^2 - 76*d^3*f^9*g*x +
7*d^3*f^10)*e^8 - 6*(52*d^4*f^4*g^6*x^5 + 344*d^4*f^5*g^5*x^4 + 571*d^4*f^6*g^4*x^3 + 308*d^4*f^7*g^3*x^2 + 19
*d^4*f^8*g^2*x - 10*d^4*f^9*g)*e^7 - 6*(55*d^5*f^3*g^7*x^5 - 46*d^5*f^4*g^6*x^4 - 497*d^5*f^5*g^5*x^3 - 649*d^
5*f^6*g^4*x^2 - 266*d^5*f^7*g^3*x - 13*d^5*f^8*g^2)*e^6 + (419*d^6*f^2*g^8*x^5 + 1828*d^6*f^3*g^7*x^4 + 1463*d
^6*f^4*g^6*x^3 - 1362*d^6*f^5*g^5*x^2 - 1896*d^6*f^6*g^4*x - 480*d^6*f^7*g^3)*e^5 - 3*(30*d^7*f*g^9*x^5 + 479*
d^7*f^2*g^8*x^4 + 1198*d^7*f^3*g^7*x^3 + 975*d^7*f^4*g^6*x^2 + 122*d^7*f^5*g^5*x - 104*d^7*f^6*g^4)*e^4 - 3*(5
*d^8*g^10*x^5 - 80*d^8*f*g^9*x^4 - 594*d^8*f^2*g^8*x^3 - 1038*d^8*f^3*g^7*x^2 - 639*d^8*f^4*g^6*x - 110*d^8*f^
5*g^5)*e^3 + (45*d^9*g^10*x^4 - 180*d^9*f*g^9*x^3 - 914*d^9*f^2*g^8*x^2 - 1108*d^9*f^3*g^7*x - 419*d^9*f^4*g^6
)*e^2 - 45*(d^10*g^10*x^3 - 3*d^10*f^2*g^8*x - 2*d^10*f^3*g^7)*e + (15*d^10*f^2*g^8 + 4*(f^8*g^2*x^4 + 2*f^9*g
*x^3 + f^10*x^2)*e^10 + 6*(5*d*f^7*g^3*x^4 + 8*d*f^8*g^2*x^3 + d*f^9*g*x^2 - 2*d*f^10*x)*e^9 + 2*(69*d^2*f^6*g
^4*x^4 + 93*d^2*f^7*g^3*x^3 - 14*d^2*f^8*g^2*x^2 - 31*d^2*f^9*g*x + 7*d^2*f^10)*e^8 - 3*(185*d^3*f^5*g^5*x^4 +
408*d^3*f^6*g^4*x^3 + 276*d^3*f^7*g^3*x^2 + 38*d^3*f^8*g^2*x - 20*d^3*f^9*g)*e^7 + 3*(54*d^4*f^4*g^6*x^4 + 51
3*d^4*f^5*g^5*x^3 + 800*d^4*f^6*g^4*x^2 + 352*d^4*f^7*g^3*x + 26*d^4*f^8*g^2)*e^6 + 3*(175*d^5*f^3*g^7*x^4 + 1
53*d^5*f^4*g^6*x^3 - 399*d^5*f^5*g^5*x^2 - 542*d^5*f^6*g^4*x - 160*d^5*f^7*g^3)*e^5 - (304*d^6*f^2*g^8*x^4 + 1
733*d^6*f^3*g^7*x^3 + 1897*d^6*f^4*g^6*x^2 + 81*d^6*f^5*g^5*x - 312*d^6*f^6*g^4)*e^4 + 3*(239*d^7*f^2*g^8*x^3
+ 673*d^7*f^3*g^7*x^2 + 569*d^7*f^4*g^6*x + 110*d^7*f^5*g^5)*e^3 - (479*d^8*f^2*g^8*x^2 + 913*d^8*f^3*g^7*x +
419*d^8*f^4*g^6)*e^2 + 45*(d^9*f^2*g^8*x + 2*d^9*f^3*g^7)*e)*sqrt(-x^2*e^2 + d^2))/(d^15*f^2*g^11*x^2 + 2*d^15
*f^3*g^10*x + d^15*f^4*g^9 + (d^3*f^11*g^2*x^5 + 2*d^3*f^12*g*x^4 + d^3*f^13*x^3)*e^12 + 3*(d^4*f^10*g^3*x^5 +
d^4*f^11*g^2*x^4 - d^4*f^12*g*x^3 - d^4*f^13*x^2)*e^11 - 3*(3*d^5*f^10*g^3*x^4 + 5*d^5*f^11*g^2*x^3 + d^5*f^1
2*g*x^2 - d^5*f^13*x)*e^10 - (8*d^6*f^8*g^5*x^5 + 16*d^6*f^9*g^4*x^4 - d^6*f^10*g^3*x^3 - 17*d^6*f^11*g^2*x^2
- 7*d^6*f^12*g*x + d^6*f^13)*e^9 - 3*(2*d^7*f^7*g^6*x^5 - 4*d^7*f^8*g^5*x^4 - 14*d^7*f^9*g^4*x^3 - 7*d^7*f^10*
g^3*x^2 + 2*d^7*f^11*g^2*x + d^7*f^12*g)*e^8 + 6*(d^8*f^6*g^7*x^5 + 5*d^8*f^7*g^6*x^4 + 3*d^8*f^8*g^5*x^3 - 5*
d^8*f^9*g^4*x^2 - 4*d^8*f^10*g^3*x)*e^7 + 2*(4*d^9*f^5*g^8*x^5 - d^9*f^6*g^7*x^4 - 23*d^9*f^7*g^6*x^3 - 23*d^9
*f^8*g^5*x^2 - d^9*f^9*g^4*x + 4*d^9*f^10*g^3)*e^6 - 6*(4*d^10*f^5*g^8*x^4 + 5*d^10*f^6*g^7*x^3 - 3*d^10*f^7*g
^6*x^2 - 5*d^10*f^8*g^5*x - d^10*f^9*g^4)*e^5 - 3*(d^11*f^3*g^10*x^5 + 2*d^11*f^4*g^9*x^4 - 7*d^11*f^5*g^8*x^3
- 14*d^11*f^6*g^7*x^2 - 4*d^11*f^7*g^6*x + 2*d^11*f^8*g^5)*e^4 - (d^12*f^2*g^11*x^5 - 7*d^12*f^3*g^10*x^4 - 1
7*d^12*f^4*g^9*x^3 - d^12*f^5*g^8*x^2 + 16*d^12*f^6*g^7*x + 8*d^12*f^7*g^6)*e^3 + 3*(d^13*f^2*g^11*x^4 - d^13*
f^3*g^10*x^3 - 5*d^13*f^4*g^9*x^2 - 3*d^13*f^5*g^8*x)*e^2 - 3*(d^14*f^2*g^11*x^3 + d^14*f^3*g^10*x^2 - d^14*f^
4*g^9*x - d^14*f^5*g^8)*e), -1/30*(15*d^11*g^10*x^2 + 30*d^11*f*g^9*x + 15*d^11*f^2*g^8 + 30*sqrt(-d^2*g^2 + f
^2*e^2)*(20*(d^3*f^4*g^5*x^5 + 2*d^3*f^5*g^4*x^4 + d^3*f^6*g^3*x^3)*e^7 - 30*(d^4*f^3*g^6*x^5 + 4*d^4*f^4*g^5*
x^4 + 5*d^4*f^5*g^4*x^3 + 2*d^4*f^6*g^3*x^2)*e^6 + (13*d^5*f^2*g^7*x^5 + 116*d^5*f^3*g^6*x^4 + 253*d^5*f^4*g^5
*x^3 + 210*d^5*f^5*g^4*x^2 + 60*d^5*f^6*g^3*x)*e^5 - (39*d^6*f^2*g^7*x^4 + 168*d^6*f^3*g^6*x^3 + 239*d^6*f^4*g
^5*x^2 + 130*d^6*f^5*g^4*x + 20*d^6*f^6*g^3)*e^4 + 3*(13*d^7*f^2*g^7*x^3 + 36*d^7*f^3*g^6*x^2 + 33*d^7*f^4*g^5
*x + 10*d^7*f^5*g^4)*e^3 - 13*(d^8*f^2*g^7*x^2 + 2*d^8*f^3*g^6*x + d^8*f^4*g^5)*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)) - 14*(f^8*g^2*x^5 + 2*f^9*g*x^4 + f^10*
x^3)*e^11 - 6*(10*d*f^7*g^3*x^5 + 13*d*f^8*g^2*x^4 - 4*d*f^9*g*x^3 - 7*d*f^10*x^2)*e^10 - 6*(13*d^2*f^6*g^4*x^
5 - 4*d^2*f^7*g^3*x^4 - 40*d^2*f^8*g^2*x^3 - 16...

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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 )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1314 vs. \(2 (373) = 746\).
time = 1.41, size = 1314, normalized size = 3.30 \begin {gather*} -\frac {{\left (13 \, d^{2} g^{5} e^{2} - 30 \, d f g^{4} e^{3} + 20 \, f^{2} g^{3} e^{4}\right )} \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^{7} g^{7} + 3 \, d^{6} f g^{6} e + d^{5} f^{2} g^{5} e^{2} - 5 \, d^{4} f^{3} g^{4} e^{3} - 5 \, d^{3} f^{4} g^{3} e^{4} + d^{2} f^{5} g^{2} e^{5} + 3 \, d f^{6} g e^{6} + f^{7} e^{7}\right )} \sqrt {-d^{2} g^{2} + f^{2} e^{2}}} + \frac {\frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} g^{8} e^{\left (-4\right )}}{x^{2}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} f g^{7} e^{\left (-1\right )}}{x} + \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} f g^{7} e^{\left (-3\right )}}{x^{2}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} f g^{7} e^{\left (-5\right )}}{x^{3}} + d^{3} f^{2} g^{6} e^{2} - \frac {19 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} f^{2} g^{6} e^{\left (-2\right )}}{x^{2}} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} f^{2} g^{6} e^{\left (-4\right )}}{x^{3}} + \frac {18 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} f^{2} g^{6}}{x} + 6 \, d^{2} f^{3} g^{5} e^{3} - \frac {29 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} f^{3} g^{5} e}{x} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} f^{3} g^{5} e^{\left (-1\right )}}{x^{2}} - \frac {11 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} f^{3} g^{5} e^{\left (-3\right )}}{x^{3}} - 10 \, d f^{4} g^{4} e^{4} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f^{4} g^{4}}{x^{2}}}{{\left (d^{7} f^{2} g^{7} + 3 \, d^{6} f^{3} g^{6} e + d^{5} f^{4} g^{5} e^{2} - 5 \, d^{4} f^{5} g^{4} e^{3} - 5 \, d^{3} f^{6} g^{3} e^{4} + d^{2} f^{7} g^{2} e^{5} + 3 \, d f^{8} g e^{6} + f^{9} e^{7}\right )} {\left (\frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d g e^{\left (-2\right )}}{x} + f e + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f e^{\left (-3\right )}}{x^{2}}\right )}^{2}} + \frac {2 \, {\left (127 \, d^{2} g^{2} e^{2} + \frac {745 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} g^{2} e^{\left (-2\right )}}{x^{2}} - \frac {525 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} g^{2} e^{\left (-4\right )}}{x^{3}} + \frac {150 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} g^{2} e^{\left (-6\right )}}{x^{4}} - \frac {485 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} g^{2}}{x} + 44 \, d f g e^{3} - \frac {145 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f g e}{x} + \frac {245 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f g e^{\left (-1\right )}}{x^{2}} - \frac {195 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f g e^{\left (-3\right )}}{x^{3}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d f g e^{\left (-5\right )}}{x^{4}} + 7 \, f^{2} e^{4} - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{2} e^{2}}{x} - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{2} e^{\left (-2\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{2} e^{\left (-4\right )}}{x^{4}} + \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{2}}{x^{2}}\right )}}{15 \, {\left (d^{8} g^{5} + 5 \, d^{7} f g^{4} e + 10 \, d^{6} f^{2} g^{3} e^{2} + 10 \, d^{5} f^{3} g^{2} e^{3} + 5 \, d^{4} f^{4} g e^{4} + d^{3} f^{5} e^{5}\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)^3/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

-(13*d^2*g^5*e^2 - 30*d*f*g^4*e^3 + 20*f^2*g^3*e^4)*arctan((d*g + (d*e + sqrt(-x^2*e^2 + d^2)*e)*f*e^(-1)/x)/s
qrt(-d^2*g^2 + f^2*e^2))/((d^7*g^7 + 3*d^6*f*g^6*e + d^5*f^2*g^5*e^2 - 5*d^4*f^3*g^4*e^3 - 5*d^3*f^4*g^3*e^4 +
d^2*f^5*g^2*e^5 + 3*d*f^6*g*e^6 + f^7*e^7)*sqrt(-d^2*g^2 + f^2*e^2)) + (2*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^
5*g^8*e^(-4)/x^2 + 2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^4*f*g^7*e^(-1)/x + 12*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d
^4*f*g^7*e^(-3)/x^2 + 2*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^4*f*g^7*e^(-5)/x^3 + d^3*f^2*g^6*e^2 - 19*(d*e + sq
rt(-x^2*e^2 + d^2)*e)^2*d^3*f^2*g^6*e^(-2)/x^2 + 6*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^3*f^2*g^6*e^(-4)/x^3 + 1
8*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^3*f^2*g^6/x + 6*d^2*f^3*g^5*e^3 - 29*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d^2*f^3
*g^5*e/x + 6*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*f^3*g^5*e^(-1)/x^2 - 11*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d^2
*f^3*g^5*e^(-3)/x^3 - 10*d*f^4*g^4*e^4 - 10*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d*f^4*g^4/x^2)/((d^7*f^2*g^7 + 3*
d^6*f^3*g^6*e + d^5*f^4*g^5*e^2 - 5*d^4*f^5*g^4*e^3 - 5*d^3*f^6*g^3*e^4 + d^2*f^7*g^2*e^5 + 3*d*f^8*g*e^6 + f^
9*e^7)*(2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d*g*e^(-2)/x + f*e + (d*e + sqrt(-x^2*e^2 + d^2)*e)^2*f*e^(-3)/x^2)^2
) + 2/15*(127*d^2*g^2*e^2 + 745*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d^2*g^2*e^(-2)/x^2 - 525*(d*e + sqrt(-x^2*e^2
+ d^2)*e)^3*d^2*g^2*e^(-4)/x^3 + 150*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*d^2*g^2*e^(-6)/x^4 - 485*(d*e + sqrt(-x
^2*e^2 + d^2)*e)*d^2*g^2/x + 44*d*f*g*e^3 - 145*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d*f*g*e/x + 245*(d*e + sqrt(-x^
2*e^2 + d^2)*e)^2*d*f*g*e^(-1)/x^2 - 195*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d*f*g*e^(-3)/x^3 + 75*(d*e + sqrt(-x
^2*e^2 + d^2)*e)^4*d*f*g*e^(-5)/x^4 + 7*f^2*e^4 - 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*f^2*e^2/x - 30*(d*e + sqrt
(-x^2*e^2 + d^2)*e)^3*f^2*e^(-2)/x^3 + 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^4*f^2*e^(-4)/x^4 + 40*(d*e + sqrt(-x^
2*e^2 + d^2)*e)^2*f^2/x^2)/((d^8*g^5 + 5*d^7*f*g^4*e + 10*d^6*f^2*g^3*e^2 + 10*d^5*f^3*g^2*e^3 + 5*d^4*f^4*g*e
^4 + d^3*f^5*e^5)*((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 )}^3\,{\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)^3*(d^2 - e^2*x^2)^(7/2)),x)

[Out]

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

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