3.1.33 \(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx\)

Optimal. Leaf size=363 \[ \frac {\left (a^2-b^2 x^2\right ) \left (2 a^2 f^2 (2 C e-B f)-b^2 e \left (f (B e-3 A f)+C e^2\right )\right )}{2 f \sqrt {a+b x} (e+f x) \sqrt {a c-b c x} \left (b^2 e^2-a^2 f^2\right )^2}+\frac {f \left (a^2-b^2 x^2\right ) \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 \sqrt {a+b x} (e+f x)^2 \sqrt {a c-b c x} \left (b^2 e^2-a^2 f^2\right )}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 \left (2 a^2 C f^2+b^2 e (C e-3 B f)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{2 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x} \left (b^2 e^2-a^2 f^2\right )^{5/2}} \]

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Rubi [A]  time = 0.59, antiderivative size = 361, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1610, 1651, 807, 725, 204} \begin {gather*} \frac {\left (a^2-b^2 x^2\right ) \left (2 a^2 f^2 (2 C e-B f)-b^2 \left (e f (B e-3 A f)+C e^3\right )\right )}{2 f \sqrt {a+b x} (e+f x) \sqrt {a c-b c x} \left (b^2 e^2-a^2 f^2\right )^2}+\frac {f \left (a^2-b^2 x^2\right ) \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 \sqrt {a+b x} (e+f x)^2 \sqrt {a c-b c x} \left (b^2 e^2-a^2 f^2\right )}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)+2 a^4 C f^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{2 \sqrt {c} \sqrt {a+b x} \sqrt {a c-b c x} \left (b^2 e^2-a^2 f^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3),x]

[Out]

(f*(A + (e*(C*e - B*f))/f^2)*(a^2 - b^2*x^2))/(2*(b^2*e^2 - a^2*f^2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)
^2) + ((2*a^2*f^2*(2*C*e - B*f) - b^2*(C*e^3 + e*f*(B*e - 3*A*f)))*(a^2 - b^2*x^2))/(2*f*(b^2*e^2 - a^2*f^2)^2
*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)) + ((2*a^4*C*f^2 + a^2*b^2*e*(C*e - 3*B*f) + A*(2*b^4*e^2 + a^2*b^2
*f^2))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(Sqrt[c]*(a^2*f + b^2*e*x))/(Sqrt[b^2*e^2 - a^2*f^2]*Sqrt[a^2*c - b^2*c*
x^2])])/(2*Sqrt[c]*(b^2*e^2 - a^2*f^2)^(5/2)*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

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 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1651

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 {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx &=\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {A+B x+C x^2}{(e+f x)^3 \sqrt {a^2 c-b^2 c x^2}} \, dx}{\sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {f \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (a^2-b^2 x^2\right )}{2 \left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \int \frac {2 c \left (A b^2 e+a^2 (C e-B f)\right )-c \left (2 a^2 C f-b^2 \left (B e+\frac {C e^2}{f}-A f\right )\right ) x}{(e+f x)^2 \sqrt {a^2 c-b^2 c x^2}} \, dx}{2 c \left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {f \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (a^2-b^2 x^2\right )}{2 \left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2}+\frac {\left (2 a^2 f^2 (2 C e-B f)-b^2 \left (C e^3+e f (B e-3 A f)\right )\right ) \left (a^2-b^2 x^2\right )}{2 f \left (b^2 e^2-a^2 f^2\right )^2 \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)}+\frac {\left (\left (2 a^4 C f^2+a^2 b^2 e (C e-3 B f)+A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}} \, dx}{2 \left (b^2 e^2-a^2 f^2\right )^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {f \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (a^2-b^2 x^2\right )}{2 \left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2}+\frac {\left (2 a^2 f^2 (2 C e-B f)-b^2 \left (C e^3+e f (B e-3 A f)\right )\right ) \left (a^2-b^2 x^2\right )}{2 f \left (b^2 e^2-a^2 f^2\right )^2 \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)}-\frac {\left (\left (2 a^4 C f^2+a^2 b^2 e (C e-3 B f)+A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-b^2 c e^2+a^2 c f^2-x^2} \, dx,x,\frac {a^2 c f+b^2 c e x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{2 \left (b^2 e^2-a^2 f^2\right )^2 \sqrt {a+b x} \sqrt {a c-b c x}}\\ &=\frac {f \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (a^2-b^2 x^2\right )}{2 \left (b^2 e^2-a^2 f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2}+\frac {\left (2 a^2 f^2 (2 C e-B f)-b^2 \left (C e^3+e f (B e-3 A f)\right )\right ) \left (a^2-b^2 x^2\right )}{2 f \left (b^2 e^2-a^2 f^2\right )^2 \sqrt {a+b x} \sqrt {a c-b c x} (e+f x)}+\frac {\left (2 a^4 C f^2+a^2 b^2 e (C e-3 B f)+A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac {\sqrt {c} \left (a^2 f+b^2 e x\right )}{\sqrt {b^2 e^2-a^2 f^2} \sqrt {a^2 c-b^2 c x^2}}\right )}{2 \sqrt {c} \left (b^2 e^2-a^2 f^2\right )^{5/2} \sqrt {a+b x} \sqrt {a c-b c x}}\\ \end {align*}

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Mathematica [A]  time = 1.31, size = 492, normalized size = 1.36 \begin {gather*} \frac {\frac {b^2 \sqrt {a-b x} \left (f (A f-B e)+C e^2\right ) \left (2 (e+f x) \left (a^2 f^2+2 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b x} \sqrt {b e-a f}}{\sqrt {a+b x} \sqrt {-a f-b e}}\right )+3 e f \sqrt {a-b x} \sqrt {a+b x} \sqrt {-a f-b e} \sqrt {b e-a f}\right )}{(e+f x) (-a f-b e)^{5/2} (b e-a f)^{5/2}}+\frac {2 f (b x-a) \sqrt {a+b x} (B f-2 C e)}{(e+f x) \left (a^2 f^2-b^2 e^2\right )}+\frac {f (b x-a) \sqrt {a+b x} \left (f (A f-B e)+C e^2\right )}{(e+f x)^2 (a f-b e) (a f+b e)}+\frac {4 b^2 e \sqrt {a-b x} (2 C e-B f) \tanh ^{-1}\left (\frac {\sqrt {a-b x} \sqrt {b e-a f}}{\sqrt {a+b x} \sqrt {-a f-b e}}\right )}{(-a f-b e)^{3/2} (b e-a f)^{3/2}}+\frac {4 C \sqrt {a-b x} \tanh ^{-1}\left (\frac {\sqrt {a-b x} \sqrt {b e-a f}}{\sqrt {a+b x} \sqrt {-a f-b e}}\right )}{\sqrt {-a f-b e} \sqrt {b e-a f}}}{2 f^2 \sqrt {c (a-b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3),x]

[Out]

((f*(C*e^2 + f*(-(B*e) + A*f))*(-a + b*x)*Sqrt[a + b*x])/((-(b*e) + a*f)*(b*e + a*f)*(e + f*x)^2) + (2*f*(-2*C
*e + B*f)*(-a + b*x)*Sqrt[a + b*x])/((-(b^2*e^2) + a^2*f^2)*(e + f*x)) + (4*C*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e
- a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/(Sqrt[-(b*e) - a*f]*Sqrt[b*e - a*f]) + (4*b^2*e*(2*
C*e - B*f)*Sqrt[a - b*x]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a - b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])])/((-(b*e)
- a*f)^(3/2)*(b*e - a*f)^(3/2)) + (b^2*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b*x]*(3*e*f*Sqrt[-(b*e) - a*f]*Sqr
t[b*e - a*f]*Sqrt[a - b*x]*Sqrt[a + b*x] + 2*(2*b^2*e^2 + a^2*f^2)*(e + f*x)*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[a -
b*x])/(Sqrt[-(b*e) - a*f]*Sqrt[a + b*x])]))/((-(b*e) - a*f)^(5/2)*(b*e - a*f)^(5/2)*(e + f*x)))/(2*f^2*Sqrt[c
*(a - b*x)])

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IntegrateAlgebraic [A]  time = 0.00, size = 610, normalized size = 1.68 \begin {gather*} \frac {\left (-2 a^4 C f^2-a^2 A b^2 f^2+3 a^2 b^2 B e f-a^2 b^2 C e^2-2 A b^4 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a c-b c x} \sqrt {a f-b e}}{\sqrt {c} \sqrt {a+b x} \sqrt {a f+b e}}\right )}{\sqrt {c} (b e-a f)^2 \sqrt {a f-b e} (a f+b e)^{5/2}}-\frac {a b \sqrt {a c-b c x} \left (-\frac {2 a^3 B f^3 (a c-b c x)}{a+b x}+\frac {4 a^3 C e f^2 (a c-b c x)}{a+b x}+2 a^3 B c f^3-4 a^3 c C e f^2+\frac {a^2 A b f^3 (a c-b c x)}{a+b x}+a^2 A b c f^3+\frac {a^2 b B e f^2 (a c-b c x)}{a+b x}+a^2 b B c e f^2-\frac {3 a^2 b C e^2 f (a c-b c x)}{a+b x}-3 a^2 b c C e^2 f-\frac {4 A b^3 e^2 f (a c-b c x)}{a+b x}+\frac {3 a A b^2 e f^2 (a c-b c x)}{a+b x}-3 a A b^2 c e f^2+\frac {2 b^3 B e^3 (a c-b c x)}{a+b x}-\frac {a b^2 B e^2 f (a c-b c x)}{a+b x}+a b^2 B c e^2 f-\frac {a b^2 C e^3 (a c-b c x)}{a+b x}+a b^2 c C e^3-4 A b^3 c e^2 f+2 b^3 B c e^3\right )}{\sqrt {a+b x} (b e-a f)^2 (a f+b e)^2 \left (\frac {b e (a c-b c x)}{a+b x}-\frac {a f (a c-b c x)}{a+b x}+a c f+b c e\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^3),x]

[Out]

-((a*b*Sqrt[a*c - b*c*x]*(2*b^3*B*c*e^3 + a*b^2*c*C*e^3 - 4*A*b^3*c*e^2*f + a*b^2*B*c*e^2*f - 3*a^2*b*c*C*e^2*
f - 3*a*A*b^2*c*e*f^2 + a^2*b*B*c*e*f^2 - 4*a^3*c*C*e*f^2 + a^2*A*b*c*f^3 + 2*a^3*B*c*f^3 + (2*b^3*B*e^3*(a*c
- b*c*x))/(a + b*x) - (a*b^2*C*e^3*(a*c - b*c*x))/(a + b*x) - (4*A*b^3*e^2*f*(a*c - b*c*x))/(a + b*x) - (a*b^2
*B*e^2*f*(a*c - b*c*x))/(a + b*x) - (3*a^2*b*C*e^2*f*(a*c - b*c*x))/(a + b*x) + (3*a*A*b^2*e*f^2*(a*c - b*c*x)
)/(a + b*x) + (a^2*b*B*e*f^2*(a*c - b*c*x))/(a + b*x) + (4*a^3*C*e*f^2*(a*c - b*c*x))/(a + b*x) + (a^2*A*b*f^3
*(a*c - b*c*x))/(a + b*x) - (2*a^3*B*f^3*(a*c - b*c*x))/(a + b*x)))/((b*e - a*f)^2*(b*e + a*f)^2*Sqrt[a + b*x]
*(b*c*e + a*c*f + (b*e*(a*c - b*c*x))/(a + b*x) - (a*f*(a*c - b*c*x))/(a + b*x))^2)) + ((-2*A*b^4*e^2 - a^2*b^
2*C*e^2 + 3*a^2*b^2*B*e*f - a^2*A*b^2*f^2 - 2*a^4*C*f^2)*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[a*c - b*c*x])/(Sqrt[
c]*Sqrt[b*e + a*f]*Sqrt[a + b*x])])/(Sqrt[c]*(b*e - a*f)^2*Sqrt[-(b*e) + a*f]*(b*e + a*f)^(5/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 9.49, size = 1658, normalized size = 4.57

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(f*x+e)^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x, algorithm="giac")

[Out]

-(2*C*a^4*sqrt(-c)*c^2*f^2 + A*a^2*b^2*sqrt(-c)*c^2*f^2 - 3*B*a^2*b^2*sqrt(-c)*c^2*f*e + C*a^2*b^2*sqrt(-c)*c^
2*e^2 + 2*A*b^4*sqrt(-c)*c^2*e^2)*arctan(1/2*(2*b*c^2*e + (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x
- a*c)*c))^2*f)/(sqrt(a^2*f^2 - b^2*e^2)*c^2))/((a^4*f^4*abs(c) - 2*a^2*b^2*f^2*abs(c)*e^2 + b^4*abs(c)*e^4)*
sqrt(a^2*f^2 - b^2*e^2)*c^2) + 2*(16*B*a^6*b*sqrt(-c)*c^8*f^5 - 32*C*a^6*b*sqrt(-c)*c^8*f^4*e - 24*A*a^4*b^3*s
qrt(-c)*c^8*f^4*e + 4*A*a^4*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*sqrt(-c)*c^6
*f^5 + 8*B*a^4*b^3*sqrt(-c)*c^8*f^3*e^2 + 20*B*a^4*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x -
a*c)*c))^2*sqrt(-c)*c^6*f^4*e + 4*B*a^4*b*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sq
rt(-c)*c^4*f^5 + 8*C*a^4*b^3*sqrt(-c)*c^8*f^2*e^3 - 44*C*a^4*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 +
(b*c*x - a*c)*c))^2*sqrt(-c)*c^6*f^3*e^2 - 40*A*a^2*b^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x
- a*c)*c))^2*sqrt(-c)*c^6*f^3*e^2 - 8*C*a^4*b*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^
4*sqrt(-c)*c^4*f^4*e - 6*A*a^2*b^3*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*
c^4*f^4*e - A*a^2*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*sqrt(-c)*c^2*f^5 + 16*
B*a^2*b^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*sqrt(-c)*c^6*f^2*e^3 + 10*B*a^2*b^
3*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*f^3*e^2 + 3*B*a^2*b^2*(sqrt(-
b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*sqrt(-c)*c^2*f^4*e + 8*C*a^2*b^4*(sqrt(-b*c*x + a*c
)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^2*sqrt(-c)*c^6*f*e^4 - 14*C*a^2*b^3*(sqrt(-b*c*x + a*c)*sqrt(-c)
- sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*f^2*e^3 - 12*A*b^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*
c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*f^2*e^3 - 5*C*a^2*b^2*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*
c*x - a*c)*c))^6*sqrt(-c)*c^2*f^3*e^2 - 2*A*b^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c)
)^6*sqrt(-c)*c^2*f^3*e^2 + 4*B*b^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*
c^4*f*e^4 + 4*C*b^5*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*sqrt(-c)*c^4*e^5 + 2*C*b
^4*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^6*sqrt(-c)*c^2*f*e^4)/((a^4*f^6*abs(c) - 2*
a^2*b^2*f^4*abs(c)*e^2 + b^4*f^2*abs(c)*e^4)*(4*a^2*c^4*f + 4*b*(sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 +
(b*c*x - a*c)*c))^2*c^2*e + (sqrt(-b*c*x + a*c)*sqrt(-c) - sqrt(2*a*c^2 + (b*c*x - a*c)*c))^4*f)^2)

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maple [B]  time = 0.00, size = 1848, normalized size = 5.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(f*x+e)^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),x)

[Out]

-1/2*(A*a^2*b^2*c*f^4*x^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/
(f*x+e))+2*A*b^4*c*e^2*f^2*x^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2
)*f)/(f*x+e))-3*B*a^2*b^2*c*e*f^3*x^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*
c)^(1/2)*f)/(f*x+e))+2*C*a^4*c*f^4*x^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)
*c)^(1/2)*f)/(f*x+e))+C*a^2*b^2*c*e^2*f^2*x^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x
^2-a^2)*c)^(1/2)*f)/(f*x+e))+2*A*a^2*b^2*c*e*f^3*x*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(
b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+4*A*b^4*c*e^3*f*x*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-
(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))-6*B*a^2*b^2*c*e^2*f^2*x*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^(
1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+4*C*a^4*c*e*f^3*x*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/f^2)^
(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+2*C*a^2*b^2*c*e^3*f*x*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/
f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+A*a^2*b^2*c*e^2*f^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)
*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+2*A*b^4*c*e^4*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)*c/
f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))-3*B*a^2*b^2*c*e^3*f*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)
*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+2*C*a^4*c*e^2*f^2*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2
)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))+C*a^2*b^2*c*e^4*ln(2*(b^2*c*e*x+a^2*c*f+((a^2*f^2-b^2*e^2)
*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*f)/(f*x+e))-3*((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*
A*b^2*e*f^3*x+2*((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*B*a^2*f^4*x+((a^2*f^2-b^2*e^2)*c/f^2)
^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*B*b^2*e^2*f^2*x-4*((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*C*a
^2*e*f^3*x+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*C*b^2*e^3*f*x+((a^2*f^2-b^2*e^2)*c/f^2)^(1
/2)*(-(b^2*x^2-a^2)*c)^(1/2)*A*a^2*f^4-4*((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*A*b^2*e^2*f^
2+((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*B*a^2*e*f^3+2*((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^
2*x^2-a^2)*c)^(1/2)*B*b^2*e^3*f-3*((a^2*f^2-b^2*e^2)*c/f^2)^(1/2)*(-(b^2*x^2-a^2)*c)^(1/2)*C*a^2*e^2*f^2)*(-(b
*x-a)*c)^(1/2)*(b*x+a)^(1/2)/(-(b^2*x^2-a^2)*c)^(1/2)/(a*f-b*e)/(a*f+b*e)/(a^2*f^2-b^2*e^2)/(f*x+e)^2/((a^2*f^
2-b^2*e^2)*c/f^2)^(1/2)/c/f

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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((C*x^2+B*x+A)/(f*x+e)^3/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2),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(a*f-b*e>0)', see `assume?` for
more details)Is a*f-b*e positive, negative or zero?

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mupad [B]  time = 0.01, size = 9344, normalized size = 25.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/((e + f*x)^3*(a*c - b*c*x)^(1/2)*(a + b*x)^(1/2)),x)

[Out]

((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(4*C*a^4*c^3*f^2 + 2*C*a^2*b^2*c^3*e^2))/(((a + b*x)^(1/2) - a^(1/2))*(b
^5*e^5 - 2*a^2*b^3*e^3*f^2 + a^4*b*e*f^4)) + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3*(68*C*a^4*c^2*f^2 - 14*C*a
^2*b^2*c^2*e^2))/(((a + b*x)^(1/2) - a^(1/2))^3*(b^5*e^5 - 2*a^2*b^3*e^3*f^2 + a^4*b*e*f^4)) - ((68*C*a^4*c*f^
2 - 14*C*a^2*b^2*c*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(((a + b*x)^(1/2) - a^(1/2))^5*(b^5*e^5 - 2*a^2
*b^3*e^3*f^2 + a^4*b*e*f^4)) - ((4*C*a^4*f^2 + 2*C*a^2*b^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(((a +
b*x)^(1/2) - a^(1/2))^7*(b^5*e^5 - 2*a^2*b^3*e^3*f^2 + a^4*b*e*f^4)) - (a^(1/2)*(a*c)^(1/2)*(48*C*a^4*c*f^3 -
24*C*a^2*b^2*c*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(((a + b*x)^(1/2) - a^(1/2))^4*(b^6*e^6 - 2*a^2*b
^4*e^4*f^2 + a^4*b^2*e^2*f^4)) + (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(24*C*a^4*f^3 + 12
*C*a^2*b^2*e^2*f))/(((a + b*x)^(1/2) - a^(1/2))^6*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a^4*b^2*e^2*f^4)) + (a^(1/2)*
(a*c)^(1/2)*(24*C*a^4*c^2*f^3 + 12*C*a^2*b^2*c^2*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(((a + b*x)^(1/
2) - a^(1/2))^2*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a^4*b^2*e^2*f^4)))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8/((a +
b*x)^(1/2) - a^(1/2))^8 + c^4 + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(16*a^2*c*f^2 + 4*b^2*c*e^2))/(b^2*e^2
*((a + b*x)^(1/2) - a^(1/2))^6) + ((16*a^2*c^3*f^2 + 4*b^2*c^3*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^
2*e^2*((a + b*x)^(1/2) - a^(1/2))^2) - ((32*a^2*c^2*f^2 - 6*b^2*c^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4
)/(b^2*e^2*((a + b*x)^(1/2) - a^(1/2))^4) - (8*a^(1/2)*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b
*e*((a + b*x)^(1/2) - a^(1/2))^7) + (8*a^(1/2)*c^3*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b*e*((a
+ b*x)^(1/2) - a^(1/2))) - (8*a^(1/2)*c*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b*e*((a + b*x)^
(1/2) - a^(1/2))^5) + (8*a^(1/2)*c^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(b*e*((a + b*x)^(1/2
) - a^(1/2))^3)) + (((4*A*a^4*f^4 - 10*A*a^2*b^2*e^2*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(((a + b*x)^(
1/2) - a^(1/2))^7*(b^5*e^7 + a^4*b*e^3*f^4 - 2*a^2*b^3*e^5*f^2)) - ((4*A*a^4*c^3*f^4 - 10*A*a^2*b^2*c^3*e^2*f^
2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1/2))*(b^5*e^7 + a^4*b*e^3*f^4 - 2*a^2*b^3*e^5*
f^2)) - ((4*A*a^4*c^2*f^4 - 58*A*a^2*b^2*c^2*e^2*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(((a + b*x)^(1/2)
- a^(1/2))^3*(b^5*e^7 + a^4*b*e^3*f^4 - 2*a^2*b^3*e^5*f^2)) + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5*(4*A*a^4
*c*f^4 - 58*A*a^2*b^2*c*e^2*f^2))/(((a + b*x)^(1/2) - a^(1/2))^5*(b^5*e^7 + a^4*b*e^3*f^4 - 2*a^2*b^3*e^5*f^2)
) + (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(16*A*b^4*e^4*f - 8*A*a^4*f^5 + 28*A*a^2*b^2*e^
2*f^3))/(((a + b*x)^(1/2) - a^(1/2))^6*(b^6*e^8 - 2*a^2*b^4*e^6*f^2 + a^4*b^2*e^4*f^4)) + (a^(1/2)*(a*c)^(1/2)
*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4*(16*A*a^4*c*f^5 + 32*A*b^4*c*e^4*f - 72*A*a^2*b^2*c*e^2*f^3))/(((a + b*
x)^(1/2) - a^(1/2))^4*(b^6*e^8 - 2*a^2*b^4*e^6*f^2 + a^4*b^2*e^4*f^4)) + (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x)^(
1/2) - (a*c)^(1/2))^2*(16*A*b^4*c^2*e^4*f - 8*A*a^4*c^2*f^5 + 28*A*a^2*b^2*c^2*e^2*f^3))/(((a + b*x)^(1/2) - a
^(1/2))^2*(b^6*e^8 - 2*a^2*b^4*e^6*f^2 + a^4*b^2*e^4*f^4)))/(((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8/((a + b*x)^
(1/2) - a^(1/2))^8 + c^4 + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(16*a^2*c*f^2 + 4*b^2*c*e^2))/(b^2*e^2*((a +
b*x)^(1/2) - a^(1/2))^6) + ((16*a^2*c^3*f^2 + 4*b^2*c^3*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(b^2*e^2*
((a + b*x)^(1/2) - a^(1/2))^2) - ((32*a^2*c^2*f^2 - 6*b^2*c^2*e^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(b^2
*e^2*((a + b*x)^(1/2) - a^(1/2))^4) - (8*a^(1/2)*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(b*e*((a
+ b*x)^(1/2) - a^(1/2))^7) + (8*a^(1/2)*c^3*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b*e*((a + b*x
)^(1/2) - a^(1/2))) - (8*a^(1/2)*c*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b*e*((a + b*x)^(1/2)
- a^(1/2))^5) + (8*a^(1/2)*c^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(b*e*((a + b*x)^(1/2) - a^
(1/2))^3)) - (((32*B*a^4*c^2*f^3 + 22*B*a^2*b^2*c^2*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/(((a + b*x)^
(1/2) - a^(1/2))^3*(b^5*e^6 + a^4*b*e^2*f^4 - 2*a^2*b^3*e^4*f^2)) - ((32*B*a^4*c*f^3 + 22*B*a^2*b^2*c*e^2*f)*(
(a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(((a + b*x)^(1/2) - a^(1/2))^5*(b^5*e^6 + a^4*b*e^2*f^4 - 2*a^2*b^3*e^4*
f^2)) + (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(8*B*a^4*c^2*f^4 + 8*B*b^4*c^2*e^4 + 20*B*a
^2*b^2*c^2*e^2*f^2))/(((a + b*x)^(1/2) - a^(1/2))^2*(b^6*e^7 - 2*a^2*b^4*e^5*f^2 + a^4*b^2*e^3*f^4)) + (a^(1/2
)*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(8*B*a^4*f^4 + 8*B*b^4*e^4 + 20*B*a^2*b^2*e^2*f^2))/(((a +
b*x)^(1/2) - a^(1/2))^6*(b^6*e^7 - 2*a^2*b^4*e^5*f^2 + a^4*b^2*e^3*f^4)) - (a^(1/2)*(a*c)^(1/2)*((a*c - b*c*x
)^(1/2) - (a*c)^(1/2))^4*(16*B*a^4*c*f^4 - 16*B*b^4*c*e^4 + 24*B*a^2*b^2*c*e^2*f^2))/(((a + b*x)^(1/2) - a^(1/
2))^4*(b^6*e^7 - 2*a^2*b^4*e^5*f^2 + a^4*b^2*e^3*f^4)) - (6*B*a^2*b*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(
((a + b*x)^(1/2) - a^(1/2))^7*(a^4*f^4 + b^4*e^4 - 2*a^2*b^2*e^2*f^2)) + (6*B*a^2*b*c^3*f*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2)))/(((a + b*x)^(1/2) - a^(1/2))*(a^4*f^4 + b^4*e^4 - 2*a^2*b^2*e^2*f^2)))/(((a*c - b*c*x)^(1/2)
- (a*c)^(1/2))^8/((a + b*x)^(1/2) - a^(1/2))^8 + c^4 + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(16*a^2*c*f^2 +
4*b^2*c*e^2))/(b^2*e^2*((a + b*x)^(1/2) - a^(1/2))^6) + ((16*a^2*c^3*f^2 + 4*b^2*c^3*e^2)*((a*c - b*c*x)^(1/2)
- (a*c)^(1/2))^2)/(b^2*e^2*((a + b*x)^(1/2) - a^(1/2))^2) - ((32*a^2*c^2*f^2 - 6*b^2*c^2*e^2)*((a*c - b*c*x)^
(1/2) - (a*c)^(1/2))^4)/(b^2*e^2*((a + b*x)^(1/2) - a^(1/2))^4) - (8*a^(1/2)*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2
) - (a*c)^(1/2))^7)/(b*e*((a + b*x)^(1/2) - a^(1/2))^7) + (8*a^(1/2)*c^3*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) -
(a*c)^(1/2)))/(b*e*((a + b*x)^(1/2) - a^(1/2))) - (8*a^(1/2)*c*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2
))^5)/(b*e*((a + b*x)^(1/2) - a^(1/2))^5) + (8*a^(1/2)*c^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3
)/(b*e*((a + b*x)^(1/2) - a^(1/2))^3)) + (C*a^2*(2*a^2*f^2 + b^2*e^2)*(2*atan(((((a*c - b*c*x)^(1/2) - (a*c)^(
1/2))*(a^2*c*f^2 - b^2*c*e^2))/((a + b*x)^(1/2) - a^(1/2)) - (a^2*c*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(
(a + b*x)^(1/2) - a^(1/2)) + 2*a^(1/2)*b*c*e*f*(a*c)^(1/2))/(2*b*c*e*(b^2*c*e^2 - a^2*c*f^2)^(1/2))) + 2*atan(
((((((4*(4*C^2*a^8*f^4 + C^2*a^4*b^4*e^4 + 4*C^2*a^6*b^2*e^2*f^2))/(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*
e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8) - (C^2*a^4*(2*a^2*f^2 + b^2*e^2)^2*(12*a^10*c*f^10 - 4*b^10*c*e
^10 + 28*a^2*b^8*c*e^8*f^2 - 72*a^4*b^6*c*e^6*f^4 + 88*a^6*b^4*c*e^4*f^6 - 52*a^8*b^2*c*e^2*f^8))/((a*f + b*e)
^4*(a*f - b*e)^4*(a^2*c*f^2 - b^2*c*e^2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^
6 + a^8*b^2*e^2*f^8)))/(4*b*c^2*e*(b^2*c*e^2 - a^2*c*f^2)^(1/2)) + (C*a^(3/2)*(2*a^2*f^2 + b^2*e^2)*(8*C*a^(17
/2)*f^7*(a*c)^(1/2) - 12*C*a^(13/2)*b^2*e^2*f^5*(a*c)^(1/2) + 4*C*a^(5/2)*b^6*e^6*f*(a*c)^(1/2)))/(2*b*c^2*e*f
*(a*c)^(1/2)*(a*f + b*e)^2*(a*f - b*e)^2*(b^2*c*e^2 - a^2*c*f^2)^(1/2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*
b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8)))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/((a + b*x)^(1/2) -
a^(1/2))^3 + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(((4*(4*C^2*a^8*c*f^4 + C^2*a^4*b^4*c*e^4 + 4*C^2*a^6*b^2*c
*e^2*f^2))/(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8) + (C^2*a^
4*(2*a^2*f^2 + b^2*e^2)^2*(4*a^10*c^2*f^10 + 4*b^10*c^2*e^10 - 12*a^2*b^8*c^2*e^8*f^2 + 8*a^4*b^6*c^2*e^6*f^4
+ 8*a^6*b^4*c^2*e^4*f^6 - 12*a^8*b^2*c^2*e^2*f^8))/((a*f + b*e)^4*(a*f - b*e)^4*(a^2*c*f^2 - b^2*c*e^2)*(b^10*
e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8)))/(4*b*c^2*e*(b^2*c*e^2 -
a^2*c*f^2)^(1/2)) + (8*C^2*a^4*(2*a^2*f^2 + b^2*e^2)^2)/(b*e*(a*f + b*e)^4*(a*f - b*e)^4*(b^2*c*e^2 - a^2*c*f^
2)^(3/2)) - (C*a^(3/2)*(2*a^2*f^2 + b^2*e^2)*(8*C*a^(17/2)*c*f^7*(a*c)^(1/2) + 4*C*a^(5/2)*b^6*c*e^6*f*(a*c)^(
1/2) - 12*C*a^(13/2)*b^2*c*e^2*f^5*(a*c)^(1/2)))/(2*b*c^2*e*f*(a*c)^(1/2)*(a*f + b*e)^2*(a*f - b*e)^2*(b^2*c*e
^2 - a^2*c*f^2)^(1/2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8
))))/((a + b*x)^(1/2) - a^(1/2)) - ((((4*(4*C^2*a^8*f^4 + C^2*a^4*b^4*e^4 + 4*C^2*a^6*b^2*e^2*f^2))/(b^10*e^10
- 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8) - (C^2*a^4*(2*a^2*f^2 + b^2*e^
2)^2*(12*a^10*c*f^10 - 4*b^10*c*e^10 + 28*a^2*b^8*c*e^8*f^2 - 72*a^4*b^6*c*e^6*f^4 + 88*a^6*b^4*c*e^4*f^6 - 52
*a^8*b^2*c*e^2*f^8))/((a*f + b*e)^4*(a*f - b*e)^4*(a^2*c*f^2 - b^2*c*e^2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a
^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8)))/(2*a^(1/2)*c*f*(a*c)^(1/2)*(b^2*c*e^2 - a^2*c*f^2)^(1/
2)) + (4*C^2*a^(9/2)*f*(a*c)^(1/2)*(2*a^2*f^2 + b^2*e^2)^2)/(b^2*c*e^2*(a*f + b*e)^4*(a*f - b*e)^4*(b^2*c*e^2
- a^2*c*f^2)^(3/2)))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 - ((4*(4*C^2*a^8*c*f
^4 + C^2*a^4*b^4*c*e^4 + 4*C^2*a^6*b^2*c*e^2*f^2))/(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*
b^4*e^4*f^6 + a^8*b^2*e^2*f^8) + (C^2*a^4*(2*a^2*f^2 + b^2*e^2)^2*(4*a^10*c^2*f^10 + 4*b^10*c^2*e^10 - 12*a^2*
b^8*c^2*e^8*f^2 + 8*a^4*b^6*c^2*e^6*f^4 + 8*a^6*b^4*c^2*e^4*f^6 - 12*a^8*b^2*c^2*e^2*f^8))/((a*f + b*e)^4*(a*f
- b*e)^4*(a^2*c*f^2 - b^2*c*e^2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8
*b^2*e^2*f^8)))/(2*a^(1/2)*c*f*(a*c)^(1/2)*(b^2*c*e^2 - a^2*c*f^2)^(1/2)))*(b^10*e^10*(a^2*c*f^2 - b^2*c*e^2)
- 4*a^2*b^8*e^8*f^2*(a^2*c*f^2 - b^2*c*e^2) + 6*a^4*b^6*e^6*f^4*(a^2*c*f^2 - b^2*c*e^2) - 4*a^6*b^4*e^4*f^6*(a
^2*c*f^2 - b^2*c*e^2) + a^8*b^2*e^2*f^8*(a^2*c*f^2 - b^2*c*e^2)))/(16*C^2*a^8*f^4 + 4*C^2*a^4*b^4*e^4 + 16*C^2
*a^6*b^2*e^2*f^2))))/(2*(a*f + b*e)^2*(a*f - b*e)^2*(b^2*c*e^2 - a^2*c*f^2)^(1/2)) + (A*b^2*(a^2*f^2 + 2*b^2*e
^2)*(2*atan(((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(a^2*c*f^2 - b^2*c*e^2))/((a + b*x)^(1/2) - a^(1/2)) - (a^2*
c*f^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x)^(1/2) - a^(1/2)) + 2*a^(1/2)*b*c*e*f*(a*c)^(1/2))/(2*b*c
*e*(b^2*c*e^2 - a^2*c*f^2)^(1/2))) + 2*atan((((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(((4*(4*A^2*b^8*c*e^4 + A^2
*a^4*b^4*c*f^4 + 4*A^2*a^2*b^6*c*e^2*f^2))/(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*
f^6 + a^8*b^2*e^2*f^8) + (A^2*b^4*(a^2*f^2 + 2*b^2*e^2)^2*(4*a^10*c^2*f^10 + 4*b^10*c^2*e^10 - 12*a^2*b^8*c^2*
e^8*f^2 + 8*a^4*b^6*c^2*e^6*f^4 + 8*a^6*b^4*c^2*e^4*f^6 - 12*a^8*b^2*c^2*e^2*f^8))/((a*f + b*e)^4*(a*f - b*e)^
4*(a^2*c*f^2 - b^2*c*e^2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2
*f^8)))/(4*b*c^2*e*(b^2*c*e^2 - a^2*c*f^2)^(1/2)) + (8*A^2*b^3*(a^2*f^2 + 2*b^2*e^2)^2)/(e*(a*f + b*e)^4*(a*f
- b*e)^4*(b^2*c*e^2 - a^2*c*f^2)^(3/2)) - (A*b*(a^2*f^2 + 2*b^2*e^2)*(4*A*a^(13/2)*b^2*c*f^7*(a*c)^(1/2) + 8*A
*a^(1/2)*b^8*c*e^6*f*(a*c)^(1/2) - 12*A*a^(5/2)*b^6*c*e^4*f^3*(a*c)^(1/2)))/(2*a^(1/2)*c^2*e*f*(a*c)^(1/2)*(a*
f + b*e)^2*(a*f - b*e)^2*(b^2*c*e^2 - a^2*c*f^2)^(1/2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*
a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8))))/((a + b*x)^(1/2) - a^(1/2)) + ((((4*(4*A^2*b^8*e^4 + A^2*a^4*b^4*f^4 + 4
*A^2*a^2*b^6*e^2*f^2))/(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^
8) - (A^2*b^4*(a^2*f^2 + 2*b^2*e^2)^2*(12*a^10*c*f^10 - 4*b^10*c*e^10 + 28*a^2*b^8*c*e^8*f^2 - 72*a^4*b^6*c*e^
6*f^4 + 88*a^6*b^4*c*e^4*f^6 - 52*a^8*b^2*c*e^2*f^8))/((a*f + b*e)^4*(a*f - b*e)^4*(a^2*c*f^2 - b^2*c*e^2)*(b^
10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8)))/(4*b*c^2*e*(b^2*c*e^2
- a^2*c*f^2)^(1/2)) + (A*b*(a^2*f^2 + 2*b^2*e^2)*(4*A*a^(13/2)*b^2*f^7*(a*c)^(1/2) - 12*A*a^(5/2)*b^6*e^4*f^3
*(a*c)^(1/2) + 8*A*a^(1/2)*b^8*e^6*f*(a*c)^(1/2)))/(2*a^(1/2)*c^2*e*f*(a*c)^(1/2)*(a*f + b*e)^2*(a*f - b*e)^2*
(b^2*c*e^2 - a^2*c*f^2)^(1/2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2
*e^2*f^8)))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3 - ((((4*(4*A^2*b^8*e^4 + A^2*
a^4*b^4*f^4 + 4*A^2*a^2*b^6*e^2*f^2))/(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 +
a^8*b^2*e^2*f^8) - (A^2*b^4*(a^2*f^2 + 2*b^2*e^2)^2*(12*a^10*c*f^10 - 4*b^10*c*e^10 + 28*a^2*b^8*c*e^8*f^2 -
72*a^4*b^6*c*e^6*f^4 + 88*a^6*b^4*c*e^4*f^6 - 52*a^8*b^2*c*e^2*f^8))/((a*f + b*e)^4*(a*f - b*e)^4*(a^2*c*f^2 -
b^2*c*e^2)*(b^10*e^10 - 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8)))/(2*a^(
1/2)*c*f*(a*c)^(1/2)*(b^2*c*e^2 - a^2*c*f^2)^(1/2)) + (4*A^2*a^(1/2)*b^2*f*(a*c)^(1/2)*(a^2*f^2 + 2*b^2*e^2)^2
)/(c*e^2*(a*f + b*e)^4*(a*f - b*e)^4*(b^2*c*e^2 - a^2*c*f^2)^(3/2)))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((
a + b*x)^(1/2) - a^(1/2))^2 - ((4*(4*A^2*b^8*c*e^4 + A^2*a^4*b^4*c*f^4 + 4*A^2*a^2*b^6*c*e^2*f^2))/(b^10*e^10
- 4*a^2*b^8*e^8*f^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8) + (A^2*b^4*(a^2*f^2 + 2*b^2*e^2
)^2*(4*a^10*c^2*f^10 + 4*b^10*c^2*e^10 - 12*a^2*b^8*c^2*e^8*f^2 + 8*a^4*b^6*c^2*e^6*f^4 + 8*a^6*b^4*c^2*e^4*f^
6 - 12*a^8*b^2*c^2*e^2*f^8))/((a*f + b*e)^4*(a*f - b*e)^4*(a^2*c*f^2 - b^2*c*e^2)*(b^10*e^10 - 4*a^2*b^8*e^8*f
^2 + 6*a^4*b^6*e^6*f^4 - 4*a^6*b^4*e^4*f^6 + a^8*b^2*e^2*f^8)))/(2*a^(1/2)*c*f*(a*c)^(1/2)*(b^2*c*e^2 - a^2*c*
f^2)^(1/2)))*(b^8*e^10*(a^2*c*f^2 - b^2*c*e^2) + a^8*e^2*f^8*(a^2*c*f^2 - b^2*c*e^2) - 4*a^2*b^6*e^8*f^2*(a^2*
c*f^2 - b^2*c*e^2) + 6*a^4*b^4*e^6*f^4*(a^2*c*f^2 - b^2*c*e^2) - 4*a^6*b^2*e^4*f^6*(a^2*c*f^2 - b^2*c*e^2)))/(
16*A^2*b^6*e^4 + 4*A^2*a^4*b^2*f^4 + 16*A^2*a^2*b^4*e^2*f^2))))/(2*(a*f + b*e)^2*(a*f - b*e)^2*(b^2*c*e^2 - a^
2*c*f^2)^(1/2)) + (3*B*a^2*b^2*e*f*(2*atan((2*b^3*c^3*e^3 + 2*b*c^2*e*(a^2*c*f^2 - b^2*c*e^2) + 2*a^2*b*c^3*e*
f^2 + (3*a^(3/2)*f^3*(a*c)^(3/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3 + (2*b^3
*c^2*e^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 - (3*a^(1/2)*f*(a*c)^(1/2)*((a*c
- b*c*x)^(1/2) - (a*c)^(1/2))^3*(a^2*c*f^2 - b^2*c*e^2))/((a + b*x)^(1/2) - a^(1/2))^3 - (a^(3/2)*c*f^3*(a*c)
^(3/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x)^(1/2) - a^(1/2)) + (2*b*c*e*((a*c - b*c*x)^(1/2) - (a*c
)^(1/2))^2*(a^2*c*f^2 - b^2*c*e^2))/((a + b*x)^(1/2) - a^(1/2))^2 + (a^(1/2)*c*f*(a*c)^(1/2)*((a*c - b*c*x)^(1
/2) - (a*c)^(1/2))*(a^2*c*f^2 - b^2*c*e^2))/((a + b*x)^(1/2) - a^(1/2)) - (10*a^2*b*c^2*e*f^2*((a*c - b*c*x)^(
1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (7*a^(1/2)*b^2*c^2*e^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/
2) - (a*c)^(1/2)))/((a + b*x)^(1/2) - a^(1/2)) - (a^(1/2)*b^2*c*e^2*f*(a*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)
^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3)/(4*a^(1/2)*b*c^2*e*f*(a*c)^(1/2)*(b^2*c*e^2 - a^2*c*f^2)^(1/2))) - 2
*atan(((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(a^2*c*f^2 - b^2*c*e^2))/((a + b*x)^(1/2) - a^(1/2)) - (a^2*c*f^2*
((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x)^(1/2) - a^(1/2)) + 2*a^(1/2)*b*c*e*f*(a*c)^(1/2))/(2*b*c*e*(b^
2*c*e^2 - a^2*c*f^2)^(1/2)))))/(2*(a*f + b*e)^2*(a*f - b*e)^2*(b^2*c*e^2 - a^2*c*f^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/(f*x+e)**3/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

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