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3.1472 \(\int (A+B x) \sqrt{d+e x} \sqrt{a+c x^2} \, dx\)

Optimal. Leaf size=438 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-7 A c d e+4 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} e^3 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{105 c e^2}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 B \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

[Out]

(-2*Sqrt[d + e*x]*(4*B*c*d^2 - 7*A*c*d*e + 5*a*B*e^2 - 3*c*e*(B*d + 7*A*e)*x)*Sq
rt[a + c*x^2])/(105*c*e^2) + (2*B*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(7*c) - (4*Sq
rt[-a]*(4*B*c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3)*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)])/(105*Sqrt[c]*e^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]
*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*(c*d^2 + a*e^2)*(4*B*c*d^2 - 7*
A*c*d*e + 5*a*B*e^2)*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)])/(105*c^(3/2)*e^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.13615, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \left (5 a B e^2-7 A c d e+4 B c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} e^3 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left (5 a B e^2-3 c e x (7 A e+B d)-7 A c d e+4 B c d^2\right )}{105 c e^2}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 \sqrt{c} e^3 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 B \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(4*B*c*d^2 - 7*A*c*d*e + 5*a*B*e^2 - 3*c*e*(B*d + 7*A*e)*x)*Sq
rt[a + c*x^2])/(105*c*e^2) + (2*B*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(7*c) - (4*Sq
rt[-a]*(4*B*c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3)*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)])/(105*Sqrt[c]*e^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]
*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*(c*d^2 + a*e^2)*(4*B*c*d^2 - 7*
A*c*d*e + 5*a*B*e^2)*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)])/(105*c^(3/2)*e^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 173.404, size = 437, normalized size = 1. \[ \frac{2 B \left (a + c x^{2}\right )^{\frac{3}{2}} \sqrt{d + e x}}{7 c} - \frac{8 \sqrt{a + c x^{2}} \sqrt{d + e x} \left (- \frac{7 A c d e}{4} + \frac{5 B a e^{2}}{4} + B c d^{2} - \frac{3 c e x \left (7 A e + B d\right )}{4}\right )}{105 c e^{2}} - \frac{4 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (21 A a e^{3} - 7 A c d^{2} e + 8 B a d e^{2} + 4 B c d^{3}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{105 \sqrt{c} e^{3} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{4 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) \left (- 7 A c d e + 5 B a e^{2} + 4 B c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{105 c^{\frac{3}{2}} e^{3} \sqrt{a + c x^{2}} \sqrt{d + e x}} \]

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*(a + c*x**2)**(3/2)*sqrt(d + e*x)/(7*c) - 8*sqrt(a + c*x**2)*sqrt(d + e*x)*(
-7*A*c*d*e/4 + 5*B*a*e**2/4 + B*c*d**2 - 3*c*e*x*(7*A*e + B*d)/4)/(105*c*e**2) -
 4*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(21*A*a*e**3 - 7*A*c*d**2*e + 8*B*a
*d*e**2 + 4*B*c*d**3)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*
e/(a*e - sqrt(c)*d*sqrt(-a)))/(105*sqrt(c)*e**3*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)
/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) + 4*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a
)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*x**2/a)*(a*e**2 + c*d**2)*(-
7*A*c*d*e + 5*B*a*e**2 + 4*B*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)))/(105*c**(3/2)*e**3*sqrt(a + c*x**2)
*sqrt(d + e*x))

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Mathematica [C]  time = 7.98242, size = 622, normalized size = 1.42 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (10 a B e^2+7 A c e (d+3 e x)+B c \left (-4 d^2+3 d e x+15 e^2 x^2\right )\right )}{c e^2}+\frac{4 \left (\sqrt{a} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (7 A \left (c d e+3 i \sqrt{a} \sqrt{c} e^2\right )+B \left (3 i \sqrt{a} \sqrt{c} d e-5 a e^2-4 c d^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right )-\sqrt{c} (d+e x)^{3/2} \left (-\sqrt{a} e+i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (21 a A e^3+8 a B d e^2-7 A c d^2 e+4 B c d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c e^4 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{105 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(10*a*B*e^2 + 7*A*c*e*(d + 3*e*x) + B*c*(-4*d^2 +
 3*d*e*x + 15*e^2*x^2)))/(c*e^2) + (4*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*B
*c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3)*(a + c*x^2) - Sqrt[c]*(I*Sqrt[c
]*d - Sqrt[a]*e)*(4*B*c*d^3 - 7*A*c*d^2*e + 8*a*B*d*e^2 + 21*a*A*e^3)*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]*e*(Sqrt
[c]*d + I*Sqrt[a]*e)*(B*(-4*c*d^2 + (3*I)*Sqrt[a]*Sqrt[c]*d*e - 5*a*e^2) + 7*A*(
c*d*e + (3*I)*Sqrt[a]*Sqrt[c]*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)*EllipticF[I*A
rcSinh[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^4*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x)
)))/(105*Sqrt[a + c*x^2])

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Maple [B]  time = 0.039, size = 2551, normalized size = 5.8 \[ \text{result too large to display} \]

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]

2/105*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(15*B*x^5*c^3*e^5-B*x*a*c^2*d^2*e^3+28*B*x^2
*a*c^2*d*e^4+28*A*x*a*c^2*d*e^4-16*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*c*d*e^4-24*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^2*d^3*e^2+10*B*x
*a^2*c*e^5+18*B*x^4*c^3*d*e^4+28*A*x^3*c^3*d*e^4+25*B*x^3*a*c^2*e^5+7*A*a*c^2*d^
2*e^3+10*B*a^2*c*d*e^4-4*B*a*c^2*d^3*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)*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^2*d^4*e+42*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^2*d^2*e^3-14*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)*E
llipticF((-(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^2*d^3*e^2-28*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^2*d^2*e^3+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*c
*d*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)*Elliptic
F((-(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^2*d^3*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^3*d^5+14*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^3*d^4*e+10*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^2*e^5+42*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)*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^2*c*e^5-4*B*x^2*c^3*d^3*e^2+18*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*c*d^2*e^3
-14*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*c*d*e^4+21*A*x^4*c^3*e^5-B*x^3*c^3*d^2*e^3+21*A*x^2*a*c^
2*e^5+7*A*x^2*c^3*d^2*e^3-42*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^2*c*e^5)/(c*e*x^3+c*d*x^2+a*e*x+a*d)/e^4/c^2

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

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. \[{\rm integral}\left (\sqrt{c x^{2} + a}{\left (B x + A\right )} \sqrt{e x + d}, x\right ) \]

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. \[ \int \left (A + B x\right ) \sqrt{a + c x^{2}} \sqrt{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(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. \[ \text{Exception raised: RuntimeError} \]

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

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