3.1456 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a-c x^2\right )^2} \, dx\)
Optimal. Leaf size=250 \[ -\frac{\left (-3 \sqrt{a} A \sqrt{c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (3 \sqrt{a} A \sqrt{c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}+\frac{\sqrt{d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \]
[Out]
(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(2*a*(c*d^2 - a*e^2)*(a - c*
x^2)) - ((2*A*c*d + a*B*e - 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(3/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2
)) + ((2*A*c*d + a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/
Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2))
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Rubi [A] time = 0.979189, antiderivative size = 250, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16
\[ -\frac{\left (-3 \sqrt{a} A \sqrt{c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2}}+\frac{\left (3 \sqrt{a} A \sqrt{c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2}}+\frac{\sqrt{d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[d + e*x]*(a - c*x^2)^2),x]
[Out]
(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(2*a*(c*d^2 - a*e^2)*(a - c*
x^2)) - ((2*A*c*d + a*B*e - 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(3/4)*(Sqrt[c]*d - Sqrt[a]*e)^(3/2
)) + ((2*A*c*d + a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/
Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(3/4)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2))
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**(1/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
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Mathematica [A] time = 0.691511, size = 277, normalized size = 1.11 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} (-a A e+a B (d-e x)+A c d x)}{c x^2-a}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \left (-3 \sqrt{a} A \sqrt{c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c} \sqrt{c d-\sqrt{a} \sqrt{c} e}}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (3 \sqrt{a} A \sqrt{c} e+a B e+2 A c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{c} \sqrt{\sqrt{a} \sqrt{c} e+c d}}}{4 a^{3/2} \left (a e^2-c d^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[d + e*x]*(a - c*x^2)^2),x]
[Out]
((2*Sqrt[a]*Sqrt[d + e*x]*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(-a + c*x^2) + (
(Sqrt[c]*d + Sqrt[a]*e)*(2*A*c*d + a*B*e - 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(Sqrt[
c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(Sqrt[c]*Sqrt[c*d - Sqrt[a]*Sq
rt[c]*e]) - ((Sqrt[c]*d - Sqrt[a]*e)*(2*A*c*d + a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*A
rcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(Sqrt[c]*Sqrt[c*d
+ Sqrt[a]*Sqrt[c]*e]))/(4*a^(3/2)*(-(c*d^2) + a*e^2))
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Maple [B] time = 0.103, size = 638, normalized size = 2.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^(1/2)/(-c*x^2+a)^2,x)
[Out]
1/4*e^2/(a*c*e^2)^(1/2)/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x+(a*c*e^2)^(1/2)
/c)*B-1/4*e/a/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)*A+1/2*
e*c^2/a/(a*c*e^2)^(1/2)/(c*d-(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*a
rctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d+1/4*e^2*c/(a*c*e^2)^
(1/2)/(c*d-(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1
/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B-3/4*e*c/a/(c*d-(a*c*e^2)^(1/2))/((-c*d+(
a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)
)*A-1/4*e^2/(a*c*e^2)^(1/2)/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x-(a*c*e^2)^(
1/2)/c)*B-1/4*e/a/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)*A+
1/2*e*c^2/a/(a*c*e^2)^(1/2)/(c*d+(a*c*e^2)^(1/2))/((c*d+(a*c*e^2)^(1/2))*c)^(1/2
)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*A*d+1/4*e^2*c/(a*c*e^
2)^(1/2)/(c*d+(a*c*e^2)^(1/2))/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)
^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*B+3/4*e*c/a/(c*d+(a*c*e^2)^(1/2))/((c*d+
(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2
))*A
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} - a\right )}^{2} \sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 - a)^2*sqrt(e*x + d)),x, algorithm="maxima")
[Out]
integrate((B*x + A)/((c*x^2 - a)^2*sqrt(e*x + d)), x)
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Fricas [A] time = 32.3839, size = 10133, normalized size = 40.53 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 - a)^2*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
-1/8*((a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^2)*sqrt((4*A^2*c^3*d^5 +
4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*
c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^
2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*
c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^
4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2
*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B
^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10
*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*
d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4
- a^6*c*e^6))*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e
^3 + 2*(9*A*B^3*a^2*c^2 - 65*A^3*B*a*c^3)*d^3*e^4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^
2*c^2 + 27*A^4*a*c^3)*d^2*e^5 - 2*(5*A*B^3*a^3*c - 81*A^3*B*a^2*c^2)*d*e^6 + (B^
4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) + (6*A^2*B*a^2*c^4*d^5*e^3 + 5*(3*A*B
^2*a^3*c^3 - A^3*a^2*c^4)*d^4*e^4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 -
12*(3*A*B^2*a^4*c^2 - 2*A^3*a^3*c^3)*d^2*e^6 + 2*(B^3*a^5*c + 30*A^2*B*a^4*c^2)*
d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*e^8 - (2*A*a^3*c^7*d^9 + B*a^4*c^6*d^8*e
- 10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5*e^4 - 14*A*a^6*
c^4*d^3*e^6 + 2*B*a^7*c^3*d^2*e^7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^2*e^9)*sqrt((36*
A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2
- 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)
*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3
*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*
e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e
^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))*sqrt((4*A^2*c^3*d
^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A
^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^6 - 3*a^4*c^3*d
^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B
^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^
4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^
2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9
+ (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8
*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8
*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2
*e^4 - a^6*c*e^6))) - (a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*e^2)*x^2)*sqrt((
4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a
^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d^6 -
3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4
+ 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3
+ 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^
3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2
*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12
- 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*
e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*
a^5*c^2*d^2*e^4 - a^6*c*e^6))*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 -
5*A^4*c^4)*d^4*e^3 + 2*(9*A*B^3*a^2*c^2 - 65*A^3*B*a*c^3)*d^3*e^4 + 3*(B^4*a^3*c
- 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5 - 2*(5*A*B^3*a^3*c - 81*A^3*B*a^2*
c^2)*d*e^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) - (6*A^2*B*a^2*c^4*d^
5*e^3 + 5*(3*A*B^2*a^3*c^3 - A^3*a^2*c^4)*d^4*e^4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3
*c^3)*d^3*e^5 - 12*(3*A*B^2*a^4*c^2 - 2*A^3*a^3*c^3)*d^2*e^6 + 2*(B^3*a^5*c + 30
*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*e^8 - (2*A*a^3*c^7*d^9 +
B*a^4*c^6*d^8*e - 10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5
*e^4 - 14*A*a^6*c^4*d^3*e^6 + 2*B*a^7*c^3*d^2*e^7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^
2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 +
(9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 -
31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2
*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c +
81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 -
20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))*s
qrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (
B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 + (a^3*c^4*d
^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6
*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a
*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B
^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*
B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*
d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5
*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2
+ 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) + (a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a^2*c*
e^2)*x^2)*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6*A*B*
a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*e^4 -
(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36*A^2*
B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 19
8*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3
*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3
*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10
)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 +
15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3*a^4*
c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))*log(-(24*A^3*B*c^4*d^5*e^2 + 4*(9*
A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e^3 + 2*(9*A*B^3*a^2*c^2 - 65*A^3*B*a*c^3)*d^3*e^
4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5 - 2*(5*A*B^3*a^3*c
- 81*A^3*B*a^2*c^2)*d*e^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqrt(e*x + d) + (6*
A^2*B*a^2*c^4*d^5*e^3 + 5*(3*A*B^2*a^3*c^3 - A^3*a^2*c^4)*d^4*e^4 + 6*(B^3*a^4*c
^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 - 12*(3*A*B^2*a^4*c^2 - 2*A^3*a^3*c^3)*d^2*e^6 + 2
*(B^3*a^5*c + 30*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a^4*c^2)*e^8 + (2
*A*a^3*c^7*d^9 + B*a^4*c^6*d^8*e - 10*A*a^4*c^6*d^7*e^2 - 2*B*a^5*c^5*d^6*e^3 +
18*A*a^5*c^5*d^5*e^4 - 14*A*a^6*c^4*d^3*e^6 + 2*B*a^7*c^3*d^2*e^7 + 4*A*a^7*c^3*
d*e^8 - B*a^8*c^2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*
B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9
*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 1
5*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*
A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^
5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a
^9*c^3*e^12)))*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*d^2*e^3 - 6
*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^2*a^2*c)*d*
e^4 - (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6)*sqrt((36
*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2
- 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3
)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^
3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)
*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*
e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*c^4*d^6 - 3
*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) - (a^2*c*d^2 - a^3*e^2 - (a*
c^2*d^2 - a^2*c*e^2)*x^2)*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a^2*c*
d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 + 5*A^
2*a^2*c)*d*e^4 - (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^
6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*
B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*
A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8
- 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A
^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a
^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/(a^3*
c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))*log(-(24*A^3*B*c^4
*d^5*e^2 + 4*(9*A^2*B^2*a*c^3 - 5*A^4*c^4)*d^4*e^3 + 2*(9*A*B^3*a^2*c^2 - 65*A^3
*B*a*c^3)*d^3*e^4 + 3*(B^4*a^3*c - 28*A^2*B^2*a^2*c^2 + 27*A^4*a*c^3)*d^2*e^5 -
2*(5*A*B^3*a^3*c - 81*A^3*B*a^2*c^2)*d*e^6 + (B^4*a^4 - 81*A^4*a^2*c^2)*e^7)*sqr
t(e*x + d) - (6*A^2*B*a^2*c^4*d^5*e^3 + 5*(3*A*B^2*a^3*c^3 - A^3*a^2*c^4)*d^4*e^
4 + 6*(B^3*a^4*c^2 - 7*A^2*B*a^3*c^3)*d^3*e^5 - 12*(3*A*B^2*a^4*c^2 - 2*A^3*a^3*
c^3)*d^2*e^6 + 2*(B^3*a^5*c + 30*A^2*B*a^4*c^2)*d*e^7 - 3*(A*B^2*a^5*c + 9*A^3*a
^4*c^2)*e^8 + (2*A*a^3*c^7*d^9 + B*a^4*c^6*d^8*e - 10*A*a^4*c^6*d^7*e^2 - 2*B*a^
5*c^5*d^6*e^3 + 18*A*a^5*c^5*d^5*e^4 - 14*A*a^6*c^4*d^3*e^6 + 2*B*a^7*c^3*d^2*e^
7 + 4*A*a^7*c^3*d*e^8 - B*a^8*c^2*e^9)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^
3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5 + (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4
)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2
*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9
+ (B^4*a^4 + 18*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*
d^10*e^2 + 15*a^5*c^7*d^8*e^4 - 20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*
c^4*d^2*e^10 + a^9*c^3*e^12)))*sqrt((4*A^2*c^3*d^5 + 4*A*B*a*c^2*d^4*e - 6*A*B*a
^2*c*d^2*e^3 - 6*A*B*a^3*e^5 + (B^2*a^2*c - 15*A^2*a*c^2)*d^3*e^2 + 3*(B^2*a^3 +
5*A^2*a^2*c)*d*e^4 - (a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6
*c*e^6)*sqrt((36*A^2*B^2*c^4*d^6*e^4 + 12*(3*A*B^3*a*c^3 - 5*A^3*B*c^4)*d^5*e^5
+ (9*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + 25*A^4*c^4)*d^4*e^6 - 8*(9*A*B^3*a^2*c^2
- 31*A^3*B*a*c^3)*d^3*e^7 + 6*(B^4*a^3*c + 40*A^2*B^2*a^2*c^2 - 15*A^4*a*c^3)*d^
2*e^8 - 28*(A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (B^4*a^4 + 18*A^2*B^2*a^3*c +
81*A^4*a^2*c^2)*e^10)/(a^3*c^9*d^12 - 6*a^4*c^8*d^10*e^2 + 15*a^5*c^7*d^8*e^4 -
20*a^6*c^6*d^6*e^6 + 15*a^7*c^5*d^4*e^8 - 6*a^8*c^4*d^2*e^10 + a^9*c^3*e^12)))/
(a^3*c^4*d^6 - 3*a^4*c^3*d^4*e^2 + 3*a^5*c^2*d^2*e^4 - a^6*c*e^6))) - 4*(B*a*d -
A*a*e + (A*c*d - B*a*e)*x)*sqrt(e*x + d))/(a^2*c*d^2 - a^3*e^2 - (a*c^2*d^2 - a
^2*c*e^2)*x^2)
_______________________________________________________________________________________
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((B*x+A)/(e*x+d)**(1/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 - a)^2*sqrt(e*x + d)),x, algorithm="giac")
[Out]
Timed out