3.34 \(\int \frac {A+B x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=221 \[ \frac {2 \sqrt {a} (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right ),-\frac {a d}{(1-e) (b c-a d)}\right )}{b^2 (1-e)^{3/2} \sqrt {c+d x}}-\frac {2 a B \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|-\frac {(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt {d} (1-e) \sqrt {c+d x}} \]

[Out]

2*(a*B*e+A*(-b*e+b))*EllipticF((1-e)^(1/2)*(b*x+a)^(1/2)/a^(1/2),(-a*d/(-a*d+b*c)/(1-e))^(1/2))*a^(1/2)*(b*(d*
x+c)/(-a*d+b*c))^(1/2)/b^2/(1-e)^(3/2)/(d*x+c)^(1/2)-2*a*B*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),(-(
-a*d+b*c)*(1-e)/a/d)^(1/2))*(a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)/b^2/(1-e)/d^(1/2)/(d*x+c)^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {158, 114, 113, 121, 119} \[ \frac {2 \sqrt {a} (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b^2 (1-e)^{3/2} \sqrt {c+d x}}-\frac {2 a B \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|-\frac {(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt {d} (1-e) \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(-2*a*B*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c
) + a*d]], -(((b*c - a*d)*(1 - e))/(a*d))])/(b^2*Sqrt[d]*(1 - e)*Sqrt[c + d*x]) + (2*Sqrt[a]*(a*B*e + A*(b - b
*e))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(Sqrt[1 - e]*Sqrt[a + b*x])/Sqrt[a]], -((a*d)/((b*c - a*
d)*(1 - e)))])/(b^2*(1 - e)^(3/2)*Sqrt[c + d*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 119

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*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

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]

Rubi steps

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

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Mathematica [C]  time = 2.25, size = 312, normalized size = 1.41 \[ -\frac {2 \sqrt {\frac {a}{e-1}} (a+b x)^{3/2} \left (\frac {i d \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} (a B e+A (b-b e)) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {a}{e-1}}}{\sqrt {a+b x}}\right ),\frac {(e-1) (b c-a d)}{a d}\right )}{\sqrt {a+b x}}-\frac {b B \sqrt {\frac {a}{e-1}} (c+d x) (a e+b (e-1) x)}{(a+b x)^2}-\frac {i a B d \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} \sqrt {\frac {b (c+d x)}{d (a+b x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {a}{e-1}}}{\sqrt {a+b x}}\right )|\frac {(b c-a d) (e-1)}{a d}\right )}{\sqrt {a+b x}}\right )}{a b^2 d \sqrt {c+d x} \sqrt {\frac {b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(-2*Sqrt[a/(-1 + e)]*(a + b*x)^(3/2)*(-((b*B*Sqrt[a/(-1 + e)]*(c + d*x)*(a*e + b*(-1 + e)*x))/(a + b*x)^2) - (
I*a*B*d*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticE[I*ArcSinh[Sqrt[a/(-1
+ e)]/Sqrt[a + b*x]], ((b*c - a*d)*(-1 + e))/(a*d)])/Sqrt[a + b*x] + (I*d*(a*B*e + A*(b - b*e))*Sqrt[(b*(c +
d*x))/(d*(a + b*x))]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticF[I*ArcSinh[Sqrt[a/(-1 + e)]/Sqrt[a + b*x]]
, ((b*c - a*d)*(-1 + e))/(a*d)])/Sqrt[a + b*x]))/(a*b^2*d*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a])

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fricas [F]  time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B a x + A a\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{2} c e + {\left (b^{2} d e - b^{2} d\right )} x^{3} - {\left (b^{2} c + a b d - {\left (b^{2} c + 2 \, a b d\right )} e\right )} x^{2} - {\left (a b c - {\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)), x)

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maple [B]  time = 0.04, size = 940, normalized size = 4.25 \[ \frac {2 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (e -1\right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) \left (e -1\right ) b}{a d e -b c e +b c}}\, \left (A a b d \,e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-A \,b^{2} c \,e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-B \,a^{2} d \,e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+B a b c \,e^{2} \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-A a b d e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+2 A \,b^{2} c e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-B \,a^{2} d e \EllipticE \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+B \,a^{2} d e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+B a b c e \EllipticE \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-2 B a b c e \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-A \,b^{2} c \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )-B a b c \EllipticE \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )+B a b c \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )\right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) \left (e -1\right )^{2} b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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