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

Optimal. Leaf size=422 \[ -\frac {2 \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} (a C (d e-c f)-b (A d f-B c f+c C e)) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {f (b c-a d)}{d (b e-a f)}\right )}{b^2 \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x} \sqrt {a d-b c}}-\frac {2 \sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} f \sqrt {c+d x} \sqrt {a d-b c} (b e-a f) \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)} \]

[Out]

-2*(A*b^2-a*(B*b-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(1/2)-2*(2*a^2*C*d*f+b^2*(A
*d*f+C*c*e)-a*b*(B*d*f+C*c*f+C*d*e))*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e
))^(1/2))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^2/f/(-a*f+b*e)/d^(1/2)/(a*d-b*c)^(1/2)/(d*x+c)^(1/2)/(b
*(f*x+e)/(-a*f+b*e))^(1/2)-2*(a*C*(-c*f+d*e)-b*(A*d*f-B*c*f+C*c*e))*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^
(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^2/f/d^(1/
2)/(a*d-b*c)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)

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Rubi [A]  time = 0.69, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1614, 158, 114, 113, 121, 120} \[ -\frac {2 \sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} f \sqrt {c+d x} \sqrt {a d-b c} (b e-a f) \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {2 \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} (a C (d e-c f)-b (A d f-B c f+c C e)) F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x} \sqrt {a d-b c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) - (2*(2*a^2
*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Elli
pticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b^2*Sqrt[d]*Sqrt[
-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (2*(a*C*(d*e - c*f) - b*(c*C*e -
B*c*f + A*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt
[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b^2*Sqrt[d]*Sqrt[-(b*c) + a*d]*f*Sqrt[c + d
*x]*Sqrt[e + f*x])

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 1614

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \int \frac {-\frac {b^2 B c e+a^2 C (d e+c f)-a b (c C e+B d e+B c f-A d f)}{2 b}+\frac {1}{2} \left (-\frac {2 a^2 C d f}{b}-b (c C e+A d f)+a (C d e+c C f+B d f)\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {(a C (d e-c f)-b (c C e-B c f+A d f)) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{b (b c-a d) f}+\frac {\left (\frac {2 a^2 C d f}{b}+b (c C e+A d f)-a (C d e+c C f+B d f)\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{(b c-a d) f (b e-a f)}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {\left ((a C (d e-c f)-b (c C e-B c f+A d f)) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}} \, dx}{b (b c-a d) f \sqrt {c+d x}}+\frac {\left (\left (\frac {2 a^2 C d f}{b}+b (c C e+A d f)-a (C d e+c C f+B d f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{(b c-a d) f (b e-a f) \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \left (2 a^2 C d f+b^2 (c C e+A d f)-a b (C d e+c C f+B d f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {\left ((a C (d e-c f)-b (c C e-B c f+A d f)) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}} \, dx}{b (b c-a d) f \sqrt {c+d x} \sqrt {e+f x}}\\ &=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}-\frac {2 \left (2 a^2 C d f+b^2 (c C e+A d f)-a b (C d e+c C f+B d f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 (a C (d e-c f)-b (c C e-B c f+A d f)) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt {d} \sqrt {-b c+a d} f \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}

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Mathematica [C]  time = 5.44, size = 477, normalized size = 1.13 \[ \frac {2 \left (\frac {i b (a+b x)^{3/2} (a d-b c) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} (a C (d e-c f)+b (A d f-B d e+c C e)) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b c}{d}-a}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{d \sqrt {\frac {b c}{d}-a}}+\frac {b^2 (c+d x) (e+f x) \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right )}{d f}+\frac {i (a+b x)^{3/2} (b c-a d) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b c}{d}-a}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{d \sqrt {\frac {b c}{d}-a}}-\left (b^2 (c+d x) (e+f x) \left (a (a C-b B)+A b^2\right )\right )\right )}{b^3 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(b^2*(A*b^2 + a*(-(b*B) + a*C))*(c + d*x)*(e + f*x)) + (b^2*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d
*e + c*C*f + B*d*f))*(c + d*x)*(e + f*x))/(d*f) + (I*(b*c - a*d)*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d
*e + c*C*f + B*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ellip
ticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(Sqrt[-a + (b*c)/d]*d) + (
I*b*(-(b*c) + a*d)*(a*C*(d*e - c*f) + b*(c*C*e - B*d*e + A*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*
x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/
(b*c*f - a*d*f)])/(Sqrt[-a + (b*c)/d]*d)))/(b^3*(b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f
*x])

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{2} d f x^{4} + a^{2} c e + {\left (b^{2} d e + {\left (b^{2} c + 2 \, a b d\right )} f\right )} x^{3} + {\left ({\left (b^{2} c + 2 \, a b d\right )} e + {\left (2 \, a b c + a^{2} d\right )} f\right )} x^{2} + {\left (a^{2} c f + {\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*x^2 + B*x + A)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(b^2*d*f*x^4 + a^2*c*e + (b^2*d*e + (b^2*
c + 2*a*b*d)*f)*x^3 + ((b^2*c + 2*a*b*d)*e + (2*a*b*c + a^2*d)*f)*x^2 + (a^2*c*f + (2*a*b*c + a^2*d)*e)*x), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError >> type

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maple [B]  time = 0.05, size = 3984, normalized size = 9.44 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(B*x^2*a*b^3*d^2*f^2+B*x*a*b^3*c*d*f^2+B*x*a*b^3*d^2*e*f-C*x*a^2*b^2*c*d*f^2-C*x*a^2*b^2*d^2*e*f-2*C*Ellipti
cE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^4*d^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x
+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-C*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*
e)/d*f)^(1/2))*b^4*c^2*e^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/
2)+B*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*c*d*e*f*((b*x+a)/(a*d-b*c)*d
)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+B*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*
d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*c*d*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/
(a*d-b*c)*b)^(1/2)+C*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*c*d*e*f*((
b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-5*C*EllipticE(((b*x+a)/(a*
d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*c*d*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*
e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+A*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/
2))*a^2*b^2*d^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-A*El
lipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*d^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/
2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+B*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c
)/(a*f-b*e)/d*f)^(1/2))*a*b^3*c^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-
b*c)*b)^(1/2)-B*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*b^4*c^2*e*f*((b*x+a)/(a
*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+B*EllipticE(((b*x+a)/(a*d-b*c)*d)^(
1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^3*b*d^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(
-(d*x+c)/(a*d-b*c)*b)^(1/2)-C*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*c
^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+C*EllipticF(((b*x
+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*d^2*e^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/
(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-C*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d
*f)^(1/2))*a^2*b^2*c^2*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/
2)-C*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*d^2*e^2*((b*x+a)/(a*d-b*c)
*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-A*x^2*b^4*d^2*f^2-C*a^2*b^2*c*d*e*f+B*a*b^
3*c*d*e*f+3*C*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^3*b*d^2*e*f*((b*x+a)/(a
*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+2*C*EllipticE(((b*x+a)/(a*d-b*c)*d)
^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*c^2*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)
*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+2*C*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3
*c*d*e^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+A*EllipticE(((b
*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*c*d*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/
(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+A*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d
*f)^(1/2))*a*b^3*d^2*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)
-A*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*b^4*c*d*e*f*((b*x+a)/(a*d-b*c)*d)^(1
/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-B*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*
c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*c*d*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a
*d-b*c)*b)^(1/2)-B*EllipticE(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*c*d*f^2*((b*
x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-B*EllipticE(((b*x+a)/(a*d-b*
c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^2*b^2*d^2*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b
)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+C*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*
a^3*b*c*d*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-A*b^4*c*d*
e*f-C*x^2*a^2*b^2*d^2*f^2-A*x*b^4*c*d*f^2-A*x*b^4*d^2*e*f-C*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(
a*f-b*e)/d*f)^(1/2))*a^3*b*d^2*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c
)*b)^(1/2)-2*C*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*c*d*e^2*((b*x+a)/(
a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+3*C*EllipticE(((b*x+a)/(a*d-b*c)*d
)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a^3*b*c*d*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2
)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-A*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*
c*d*f^2*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)-A*EllipticF(((b*
x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*a*b^3*d^2*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(
a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+A*EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*
f)^(1/2))*b^4*c*d*e*f*((b*x+a)/(a*d-b*c)*d)^(1/2)*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2)+C*
EllipticF(((b*x+a)/(a*d-b*c)*d)^(1/2),((a*d-b*c)/(a*f-b*e)/d*f)^(1/2))*b^4*c^2*e^2*((b*x+a)/(a*d-b*c)*d)^(1/2)
*(-(f*x+e)/(a*f-b*e)*b)^(1/2)*(-(d*x+c)/(a*d-b*c)*b)^(1/2))*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/f/d/b^3/
(a*f-b*e)/(a*d-b*c)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*x^2 + B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {C\,x^2+B\,x+A}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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