3.787 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{\sqrt{e x}} \, dx\)

Optimal. Leaf size=176 \[ \frac{2 a^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (7 A b-a B)}{21 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{3/2}}{7 b e} \]

[Out]

(2*(7*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b*e) + (2*B*Sqrt[e*x]*(a + b*x^2
)^(3/2))/(7*b*e) + (2*a^(3/4)*(7*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^
2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt
[e])], 1/2])/(21*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.292616, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 a^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (7 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{e} \sqrt{a+b x^2}}+\frac{2 \sqrt{e x} \sqrt{a+b x^2} (7 A b-a B)}{21 b e}+\frac{2 B \sqrt{e x} \left (a+b x^2\right )^{3/2}}{7 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(7*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b*e) + (2*B*Sqrt[e*x]*(a + b*x^2
)^(3/2))/(7*b*e) + (2*a^(3/4)*(7*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^
2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt
[e])], 1/2])/(21*b^(5/4)*Sqrt[e]*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 27.9703, size = 158, normalized size = 0.9 \[ \frac{2 B \sqrt{e x} \left (a + b x^{2}\right )^{\frac{3}{2}}}{7 b e} + \frac{2 a^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (7 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{5}{4}} \sqrt{e} \sqrt{a + b x^{2}}} + \frac{2 \sqrt{e x} \sqrt{a + b x^{2}} \left (7 A b - B a\right )}{21 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(e*x)*(a + b*x**2)**(3/2)/(7*b*e) + 2*a**(3/4)*sqrt((a + b*x**2)/(sqrt(a
) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(7*A*b - B*a)*elliptic_f(2*atan(b**(1/4
)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(21*b**(5/4)*sqrt(e)*sqrt(a + b*x**2)) + 2
*sqrt(e*x)*sqrt(a + b*x**2)*(7*A*b - B*a)/(21*b*e)

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Mathematica [C]  time = 0.360139, size = 132, normalized size = 0.75 \[ \frac{2 x \left (\left (a+b x^2\right ) \left (2 a B+7 A b+3 b B x^2\right )-\frac{2 i a \sqrt{x} \sqrt{\frac{a}{b x^2}+1} (a B-7 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{21 b \sqrt{e x} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*((a + b*x^2)*(7*A*b + 2*a*B + 3*b*B*x^2) - ((2*I)*a*(-7*A*b + a*B)*Sqrt[1 +
 a/(b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])
/Sqrt[(I*Sqrt[a])/Sqrt[b]]))/(21*b*Sqrt[e*x]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.037, size = 246, normalized size = 1.4 \[{\frac{2}{21\,{b}^{2}} \left ( 7\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}ab-B\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}{a}^{2}+3\,B{x}^{5}{b}^{3}+7\,A{x}^{3}{b}^{3}+5\,B{x}^{3}a{b}^{2}+7\,Axa{b}^{2}+2\,Bx{a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21/(b*x^2+a)^(1/2)*(7*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x
+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a
*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a*b-B*((b*x+(-a*b)^(1/2
))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-
a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))
*(-a*b)^(1/2)*a^2+3*B*x^5*b^3+7*A*x^3*b^3+5*B*x^3*a*b^2+7*A*x*a*b^2+2*B*x*a^2*b)
/(e*x)^(1/2)/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.7972, size = 97, normalized size = 0.55 \[ \frac{A \sqrt{a} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{B \sqrt{a} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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