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*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]*Sq
rt[c]*d - a*e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.53298, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} (B d-A e) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} B \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

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]*Sq
rt[c]*d - a*e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 94.3175, size = 270, normalized size = 0.94 \[ - \frac{2 B \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} e \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{2 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (A e - B d\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} e \sqrt{a + c x^{2}} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**(1/2)/(c*x**2+a)**(1/2),x)

[Out]

-2*B*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*elliptic_e(asin(sqrt(-sqrt(c)*x/(
2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(sqrt(c)*e*sqrt(sqrt(c)*s
qrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) - 2*sqrt(-a)*sq
rt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*x**2/a)*(A
*e - B*d)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqr
t(c)*d*sqrt(-a)))/(sqrt(c)*e*sqrt(a + c*x**2)*sqrt(d + e*x))

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Mathematica [C]  time = 2.68375, 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)/Sqrt[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]*Sqrt[a + c*x^2])

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Maple [B]  time = 0.053, size = 520, normalized size = 1.8 \[ 2\,{\frac{\sqrt{ex+d}\sqrt{c{x}^{2}+a}}{{e}^{2}c \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ) } \left ( A{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) cde-A{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) \sqrt{-ac}{e}^{2}+B{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) a{e}^{2}+B{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) \sqrt{-ac}de-B{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) a{e}^{2}-B{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) c{d}^{2} \right ) \sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}}} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError