3.22 \(\int \frac{A+B x}{\left (a+c x^2\right ) \sqrt{d+e x+f x^2}} \, dx\)

Optimal. Leaf size=780 \[ \frac{\sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac{\sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}} \]

[Out]

(Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[
-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[
(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])
) - c*(a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/
(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*
c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(
e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c
^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]) - (Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^
2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2
+ a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f - Sqr
t[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f + Sqrt[c^2*
d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[-(A*c*e)
+ B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c
*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])]
)/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])

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Rubi [A]  time = 10.709, antiderivative size = 780, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{\sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{A \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \sqrt{B \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}}-\frac{\sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e} \tanh ^{-1}\left (\frac{\sqrt{e} \left (a \left (A c e-B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )\right )-c x \left (A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e\right )\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{d+e x+f x^2} \sqrt{B \left (-\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )-A c e} \sqrt{A \left (\sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}-a f+c d\right )+a B e}}\right )}{\sqrt{2} \sqrt{a} \sqrt{c} \sqrt{e} \sqrt{a^2 f^2+a c \left (e^2-2 d f\right )+c^2 d^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]),x]

[Out]

(Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[
-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[
(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])
) - c*(a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/
(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a*B*e + A*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*
c*(e^2 - 2*d*f)])]*Sqrt[-(A*c*e) + B*(c*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(
e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c
^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]) - (Sqrt[-(A*c*e) + B*(c*d - a*f - Sqrt[c^
2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c*d - a*f + Sqrt[c^2*d^2
+ a^2*f^2 + a*c*(e^2 - 2*d*f)])]*ArcTanh[(Sqrt[e]*(a*(A*c*e - B*(c*d - a*f - Sqr
t[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])) - c*(a*B*e + A*(c*d - a*f + Sqrt[c^2*
d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)]))*x))/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[-(A*c*e)
+ B*(c*d - a*f - Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[a*B*e + A*(c
*d - a*f + Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])]*Sqrt[d + e*x + f*x^2])]
)/(Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[e]*Sqrt[c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)])

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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)/(c*x**2+a)/(f*x**2+e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.73241, size = 411, normalized size = 0.53 \[ \frac{\frac{\left (-\sqrt{a} B+i A \sqrt{c}\right ) \log \left (\frac{i \sqrt{a} \sqrt{c} \left (2 \sqrt{d+x (e+f x)} \sqrt{i \sqrt{a} \sqrt{c} e-a f+c d}+i \sqrt{a} (e+2 f x)+\sqrt{c} (2 d+e x)\right )}{\left (\sqrt{a}+i \sqrt{c} x\right ) \left (\sqrt{a} B-i A \sqrt{c}\right ) \sqrt{i \sqrt{a} \sqrt{c} e-a f+c d}}\right )}{\sqrt{i \sqrt{a} \sqrt{c} e-a f+c d}}-\frac{\left (\sqrt{a} B+i A \sqrt{c}\right ) \log \left (-\frac{\sqrt{a} \sqrt{c} \left (2 i \sqrt{d+x (e+f x)} \sqrt{-i \sqrt{a} \sqrt{c} e-a f+c d}+\sqrt{a} (e+2 f x)+i \sqrt{c} (2 d+e x)\right )}{\left (\sqrt{a}-i \sqrt{c} x\right ) \left (\sqrt{a} B+i A \sqrt{c}\right ) \sqrt{-i \sqrt{a} \sqrt{c} e-a f+c d}}\right )}{\sqrt{-i \sqrt{a} \sqrt{c} e-a f+c d}}}{2 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]),x]

[Out]

(-(((Sqrt[a]*B + I*A*Sqrt[c])*Log[-((Sqrt[a]*Sqrt[c]*(I*Sqrt[c]*(2*d + e*x) + Sq
rt[a]*(e + 2*f*x) + (2*I)*Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e - a*f]*Sqrt[d + x*(e +
f*x)]))/((Sqrt[a]*B + I*A*Sqrt[c])*Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e - a*f]*(Sqrt[a
] - I*Sqrt[c]*x)))])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e - a*f]) + ((-(Sqrt[a]*B) + I
*A*Sqrt[c])*Log[(I*Sqrt[a]*Sqrt[c]*(Sqrt[c]*(2*d + e*x) + I*Sqrt[a]*(e + 2*f*x)
+ 2*Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e - a*f]*Sqrt[d + x*(e + f*x)]))/((Sqrt[a]*B -
I*A*Sqrt[c])*Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e - a*f]*(Sqrt[a] + I*Sqrt[c]*x))])/Sq
rt[c*d + I*Sqrt[a]*Sqrt[c]*e - a*f])/(2*Sqrt[a]*Sqrt[c])

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Maple [A]  time = 0.065, size = 784, normalized size = 1. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+a)/(f*x^2+e*x+d)^(1/2),x)

[Out]

1/2/(-a*c)^(1/2)/(-((-a*c)^(1/2)*e+f*a-c*d)/c)^(1/2)*ln((-2*((-a*c)^(1/2)*e+f*a-
c*d)/c+1/c*(-2*f*(-a*c)^(1/2)+c*e)*(x+(-a*c)^(1/2)/c)+2*(-((-a*c)^(1/2)*e+f*a-c*
d)/c)^(1/2)*((x+(-a*c)^(1/2)/c)^2*f+1/c*(-2*f*(-a*c)^(1/2)+c*e)*(x+(-a*c)^(1/2)/
c)-((-a*c)^(1/2)*e+f*a-c*d)/c)^(1/2))/(x+(-a*c)^(1/2)/c))*A-1/2/c/(-((-a*c)^(1/2
)*e+f*a-c*d)/c)^(1/2)*ln((-2*((-a*c)^(1/2)*e+f*a-c*d)/c+1/c*(-2*f*(-a*c)^(1/2)+c
*e)*(x+(-a*c)^(1/2)/c)+2*(-((-a*c)^(1/2)*e+f*a-c*d)/c)^(1/2)*((x+(-a*c)^(1/2)/c)
^2*f+1/c*(-2*f*(-a*c)^(1/2)+c*e)*(x+(-a*c)^(1/2)/c)-((-a*c)^(1/2)*e+f*a-c*d)/c)^
(1/2))/(x+(-a*c)^(1/2)/c))*B-1/2/(-a*c)^(1/2)/(-(-(-a*c)^(1/2)*e+f*a-c*d)/c)^(1/
2)*ln((-2*(-(-a*c)^(1/2)*e+f*a-c*d)/c+(2*f*(-a*c)^(1/2)+c*e)/c*(x-(-a*c)^(1/2)/c
)+2*(-(-(-a*c)^(1/2)*e+f*a-c*d)/c)^(1/2)*((x-(-a*c)^(1/2)/c)^2*f+(2*f*(-a*c)^(1/
2)+c*e)/c*(x-(-a*c)^(1/2)/c)-(-(-a*c)^(1/2)*e+f*a-c*d)/c)^(1/2))/(x-(-a*c)^(1/2)
/c))*A-1/2/c/(-(-(-a*c)^(1/2)*e+f*a-c*d)/c)^(1/2)*ln((-2*(-(-a*c)^(1/2)*e+f*a-c*
d)/c+(2*f*(-a*c)^(1/2)+c*e)/c*(x-(-a*c)^(1/2)/c)+2*(-(-(-a*c)^(1/2)*e+f*a-c*d)/c
)^(1/2)*((x-(-a*c)^(1/2)/c)^2*f+(2*f*(-a*c)^(1/2)+c*e)/c*(x-(-a*c)^(1/2)/c)-(-(-
a*c)^(1/2)*e+f*a-c*d)/c)^(1/2))/(x-(-a*c)^(1/2)/c))*B

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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((B*x + A)/((c*x^2 + a)*sqrt(f*x^2 + e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 45.8573, size = 9262, normalized size = 11.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*sqrt(f*x^2 + e*x + d)),x, algorithm="fricas")

[Out]

-1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f + (a*c^3
*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2
*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c
^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*
c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 - A^4*
c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2
*(A*B^3*a*c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x + 2*(2*A^2*B*c^3*d^2 +
2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^
2 - (4*A^2*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f - (B*a*c^4*d^3 - A*a*c^4
*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*
c^2*e)*f^2 - (3*B*a^2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2
*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/
(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(
3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x
^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f
+ (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^
2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c
+ A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4
+ 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d
^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^
2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)) - (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*
c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f + 2*(B^2*a^3*c + A^2*
a^2*c^2)*d*f^2 + ((B^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 -
2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4
*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2
- 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)
*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 +
2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/x) +
1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f + (a*c^3
*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2
*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c
^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*
c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 - A^4*
c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2
*(A*B^3*a*c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x - 2*(2*A^2*B*c^3*d^2 +
2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^
2 - (4*A^2*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f - (B*a*c^4*d^3 - A*a*c^4
*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*
c^2*e)*f^2 - (3*B*a^2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2
*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/
(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(
3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x
^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f
+ (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^
2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c
+ A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4
+ 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d
^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^
2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)) - (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*
c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f + 2*(B^2*a^3*c + A^2*
a^2*c^2)*d*f^2 + ((B^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 -
2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4
*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2
- 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)
*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 +
2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/x) -
1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f - (a*c^3
*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2
*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c
^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*
c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 - A^4*
c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2
*(A*B^3*a*c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x + 2*(2*A^2*B*c^3*d^2 +
2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^
2 - (4*A^2*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f + (B*a*c^4*d^3 - A*a*c^4
*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*
c^2*e)*f^2 - (3*B*a^2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2
*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/
(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(
3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x
^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f
- (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^
2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c
+ A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4
+ 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d
^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^
2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)) + (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*
c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f + 2*(B^2*a^3*c + A^2*
a^2*c^2)*d*f^2 + ((B^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 -
2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4
*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2
- 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)
*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 +
2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/x) +
1/4*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f - (a*c^3
*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^2 + 4*A^2
*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^
2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4 + 2*a^2*c
^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d^2 + a^4*
c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^2*e^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*log(-(2*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 - A^4*
c^2)*e^2 - 2*(A*B^3*a^2 + A^3*B*a*c)*e*f - 2*(2*(A*B^3*a^2 + A^3*B*a*c)*f^2 - (2
*(A*B^3*a*c + A^3*B*c^2)*d + (B^4*a^2 - A^4*c^2)*e)*f)*x - 2*(2*A^2*B*c^3*d^2 +
2*A^2*B*a^2*c*f^2 + (3*A*B^2*a*c^2 - A^3*c^3)*d*e + (B^3*a^2*c - A^2*B*a*c^2)*e^
2 - (4*A^2*B*a*c^2*d + (3*A*B^2*a^2*c - A^3*a*c^2)*e)*f + (B*a*c^4*d^3 - A*a*c^4
*d^2*e + B*a^2*c^3*d*e^2 - A*a^2*c^3*e^3 - B*a^4*c*f^3 + (3*B*a^3*c^2*d - A*a^3*
c^2*e)*f^2 - (3*B*a^2*c^3*d^2 - 2*A*a^2*c^3*d*e + B*a^3*c^2*e^2)*f)*sqrt(-(4*A^2
*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/
(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(
3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))*sqrt(f*x
^2 + e*x + d)*sqrt(-(2*A*B*a*c*e - (B^2*a*c - A^2*c^2)*d + (B^2*a^2 - A^2*a*c)*f
- (a*c^3*d^2 + a^2*c^2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)*sqrt(-(4*A^2*B^2*c^2*d^
2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2 - 2*A^2*B^2*a*c
+ A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)*f)/(a*c^5*d^4
+ 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 + 2*(3*a^3*c^3*d
^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/(a*c^3*d^2 + a^2*c^
2*e^2 - 2*a^2*c^2*d*f + a^3*c*f^2)) + (2*(B^2*a*c^3 + A^2*c^4)*d^3 + 2*(B^2*a^2*
c^2 + A^2*a*c^3)*d*e^2 - 4*(B^2*a^2*c^2 + A^2*a*c^3)*d^2*f + 2*(B^2*a^3*c + A^2*
a^2*c^2)*d*f^2 + ((B^2*a*c^3 + A^2*c^4)*d^2*e + (B^2*a^2*c^2 + A^2*a*c^3)*e^3 -
2*(B^2*a^2*c^2 + A^2*a*c^3)*d*e*f + (B^2*a^3*c + A^2*a^2*c^2)*e*f^2)*x)*sqrt(-(4
*A^2*B^2*c^2*d^2 + 4*A^2*B^2*a^2*f^2 + 4*(A*B^3*a*c - A^3*B*c^2)*d*e + (B^4*a^2
- 2*A^2*B^2*a*c + A^4*c^2)*e^2 - 4*(2*A^2*B^2*a*c*d + (A*B^3*a^2 - A^3*B*a*c)*e)
*f)/(a*c^5*d^4 + 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4 - 4*a^4*c^2*d*f^3 + a^5*c*f^4 +
2*(3*a^3*c^3*d^2 + a^4*c^2*e^2)*f^2 - 4*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f)))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + c x^{2}\right ) \sqrt{d + e x + f x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(c*x**2+a)/(f*x**2+e*x+d)**(1/2),x)

[Out]

Integral((A + B*x)/((a + c*x**2)*sqrt(d + e*x + f*x**2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)*sqrt(f*x^2 + e*x + d)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError