3.1482 \(\int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx\)

Optimal. Leaf size=288 \[ \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-a} B \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {844, 719, 424, 419} \[ \frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-a} B \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} e \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

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

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 1.59, size = 439, normalized size = 1.52 \[ -\frac {2 \left (\sqrt {c} e (d+e x)^{3/2} \left (\sqrt {a} B-i A \sqrt {c}\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-B e^2 \left (a+c x^2\right ) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}+i B \sqrt {c} (d+e x)^{3/2} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^2 \sqrt {a+c x^2} \sqrt {d+e x} \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x + d}}{c e x^{3} + c d x^{2} + a e x + a d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {c x^{2} + a} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.15, size = 520, normalized size = 1.81 \[ \frac {2 \left (A c d e \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )-B a \,e^{2} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+B a \,e^{2} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )-B c \,d^{2} \EllipticE \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )-\sqrt {-a c}\, A \,e^{2} \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )+\sqrt {-a c}\, B d e \EllipticF \left (\sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}, \sqrt {-\frac {-c d +\sqrt {-a c}\, e}{c d +\sqrt {-a c}\, e}}\right )\right ) \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{-c d +\sqrt {-a c}\, e}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{c d +\sqrt {-a c}\, e}}\, \sqrt {-\frac {\left (e x +d \right ) c}{-c d +\sqrt {-a c}\, e}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{\left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right ) c \,e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B x + A}{\sqrt {c x^{2} + a} \sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{\sqrt {c\,x^2+a}\,\sqrt {d+e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B x}{\sqrt {a + c x^{2}} \sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________