3.586 \(\int \frac{(d+e x)^3}{(f+g x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\)
Optimal. Leaf size=311 \[ \frac{e g^3 (4 e f-3 d g) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{(e f-d g) (d g+e f)^4 \sqrt{e^2 f^2-d^2 g^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(f+g x) (e f-d g) (d g+e f)^4}-\frac{e (5 d (e f-3 d g)-e x (21 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^3}+\frac{4 d e (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^2}+\frac{e \left (45 d^3 g^2+e x \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^4} \]
[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])
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Rubi [A] time = 2.41898, antiderivative size = 311, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129
\[ \frac{e g^3 (4 e f-3 d g) \tan ^{-1}\left (\frac{d^2 g+e^2 f x}{\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}}\right )}{(e f-d g) (d g+e f)^4 \sqrt{e^2 f^2-d^2 g^2}}+\frac{g^4 \sqrt{d^2-e^2 x^2}}{(f+g x) (e f-d g) (d g+e f)^4}-\frac{e (5 d (e f-3 d g)-e x (21 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^3}+\frac{4 d e (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^2}+\frac{e \left (45 d^3 g^2+e x \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt{d^2-e^2 x^2} (d g+e f)^4} \]
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])
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Rubi in Sympy [A] time = 129.43, size = 405, normalized size = 1.3 \[ \frac{e^{2} f g^{3} \operatorname{atanh}{\left (\frac{d^{2} g + e^{2} f x}{\sqrt{d^{2} - e^{2} x^{2}} \sqrt{d g - e f} \sqrt{d g + e f}} \right )}}{\left (d g - e f\right )^{\frac{3}{2}} \left (d g + e f\right )^{\frac{9}{2}}} - \frac{3 e g^{3} \operatorname{atanh}{\left (\frac{d^{2} g + e^{2} f x}{\sqrt{d^{2} - e^{2} x^{2}} \sqrt{d g - e f} \sqrt{d g + e f}} \right )}}{\sqrt{d g - e f} \left (d g + e f\right )^{\frac{9}{2}}} - \frac{g^{4} \sqrt{d^{2} - e^{2} x^{2}}}{\left (f + g x\right ) \left (d g - e f\right ) \left (d g + e f\right )^{4}} + \frac{3 e g^{2} \sqrt{d^{2} - e^{2} x^{2}}}{d \left (d - e x\right ) \left (d g + e f\right )^{4}} + \frac{2 e g \sqrt{d^{2} - e^{2} x^{2}}}{3 d \left (d - e x\right )^{2} \left (d g + e f\right )^{3}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{5 d \left (d - e x\right )^{3} \left (d g + e f\right )^{2}} + \frac{2 e g \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{2} \left (d - e x\right ) \left (d g + e f\right )^{3}} + \frac{2 e \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{2} \left (d - e x\right )^{2} \left (d g + e f\right )^{2}} + \frac{2 e \sqrt{d^{2} - e^{2} x^{2}}}{15 d^{3} \left (d - e x\right ) \left (d g + e f\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(g*x+f)**2/(-e**2*x**2+d**2)**(7/2),x)
[Out]
e**2*f*g**3*atanh((d**2*g + e**2*f*x)/(sqrt(d**2 - e**2*x**2)*sqrt(d*g - e*f)*sq
rt(d*g + e*f)))/((d*g - e*f)**(3/2)*(d*g + e*f)**(9/2)) - 3*e*g**3*atanh((d**2*g
+ e**2*f*x)/(sqrt(d**2 - e**2*x**2)*sqrt(d*g - e*f)*sqrt(d*g + e*f)))/(sqrt(d*g
- e*f)*(d*g + e*f)**(9/2)) - g**4*sqrt(d**2 - e**2*x**2)/((f + g*x)*(d*g - e*f)
*(d*g + e*f)**4) + 3*e*g**2*sqrt(d**2 - e**2*x**2)/(d*(d - e*x)*(d*g + e*f)**4)
+ 2*e*g*sqrt(d**2 - e**2*x**2)/(3*d*(d - e*x)**2*(d*g + e*f)**3) + e*sqrt(d**2 -
e**2*x**2)/(5*d*(d - e*x)**3*(d*g + e*f)**2) + 2*e*g*sqrt(d**2 - e**2*x**2)/(3*
d**2*(d - e*x)*(d*g + e*f)**3) + 2*e*sqrt(d**2 - e**2*x**2)/(15*d**2*(d - e*x)**
2*(d*g + e*f)**2) + 2*e*sqrt(d**2 - e**2*x**2)/(15*d**3*(d - e*x)*(d*g + e*f)**2
)
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Mathematica [C] time = 1.00998, size = 308, normalized size = 0.99 \[ \frac{\sqrt{d^2-e^2 x^2} \left (\frac{2 e (d g+e f) (6 d g+e f)}{d^2 (d-e x)^2}+\frac{e \left (57 d^2 g^2+14 d e f g+2 e^2 f^2\right )}{d^3 (d-e x)}+\frac{15 g^4}{(f+g x) (e f-d g)}+\frac{3 e (d g+e f)^2}{d (d-e x)^3}\right )-\frac{15 i e g^3 (4 e f-3 d g) \log \left (\frac{2 (e f-d g) (d g+e f)^4 \left (\sqrt{d^2-e^2 x^2} \sqrt{e^2 f^2-d^2 g^2}+i d^2 g+i e^2 f x\right )}{e g^2 (f+g x) (4 e f-3 d g) \sqrt{e^2 f^2-d^2 g^2}}\right )}{(e f-d g) \sqrt{e^2 f^2-d^2 g^2}}}{15 (d g+e f)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/((f + g*x)^2*(d^2 - e^2*x^2)^(7/2)),x]
[Out]
(Sqrt[d^2 - e^2*x^2]*((3*e*(e*f + d*g)^2)/(d*(d - e*x)^3) + (2*e*(e*f + d*g)*(e*
f + 6*d*g))/(d^2*(d - e*x)^2) + (e*(2*e^2*f^2 + 14*d*e*f*g + 57*d^2*g^2))/(d^3*(
d - e*x)) + (15*g^4)/((e*f - d*g)*(f + g*x))) - ((15*I)*e*g^3*(4*e*f - 3*d*g)*Lo
g[(2*(e*f - d*g)*(e*f + d*g)^4*(I*d^2*g + I*e^2*f*x + Sqrt[e^2*f^2 - d^2*g^2]*Sq
rt[d^2 - e^2*x^2]))/(e*g^2*(4*e*f - 3*d*g)*Sqrt[e^2*f^2 - d^2*g^2]*(f + g*x))])/
((e*f - d*g)*Sqrt[e^2*f^2 - d^2*g^2]))/(15*(e*f + d*g)^4)
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Maple [B] time = 0.038, size = 6760, normalized size = 21.7 \[ \text{output too large to display} \]
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] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*(g*x + f)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.471573, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*(g*x + f)^2),x, algorithm="fricas")
[Out]
[1/15*(15*(32*d^9*e^2*f^3*g^3 - 24*d^10*e*f^2*g^4 + (4*d^3*e^8*f^2*g^4 - 3*d^4*e
^7*f*g^5)*x^7 + (4*d^3*e^8*f^3*g^3 + d^4*e^7*f^2*g^4 - 3*d^5*e^6*f*g^5)*x^6 + (4
*d^4*e^7*f^3*g^3 - 55*d^5*e^6*f^2*g^4 + 39*d^6*e^5*f*g^5)*x^5 - (52*d^5*e^6*f^3*
g^3 - 99*d^6*e^5*f^2*g^4 + 45*d^7*e^4*f*g^5)*x^4 + (60*d^6*e^5*f^3*g^3 - 13*d^7*
e^4*f^2*g^4 - 24*d^8*e^3*f*g^5)*x^3 + 4*(8*d^7*e^4*f^3*g^3 - 26*d^8*e^3*f^2*g^4
+ 15*d^9*e^2*f*g^5)*x^2 - 4*(20*d^8*e^3*f^3*g^3 - 23*d^9*e^2*f^2*g^4 + 6*d^10*e*
f*g^5)*x - (32*d^8*e^2*f^3*g^3 - 24*d^9*e*f^2*g^4 + (4*d^3*e^7*f^2*g^4 - 3*d^4*e
^6*f*g^5)*x^6 + (4*d^3*e^7*f^3*g^3 - 27*d^4*e^6*f^2*g^4 + 18*d^5*e^5*f*g^5)*x^5
- (24*d^4*e^6*f^3*g^3 - 38*d^5*e^5*f^2*g^4 + 15*d^6*e^4*f*g^5)*x^4 + (20*d^5*e^5
*f^3*g^3 + 33*d^6*e^4*f^2*g^4 - 36*d^7*e^3*f*g^5)*x^3 + 4*(12*d^6*e^4*f^3*g^3 -
29*d^7*e^3*f^2*g^4 + 15*d^8*e^2*f*g^5)*x^2 - 4*(20*d^7*e^3*f^3*g^3 - 23*d^8*e^2*
f^2*g^4 + 6*d^9*e*f*g^5)*x)*sqrt(-e^2*x^2 + d^2))*log(((d*e^2*f^2*g - d^3*g^3)*x
^2 - (e^2*f^3 - d^2*f*g^2)*sqrt(-e^2*x^2 + d^2)*x + (d*e^2*f^3 - d^3*f*g^2)*x +
(d^2*f*g*x + d^2*f^2 - (e^2*f^2 - d^2*g^2)*x^2 - (d*f*g*x + d*f^2)*sqrt(-e^2*x^2
+ d^2))*sqrt(-e^2*f^2 + d^2*g^2))/(d*g*x + d*f - sqrt(-e^2*x^2 + d^2)*(g*x + f)
)) + (5*(e^10*f^4*g + 3*d*e^9*f^3*g^2 - d^2*e^8*f^2*g^3 + 3*d^4*e^6*g^5)*x^7 + (
5*e^10*f^5 + 34*d*e^9*f^4*g + 94*d^2*e^8*f^3*g^2 + 236*d^3*e^7*f^2*g^3 - 444*d^4
*e^6*f*g^4 + 15*d^5*e^5*g^5)*x^6 + (19*d*e^9*f^5 - d^2*e^8*f^4*g - 139*d^3*e^7*f
^3*g^2 - 809*d^4*e^6*f^2*g^3 + 1005*d^5*e^5*f*g^4 - 195*d^6*e^4*g^5)*x^5 - 5*(20
*d^2*e^8*f^5 + 64*d^3*e^7*f^4*g + 58*d^4*e^6*f^3*g^2 - 154*d^5*e^5*f^2*g^3 - 3*d
^6*e^4*f*g^4 - 45*d^7*e^3*g^5)*x^4 + 10*(7*d^3*e^7*f^5 + 27*d^4*e^6*f^4*g + 32*d
^5*e^5*f^3*g^2 + 54*d^6*e^4*f^2*g^3 - 135*d^7*e^3*f*g^4 + 12*d^8*e^2*g^5)*x^3 +
60*(2*d^4*e^6*f^5 + 6*d^5*e^5*f^4*g + 4*d^6*e^4*f^3*g^2 - 24*d^7*e^3*f^2*g^3 + 1
3*d^8*e^2*f*g^4 - 5*d^9*e*g^5)*x^2 - 120*(d^5*e^5*f^5 + 3*d^6*e^4*f^4*g + 2*d^7*
e^3*f^3*g^2 - 6*d^8*e^2*f^2*g^3 - d^10*g^5)*x - (3*(3*e^9*f^4*g + 13*d*e^8*f^3*g
^2 + 27*d^2*e^7*f^2*g^3 - 48*d^3*e^6*f*g^4 + 5*d^4*e^5*g^5)*x^6 + (9*e^9*f^5 - d
*e^8*f^4*g - 69*d^2*e^7*f^3*g^2 - 269*d^3*e^6*f^2*g^3 + 330*d^4*e^5*f*g^4 - 90*d
^5*e^4*g^5)*x^5 - 5*(8*d*e^8*f^5 + 28*d^2*e^7*f^4*g + 34*d^3*e^6*f^3*g^2 - 10*d^
4*e^5*f^2*g^3 - 81*d^5*e^4*f*g^4 - 15*d^6*e^3*g^5)*x^4 + 10*(d^2*e^7*f^5 + 9*d^3
*e^6*f^4*g + 20*d^4*e^5*f^3*g^2 + 90*d^5*e^4*f^2*g^3 - 135*d^6*e^3*f*g^4 + 18*d^
7*e^2*g^5)*x^3 + 60*(2*d^3*e^6*f^5 + 6*d^4*e^5*f^4*g + 4*d^5*e^4*f^3*g^2 - 24*d^
6*e^3*f^2*g^3 + 13*d^7*e^2*f*g^4 - 5*d^8*e*g^5)*x^2 - 120*(d^4*e^5*f^5 + 3*d^5*e
^4*f^4*g + 2*d^6*e^3*f^3*g^2 - 6*d^7*e^2*f^2*g^3 - d^9*g^5)*x)*sqrt(-e^2*x^2 + d
^2))*sqrt(-e^2*f^2 + d^2*g^2))/((8*d^9*e^5*f^7 + 24*d^10*e^4*f^6*g + 16*d^11*e^3
*f^5*g^2 - 16*d^12*e^2*f^4*g^3 - 24*d^13*e*f^3*g^4 - 8*d^14*f^2*g^5 + (d^3*e^11*
f^6*g + 3*d^4*e^10*f^5*g^2 + 2*d^5*e^9*f^4*g^3 - 2*d^6*e^8*f^3*g^4 - 3*d^7*e^7*f
^2*g^5 - d^8*e^6*f*g^6)*x^7 + (d^3*e^11*f^7 + 4*d^4*e^10*f^6*g + 5*d^5*e^9*f^5*g
^2 - 5*d^7*e^7*f^3*g^4 - 4*d^8*e^6*f^2*g^5 - d^9*e^5*f*g^6)*x^6 + (d^4*e^10*f^7
- 10*d^5*e^9*f^6*g - 37*d^6*e^8*f^5*g^2 - 28*d^7*e^7*f^4*g^3 + 23*d^8*e^6*f^3*g^
4 + 38*d^9*e^5*f^2*g^5 + 13*d^10*e^4*f*g^6)*x^5 - (13*d^5*e^9*f^7 + 24*d^6*e^8*f
^6*g - 19*d^7*e^7*f^5*g^2 - 56*d^8*e^6*f^4*g^3 - 9*d^9*e^5*f^3*g^4 + 32*d^10*e^4
*f^2*g^5 + 15*d^11*e^3*f*g^6)*x^4 + (15*d^6*e^8*f^7 + 53*d^7*e^7*f^6*g + 54*d^8*
e^6*f^5*g^2 - 14*d^9*e^5*f^4*g^3 - 61*d^10*e^4*f^3*g^4 - 39*d^11*e^3*f^2*g^5 - 8
*d^12*e^2*f*g^6)*x^3 + 4*(2*d^7*e^7*f^7 + d^8*e^6*f^6*g - 11*d^9*e^5*f^5*g^2 - 1
4*d^10*e^4*f^4*g^3 + 4*d^11*e^3*f^3*g^4 + 13*d^12*e^2*f^2*g^5 + 5*d^13*e*f*g^6)*
x^2 - 4*(5*d^8*e^6*f^7 + 13*d^9*e^5*f^6*g + 4*d^10*e^4*f^5*g^2 - 14*d^11*e^3*f^4
*g^3 - 11*d^12*e^2*f^3*g^4 + d^13*e*f^2*g^5 + 2*d^14*f*g^6)*x - (8*d^8*e^5*f^7 +
24*d^9*e^4*f^6*g + 16*d^10*e^3*f^5*g^2 - 16*d^11*e^2*f^4*g^3 - 24*d^12*e*f^3*g^
4 - 8*d^13*f^2*g^5 + (d^3*e^10*f^6*g + 3*d^4*e^9*f^5*g^2 + 2*d^5*e^8*f^4*g^3 - 2
*d^6*e^7*f^3*g^4 - 3*d^7*e^6*f^2*g^5 - d^8*e^5*f*g^6)*x^6 + (d^3*e^10*f^7 - 3*d^
4*e^9*f^6*g - 16*d^5*e^8*f^5*g^2 - 14*d^6*e^7*f^4*g^3 + 9*d^7*e^6*f^3*g^4 + 17*d
^8*e^5*f^2*g^5 + 6*d^9*e^4*f*g^6)*x^5 - (6*d^4*e^9*f^7 + 13*d^5*e^8*f^6*g - 3*d^
6*e^7*f^5*g^2 - 22*d^7*e^6*f^4*g^3 - 8*d^8*e^5*f^3*g^4 + 9*d^9*e^4*f^2*g^5 + 5*d
^10*e^3*f*g^6)*x^4 + (5*d^5*e^8*f^7 + 27*d^6*e^7*f^6*g + 46*d^7*e^6*f^5*g^2 + 14
*d^8*e^5*f^4*g^3 - 39*d^9*e^4*f^3*g^4 - 41*d^10*e^3*f^2*g^5 - 12*d^11*e^2*f*g^6)
*x^3 + 4*(3*d^6*e^7*f^7 + 4*d^7*e^6*f^6*g - 9*d^8*e^5*f^5*g^2 - 16*d^9*e^4*f^4*g
^3 + d^10*e^3*f^3*g^4 + 12*d^11*e^2*f^2*g^5 + 5*d^12*e*f*g^6)*x^2 - 4*(5*d^7*e^6
*f^7 + 13*d^8*e^5*f^6*g + 4*d^9*e^4*f^5*g^2 - 14*d^10*e^3*f^4*g^3 - 11*d^11*e^2*
f^3*g^4 + d^12*e*f^2*g^5 + 2*d^13*f*g^6)*x)*sqrt(-e^2*x^2 + d^2))*sqrt(-e^2*f^2
+ d^2*g^2)), 1/15*(30*(32*d^9*e^2*f^3*g^3 - 24*d^10*e*f^2*g^4 + (4*d^3*e^8*f^2*g
^4 - 3*d^4*e^7*f*g^5)*x^7 + (4*d^3*e^8*f^3*g^3 + d^4*e^7*f^2*g^4 - 3*d^5*e^6*f*g
^5)*x^6 + (4*d^4*e^7*f^3*g^3 - 55*d^5*e^6*f^2*g^4 + 39*d^6*e^5*f*g^5)*x^5 - (52*
d^5*e^6*f^3*g^3 - 99*d^6*e^5*f^2*g^4 + 45*d^7*e^4*f*g^5)*x^4 + (60*d^6*e^5*f^3*g
^3 - 13*d^7*e^4*f^2*g^4 - 24*d^8*e^3*f*g^5)*x^3 + 4*(8*d^7*e^4*f^3*g^3 - 26*d^8*
e^3*f^2*g^4 + 15*d^9*e^2*f*g^5)*x^2 - 4*(20*d^8*e^3*f^3*g^3 - 23*d^9*e^2*f^2*g^4
+ 6*d^10*e*f*g^5)*x - (32*d^8*e^2*f^3*g^3 - 24*d^9*e*f^2*g^4 + (4*d^3*e^7*f^2*g
^4 - 3*d^4*e^6*f*g^5)*x^6 + (4*d^3*e^7*f^3*g^3 - 27*d^4*e^6*f^2*g^4 + 18*d^5*e^5
*f*g^5)*x^5 - (24*d^4*e^6*f^3*g^3 - 38*d^5*e^5*f^2*g^4 + 15*d^6*e^4*f*g^5)*x^4 +
(20*d^5*e^5*f^3*g^3 + 33*d^6*e^4*f^2*g^4 - 36*d^7*e^3*f*g^5)*x^3 + 4*(12*d^6*e^
4*f^3*g^3 - 29*d^7*e^3*f^2*g^4 + 15*d^8*e^2*f*g^5)*x^2 - 4*(20*d^7*e^3*f^3*g^3 -
23*d^8*e^2*f^2*g^4 + 6*d^9*e*f*g^5)*x)*sqrt(-e^2*x^2 + d^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)) + (5*(e^10*f^4*g + 3*d*
e^9*f^3*g^2 - d^2*e^8*f^2*g^3 + 3*d^4*e^6*g^5)*x^7 + (5*e^10*f^5 + 34*d*e^9*f^4*
g + 94*d^2*e^8*f^3*g^2 + 236*d^3*e^7*f^2*g^3 - 444*d^4*e^6*f*g^4 + 15*d^5*e^5*g^
5)*x^6 + (19*d*e^9*f^5 - d^2*e^8*f^4*g - 139*d^3*e^7*f^3*g^2 - 809*d^4*e^6*f^2*g
^3 + 1005*d^5*e^5*f*g^4 - 195*d^6*e^4*g^5)*x^5 - 5*(20*d^2*e^8*f^5 + 64*d^3*e^7*
f^4*g + 58*d^4*e^6*f^3*g^2 - 154*d^5*e^5*f^2*g^3 - 3*d^6*e^4*f*g^4 - 45*d^7*e^3*
g^5)*x^4 + 10*(7*d^3*e^7*f^5 + 27*d^4*e^6*f^4*g + 32*d^5*e^5*f^3*g^2 + 54*d^6*e^
4*f^2*g^3 - 135*d^7*e^3*f*g^4 + 12*d^8*e^2*g^5)*x^3 + 60*(2*d^4*e^6*f^5 + 6*d^5*
e^5*f^4*g + 4*d^6*e^4*f^3*g^2 - 24*d^7*e^3*f^2*g^3 + 13*d^8*e^2*f*g^4 - 5*d^9*e*
g^5)*x^2 - 120*(d^5*e^5*f^5 + 3*d^6*e^4*f^4*g + 2*d^7*e^3*f^3*g^2 - 6*d^8*e^2*f^
2*g^3 - d^10*g^5)*x - (3*(3*e^9*f^4*g + 13*d*e^8*f^3*g^2 + 27*d^2*e^7*f^2*g^3 -
48*d^3*e^6*f*g^4 + 5*d^4*e^5*g^5)*x^6 + (9*e^9*f^5 - d*e^8*f^4*g - 69*d^2*e^7*f^
3*g^2 - 269*d^3*e^6*f^2*g^3 + 330*d^4*e^5*f*g^4 - 90*d^5*e^4*g^5)*x^5 - 5*(8*d*e
^8*f^5 + 28*d^2*e^7*f^4*g + 34*d^3*e^6*f^3*g^2 - 10*d^4*e^5*f^2*g^3 - 81*d^5*e^4
*f*g^4 - 15*d^6*e^3*g^5)*x^4 + 10*(d^2*e^7*f^5 + 9*d^3*e^6*f^4*g + 20*d^4*e^5*f^
3*g^2 + 90*d^5*e^4*f^2*g^3 - 135*d^6*e^3*f*g^4 + 18*d^7*e^2*g^5)*x^3 + 60*(2*d^3
*e^6*f^5 + 6*d^4*e^5*f^4*g + 4*d^5*e^4*f^3*g^2 - 24*d^6*e^3*f^2*g^3 + 13*d^7*e^2
*f*g^4 - 5*d^8*e*g^5)*x^2 - 120*(d^4*e^5*f^5 + 3*d^5*e^4*f^4*g + 2*d^6*e^3*f^3*g
^2 - 6*d^7*e^2*f^2*g^3 - d^9*g^5)*x)*sqrt(-e^2*x^2 + d^2))*sqrt(e^2*f^2 - d^2*g^
2))/((8*d^9*e^5*f^7 + 24*d^10*e^4*f^6*g + 16*d^11*e^3*f^5*g^2 - 16*d^12*e^2*f^4*
g^3 - 24*d^13*e*f^3*g^4 - 8*d^14*f^2*g^5 + (d^3*e^11*f^6*g + 3*d^4*e^10*f^5*g^2
+ 2*d^5*e^9*f^4*g^3 - 2*d^6*e^8*f^3*g^4 - 3*d^7*e^7*f^2*g^5 - d^8*e^6*f*g^6)*x^7
+ (d^3*e^11*f^7 + 4*d^4*e^10*f^6*g + 5*d^5*e^9*f^5*g^2 - 5*d^7*e^7*f^3*g^4 - 4*
d^8*e^6*f^2*g^5 - d^9*e^5*f*g^6)*x^6 + (d^4*e^10*f^7 - 10*d^5*e^9*f^6*g - 37*d^6
*e^8*f^5*g^2 - 28*d^7*e^7*f^4*g^3 + 23*d^8*e^6*f^3*g^4 + 38*d^9*e^5*f^2*g^5 + 13
*d^10*e^4*f*g^6)*x^5 - (13*d^5*e^9*f^7 + 24*d^6*e^8*f^6*g - 19*d^7*e^7*f^5*g^2 -
56*d^8*e^6*f^4*g^3 - 9*d^9*e^5*f^3*g^4 + 32*d^10*e^4*f^2*g^5 + 15*d^11*e^3*f*g^
6)*x^4 + (15*d^6*e^8*f^7 + 53*d^7*e^7*f^6*g + 54*d^8*e^6*f^5*g^2 - 14*d^9*e^5*f^
4*g^3 - 61*d^10*e^4*f^3*g^4 - 39*d^11*e^3*f^2*g^5 - 8*d^12*e^2*f*g^6)*x^3 + 4*(2
*d^7*e^7*f^7 + d^8*e^6*f^6*g - 11*d^9*e^5*f^5*g^2 - 14*d^10*e^4*f^4*g^3 + 4*d^11
*e^3*f^3*g^4 + 13*d^12*e^2*f^2*g^5 + 5*d^13*e*f*g^6)*x^2 - 4*(5*d^8*e^6*f^7 + 13
*d^9*e^5*f^6*g + 4*d^10*e^4*f^5*g^2 - 14*d^11*e^3*f^4*g^3 - 11*d^12*e^2*f^3*g^4
+ d^13*e*f^2*g^5 + 2*d^14*f*g^6)*x - (8*d^8*e^5*f^7 + 24*d^9*e^4*f^6*g + 16*d^10
*e^3*f^5*g^2 - 16*d^11*e^2*f^4*g^3 - 24*d^12*e*f^3*g^4 - 8*d^13*f^2*g^5 + (d^3*e
^10*f^6*g + 3*d^4*e^9*f^5*g^2 + 2*d^5*e^8*f^4*g^3 - 2*d^6*e^7*f^3*g^4 - 3*d^7*e^
6*f^2*g^5 - d^8*e^5*f*g^6)*x^6 + (d^3*e^10*f^7 - 3*d^4*e^9*f^6*g - 16*d^5*e^8*f^
5*g^2 - 14*d^6*e^7*f^4*g^3 + 9*d^7*e^6*f^3*g^4 + 17*d^8*e^5*f^2*g^5 + 6*d^9*e^4*
f*g^6)*x^5 - (6*d^4*e^9*f^7 + 13*d^5*e^8*f^6*g - 3*d^6*e^7*f^5*g^2 - 22*d^7*e^6*
f^4*g^3 - 8*d^8*e^5*f^3*g^4 + 9*d^9*e^4*f^2*g^5 + 5*d^10*e^3*f*g^6)*x^4 + (5*d^5
*e^8*f^7 + 27*d^6*e^7*f^6*g + 46*d^7*e^6*f^5*g^2 + 14*d^8*e^5*f^4*g^3 - 39*d^9*e
^4*f^3*g^4 - 41*d^10*e^3*f^2*g^5 - 12*d^11*e^2*f*g^6)*x^3 + 4*(3*d^6*e^7*f^7 + 4
*d^7*e^6*f^6*g - 9*d^8*e^5*f^5*g^2 - 16*d^9*e^4*f^4*g^3 + d^10*e^3*f^3*g^4 + 12*
d^11*e^2*f^2*g^5 + 5*d^12*e*f*g^6)*x^2 - 4*(5*d^7*e^6*f^7 + 13*d^8*e^5*f^6*g + 4
*d^9*e^4*f^5*g^2 - 14*d^10*e^3*f^4*g^3 - 11*d^11*e^2*f^3*g^4 + d^12*e*f^2*g^5 +
2*d^13*f*g^6)*x)*sqrt(-e^2*x^2 + d^2))*sqrt(e^2*f^2 - d^2*g^2))]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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]
Timed out
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/((-e^2*x^2 + d^2)^(7/2)*(g*x + f)^2),x, algorithm="giac")
[Out]
Exception raised: TypeError