3.1473 \(\int \frac{(A+B x) \sqrt{a+c x^2}}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=365 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) (4 B d-5 A e) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 a B e^2-5 A c d e+4 B c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2} \]

[Out]

(-2*Sqrt[d + e*x]*(4*B*d - 5*A*e - 3*B*e*x)*Sqrt[a + c*x^2])/(15*e^2) - (4*Sqrt[
-a]*(4*B*c*d^2 - 5*A*c*d*e + 3*a*B*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ellipt
icE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d
- a*e)])/(15*Sqrt[c]*e^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqr
t[a + c*x^2]) + (4*Sqrt[-a]*(4*B*d - 5*A*e)*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e
*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sq
rt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*Sqrt[c]*e
^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.881061, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) (4 B d-5 A e) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 a B e^2-5 A c d e+4 B c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(4*B*d - 5*A*e - 3*B*e*x)*Sqrt[a + c*x^2])/(15*e^2) - (4*Sqrt[
-a]*(4*B*c*d^2 - 5*A*c*d*e + 3*a*B*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ellipt
icE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d
- a*e)])/(15*Sqrt[c]*e^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqr
t[a + c*x^2]) + (4*Sqrt[-a]*(4*B*d - 5*A*e)*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e
*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sq
rt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*Sqrt[c]*e
^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 139.286, size = 359, normalized size = 0.98 \[ \frac{4 \sqrt{a + c x^{2}} \sqrt{d + e x} \left (\frac{5 A e}{2} - 2 B d + \frac{3 B e x}{2}\right )}{15 e^{2}} - \frac{4 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (5 A e - 4 B d\right ) \left (a e^{2} + c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 \sqrt{c} e^{3} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{4 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (- 5 A c d e + 3 B a e^{2} + 4 B c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 \sqrt{c} e^{3} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(a + c*x**2)*sqrt(d + e*x)*(5*A*e/2 - 2*B*d + 3*B*e*x/2)/(15*e**2) - 4*sqr
t(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*x*
*2/a)*(5*A*e - 4*B*d)*(a*e**2 + c*d**2)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(
-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(15*sqrt(c)*e**3*sqrt(a + c*x**2
)*sqrt(d + e*x)) - 4*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(-5*A*c*d*e + 3*B
*a*e**2 + 4*B*c*d**2)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*
e/(a*e - sqrt(c)*d*sqrt(-a)))/(15*sqrt(c)*e**3*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/
(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2))

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Mathematica [C]  time = 6.94807, size = 549, normalized size = 1.5 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) (5 A e-4 B d+3 B e x)}{e^2}-\frac{4 \left (e^2 \left (a+c x^2\right ) \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (3 a B e^2-5 A c d e+4 B c d^2\right )+\sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (-3 a B e^2+5 A c d e-4 B c d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (-3 i \sqrt{a} B e-5 A \sqrt{c} e+4 B \sqrt{c} d\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e^4 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{15 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((2*(-4*B*d + 5*A*e + 3*B*e*x)*(a + c*x^2))/e^2 - (4*(-(e^2*Sqrt[
-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*B*c*d^2 - 5*A*c*d*e + 3*a*B*e^2)*(a + c*x^2)) + S
qrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(-4*B*c*d^2 + 5*A*c*d*e - 3*a*B*e^2)*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]]/Sqr
t[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt
[c]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*(4*B*Sqrt[c]*d - (3*I)*Sqrt[a]*B*e - 5*A*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)/S
qrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(
c*e^4*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])

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Maple [B]  time = 0.032, size = 1826, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)/c*(10*A*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-
a*c)^(1/2)*e-c*d))^(1/2)*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)*a*e^4+10*A*(-(e*x+d)
*c/((-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
)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*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)*c*d^2*e^2-10*A*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/
2)*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*c*d*e^3-10*A*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-a*c)
^(1/2)*e-c*d))^(1/2)*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^2*d^3*e-8*B*(-(e*x+d)*c/((-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)*((c*x+(-a*c)
^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*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)*a*d*e^3
-8*B*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*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)*c*d^3*e-6*B*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*
e-c*d))^(1/2)*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^2*e^4-6*B*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e
/((-a*c)^(1/2)*e-c*d))^(1/2)*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*d^2*e^2+6*B*(-(e*x+d)*c/(
(-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)*((
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*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^2*e^4+14
*B*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*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*c*d^2*e^2+8*B*(-(e*x+d)*c/((-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)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/
2)*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^2*d^4-3*B*x^4*c^2*e^4-5*A*x^3*c^2*e^4+B*x^3*c^2*d*e^3
-5*A*x^2*c^2*d*e^3-3*B*x^2*a*c*e^4+4*B*x^2*c^2*d^2*e^2-5*A*x*a*c*e^4+B*x*a*c*d*e
^3-5*A*a*c*d*e^3+4*B*a*c*d^2*e^2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)/e^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(1/2)/(e*x+d)**(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. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError