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

Optimal. Leaf size=328 \[ -\frac{2 a^{5/4} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B-21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 a B-21 A c x)}{105 c}+\frac{4 a A e x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c} \]

[Out]

(4*a*A*e*x*Sqrt[a + c*x^2])/(5*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*Sqr
t[e*x]*(5*a*B - 21*A*c*x)*Sqrt[a + c*x^2])/(105*c) + (2*B*Sqrt[e*x]*(a + c*x^2)^
(3/2))/(7*c) - (4*a^(5/4)*A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*c^
(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - (2*a^(5/4)*(5*Sqrt[a]*B - 21*A*Sqrt[c])*e*Sqr
t[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2
*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(105*c^(5/4)*Sqrt[e*x]*Sqrt[a + c*x^2]
)

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Rubi [A]  time = 0.796806, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{2 a^{5/4} e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B-21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 a^{5/4} A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{e x} \sqrt{a+c x^2} (5 a B-21 A c x)}{105 c}+\frac{4 a A e x \sqrt{a+c x^2}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 B \sqrt{e x} \left (a+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*a*A*e*x*Sqrt[a + c*x^2])/(5*Sqrt[c]*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (2*Sqr
t[e*x]*(5*a*B - 21*A*c*x)*Sqrt[a + c*x^2])/(105*c) + (2*B*Sqrt[e*x]*(a + c*x^2)^
(3/2))/(7*c) - (4*a^(5/4)*A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*c^
(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) - (2*a^(5/4)*(5*Sqrt[a]*B - 21*A*Sqrt[c])*e*Sqr
t[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2
*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(105*c^(5/4)*Sqrt[e*x]*Sqrt[a + c*x^2]
)

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Rubi in Sympy [A]  time = 99.7192, size = 313, normalized size = 0.95 \[ - \frac{4 A a^{\frac{5}{4}} e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{4 A a e x \sqrt{a + c x^{2}}}{5 \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{2 B \sqrt{e x} \left (a + c x^{2}\right )^{\frac{3}{2}}}{7 c} + \frac{2 a^{\frac{5}{4}} e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (21 A \sqrt{c} - 5 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 c^{\frac{5}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{8 \sqrt{e x} \sqrt{a + c x^{2}} \left (- \frac{21 A c x}{4} + \frac{5 B a}{4}\right )}{105 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.07395, size = 236, normalized size = 0.72 \[ \frac{2 e \left (2 a^{3/2} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (21 A \sqrt{c}-5 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-42 a^{3/2} A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (2 a (21 A+5 B x)+3 c x^2 (7 A+5 B x)\right )\right )}{105 c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(3*c*x^2*(7*A + 5*B*x) + 2*a*(21*A +
5*B*x)) - 42*a^(3/2)*A*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[
Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 2*a^(3/2)*((-5*I)*Sqrt[a]*B + 21*A*Sqr
t[c])*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/
Sqrt[x]], -1]))/(105*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c*Sqrt[e*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.036, size = 333, normalized size = 1. \[{\frac{2}{105\,{c}^{2}x}\sqrt{ex} \left ( 42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-21\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-5\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}+15\,B{c}^{3}{x}^{5}+21\,A{c}^{3}{x}^{4}+25\,aB{c}^{2}{x}^{3}+21\,aA{c}^{2}{x}^{2}+10\,{a}^{2}Bcx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(42*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*
2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*Ellip
ticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c-21*A*((c*x+(-a*c
)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-
x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^
(1/2))*a^2*c-5*B*(-a*c)^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((
-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x
+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2+15*B*c^3*x^5+21*A*c^3*x^4+25
*a*B*c^2*x^3+21*a*A*c^2*x^2+10*a^2*B*c*x)/x/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.53715, size = 95, normalized size = 0.29 \[ \frac{A \sqrt{a} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{B \sqrt{a} \left (e x\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{2} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*sqrt(a)*(e*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**2*exp_polar(I*
pi)/a)/(2*e*gamma(7/4)) + B*sqrt(a)*(e*x)**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (
9/4,), c*x**2*exp_polar(I*pi)/a)/(2*e**2*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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