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

Optimal. Leaf size=145 \[ \frac {2 \sqrt {a} (a B e+A (b-b e)) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right ),\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}-\frac {2 a^{3/2} B E\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {158, 113, 119} \[ \frac {2 \sqrt {a} (a B e+A (b-b e)) F\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)}-\frac {2 a^{3/2} B E\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b^2 \sqrt {1-c} (1-e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 1.51, size = 309, normalized size = 2.13 \[ -\frac {2 \sqrt {\frac {a}{c-1}} (a+b x)^{3/2} \left (\frac {i (e-1) \sqrt {\frac {\frac {a}{a+b x}+c-1}{c-1}} \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} (a B c+A (b-b c)) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {a}{c-1}}}{\sqrt {a+b x}}\right ),\frac {c-1}{e-1}\right )}{\sqrt {a+b x}}-B \sqrt {\frac {a}{c-1}} \left (\frac {a}{a+b x}+c-1\right ) \left (\frac {a}{a+b x}+e-1\right )-\frac {i a B (e-1) \sqrt {\frac {\frac {a}{a+b x}+c-1}{c-1}} \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {a}{c-1}}}{\sqrt {a+b x}}\right )|\frac {c-1}{e-1}\right )}{\sqrt {a+b x}}\right )}{a b^2 (e-1) \sqrt {\frac {b (c-1) x}{a}+c} \sqrt {\frac {b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B a^{2} x + A a^{2}\right )} \sqrt {b x + a} \sqrt {\frac {a c + {\left (b c - b\right )} x}{a}} \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{3} c e - {\left (b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e\right )} x^{3} - {\left (2 \, a b^{2} c - a b^{2} - {\left (3 \, a b^{2} c - 2 \, a b^{2}\right )} e\right )} x^{2} - {\left (a^{2} b c - {\left (3 \, a^{2} b c - a^{2} b\right )} e\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 624, normalized size = 4.30 \[ -\frac {2 \left (A b \,c^{2} \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )-A b c e \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )-B a \,c^{2} \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )+B a c e \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )-A b c \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )+A b e \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )-B a c \EllipticE \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )+B a c \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )+B a e \EllipticE \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )-B a e \EllipticF \left (\sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}, \sqrt {-\frac {c -e}{e -1}}\right )\right ) \sqrt {\frac {\left (c -1\right ) \left (b e x +a e -b x \right )}{\left (c -e \right ) a}}\, \sqrt {-\frac {\left (b x +a \right ) \left (c -1\right )}{a}}\, \sqrt {-\frac {\left (e -1\right ) \left (b c x +a c -b x \right )}{\left (c -e \right ) a}}\, a}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \left (e -1\right ) \left (c -1\right )^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,x}{\sqrt {c+\frac {b\,x\,\left (c-1\right )}{a}}\,\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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