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

Optimal. Leaf size=188 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^2 \sqrt{b c-a d} \sqrt{b e-a f}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (2 a C d f+b (-2 B d f+c C f+C d e))}{b^2 d^{3/2} f^{3/2}}+\frac{C \sqrt{c+d x} \sqrt{e+f x}}{b d f} \]

[Out]

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

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Rubi [A]  time = 0.340917, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {1615, 157, 63, 217, 206, 93, 208} \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^2 \sqrt{b c-a d} \sqrt{b e-a f}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (2 a C d f+b (-2 B d f+c C f+C d e))}{b^2 d^{3/2} f^{3/2}}+\frac{C \sqrt{c+d x} \sqrt{e+f x}}{b d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.999701, size = 304, normalized size = 1.62 \[ \frac{2 \left (\frac{\left (a (a C-b B)+A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{\sqrt{a d-b c} \sqrt{b e-a f}}-\frac{\sqrt{e+f x} (a C f-b B f+b C e) \sinh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{f^{3/2} \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}}+\frac{b C \sqrt{e+f x} \left (\sqrt{f} \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{d e-c f}}+\sqrt{d e-c f} \sinh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )\right )}{2 d f^{3/2} \sqrt{\frac{d (e+f x)}{d e-c f}}}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.031, size = 746, normalized size = 4. \begin{align*} -{\frac{1}{2\,df{b}^{3}} \left ( 2\,A\ln \left ({\frac{1}{bx+a} \left ( -2\,adfx+bcfx+bdex+2\,\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }b-acf-ade+2\,bce \right ) } \right ){b}^{2}df\sqrt{df}-2\,B\ln \left ({\frac{1}{bx+a} \left ( -2\,adfx+bcfx+bdex+2\,\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }b-acf-ade+2\,bce \right ) } \right ) abdf\sqrt{df}-2\,B\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{df}+cf+de}{\sqrt{df}}} \right ){b}^{2}df\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}+2\,C\ln \left ({\frac{1}{bx+a} \left ( -2\,adfx+bcfx+bdex+2\,\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }b-acf-ade+2\,bce \right ) } \right ){a}^{2}df\sqrt{df}+2\,C\ln \left ( 1/2\,{\frac{2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{df}+cf+de}{\sqrt{df}}} \right ) abdf\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}+C\ln \left ({\frac{1}{2} \left ( 2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{df}+cf+de \right ){\frac{1}{\sqrt{df}}}} \right ){b}^{2}cf\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}+C\ln \left ({\frac{1}{2} \left ( 2\,dfx+2\,\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{df}+cf+de \right ){\frac{1}{\sqrt{df}}}} \right ){b}^{2}de\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}-2\,C{b}^{2}\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }\sqrt{df}\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}} \right ) \sqrt{fx+e}\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( dx+c \right ) \left ( fx+e \right ) }}}{\frac{1}{\sqrt{df}}}{\frac{1}{\sqrt{{\frac{{a}^{2}df-abcf-abde+ce{b}^{2}}{{b}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.6315, size = 467, normalized size = 2.48 \begin{align*} \frac{\sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \sqrt{d x + c} C{\left | d \right |}}{b d^{3} f} - \frac{2 \,{\left (\sqrt{d f} C a^{2} d^{2} - \sqrt{d f} B a b d^{2} + \sqrt{d f} A b^{2} d^{2}\right )} \arctan \left (-\frac{b c d f - 2 \, a d^{2} f + b d^{2} e -{\left (\sqrt{d f} \sqrt{d x + c} - \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}\right )}^{2} b}{2 \, \sqrt{a b c d f^{2} - a^{2} d^{2} f^{2} - b^{2} c d f e + a b d^{2} f e} d}\right )}{\sqrt{a b c d f^{2} - a^{2} d^{2} f^{2} - b^{2} c d f e + a b d^{2} f e} b^{2} d{\left | d \right |}} + \frac{{\left (\sqrt{d f} C b c f + 2 \, \sqrt{d f} C a d f - 2 \, \sqrt{d f} B b d f + \sqrt{d f} C b d e\right )} \log \left ({\left (\sqrt{d f} \sqrt{d x + c} - \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}\right )}^{2}\right )}{2 \, b^{2} d f^{2}{\left | d \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt((d*x + c)*d*f - c*d*f + d^2*e)*sqrt(d*x + c)*C*abs(d)/(b*d^3*f) - 2*(sqrt(d*f)*C*a^2*d^2 - sqrt(d*f)*B*a*
b*d^2 + sqrt(d*f)*A*b^2*d^2)*arctan(-1/2*(b*c*d*f - 2*a*d^2*f + b*d^2*e - (sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x
+ c)*d*f - c*d*f + d^2*e))^2*b)/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d))/(sqrt(a*b*c*
d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*b^2*d*abs(d)) + 1/2*(sqrt(d*f)*C*b*c*f + 2*sqrt(d*f)*C*a*d*f
- 2*sqrt(d*f)*B*b*d*f + sqrt(d*f)*C*b*d*e)*log((sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))
^2)/(b^2*d*f^2*abs(d))