3.824 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{\sqrt{e x}} \, dx\)

Optimal. Leaf size=244 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right )}{231 d^2 e}+\frac{2 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3} \]

[Out]

(2*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2*d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^2*e)
- (2*b*(5*b*c - 22*a*d)*Sqrt[e*x]*(c + d*x^2)^(3/2))/(77*d^2*e) + (2*b^2*(e*x)^(
5/2)*(c + d*x^2)^(3/2))/(11*d*e^3) + (2*c^(3/4)*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2
*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[
2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*d^(9/4)*Sqrt[e]*Sqrt
[c + d*x^2])

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Rubi [A]  time = 0.513392, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right )}{231 d^2 e}+\frac{2 c^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{2 b \sqrt{e x} \left (c+d x^2\right )^{3/2} (5 b c-22 a d)}{77 d^2 e}+\frac{2 b^2 (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{11 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2*d^2)*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^2*e)
- (2*b*(5*b*c - 22*a*d)*Sqrt[e*x]*(c + d*x^2)^(3/2))/(77*d^2*e) + (2*b^2*(e*x)^(
5/2)*(c + d*x^2)^(3/2))/(11*d*e^3) + (2*c^(3/4)*(5*b^2*c^2 - 22*a*b*c*d + 77*a^2
*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[
2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(231*d^(9/4)*Sqrt[e]*Sqrt
[c + d*x^2])

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Rubi in Sympy [A]  time = 48.9388, size = 228, normalized size = 0.93 \[ \frac{2 b^{2} \left (e x\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}{11 d e^{3}} + \frac{2 b \sqrt{e x} \left (c + d x^{2}\right )^{\frac{3}{2}} \left (22 a d - 5 b c\right )}{77 d^{2} e} + \frac{2 c^{\frac{3}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (77 a^{2} d^{2} - b c \left (22 a d - 5 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 d^{\frac{9}{4}} \sqrt{e} \sqrt{c + d x^{2}}} + \frac{2 \sqrt{e x} \sqrt{c + d x^{2}} \left (77 a^{2} d^{2} - b c \left (22 a d - 5 b c\right )\right )}{231 d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(5/2)*(c + d*x**2)**(3/2)/(11*d*e**3) + 2*b*sqrt(e*x)*(c + d*x**2)
**(3/2)*(22*a*d - 5*b*c)/(77*d**2*e) + 2*c**(3/4)*sqrt((c + d*x**2)/(sqrt(c) + s
qrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(77*a**2*d**2 - b*c*(22*a*d - 5*b*c))*ellipt
ic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(231*d**(9/4)*sqrt(e)*s
qrt(c + d*x**2)) + 2*sqrt(e*x)*sqrt(c + d*x**2)*(77*a**2*d**2 - b*c*(22*a*d - 5*
b*c))/(231*d**2*e)

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Mathematica [C]  time = 0.315699, size = 189, normalized size = 0.77 \[ \frac{\sqrt{x} \left (\frac{2 \sqrt{x} \left (c+d x^2\right ) \left (77 a^2 d^2+22 a b d \left (2 c+3 d x^2\right )+b^2 \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )\right )}{d^2}+\frac{4 i c x \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2-22 a b c d+5 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{231 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*((2*Sqrt[x]*(c + d*x^2)*(77*a^2*d^2 + 22*a*b*d*(2*c + 3*d*x^2) + b^2*(-
10*c^2 + 6*c*d*x^2 + 21*d^2*x^4)))/d^2 + ((4*I)*c*(5*b^2*c^2 - 22*a*b*c*d + 77*a
^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt
[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^2)))/(231*Sqrt[e*x]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.041, size = 401, normalized size = 1.6 \[{\frac{2}{231\,{d}^{3}} \left ( 21\,{x}^{7}{b}^{2}{d}^{4}+77\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-22\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+5\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+66\,{x}^{5}ab{d}^{4}+27\,{x}^{5}{b}^{2}c{d}^{3}+77\,{x}^{3}{a}^{2}{d}^{4}+110\,{x}^{3}abc{d}^{3}-4\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}+77\,x{a}^{2}c{d}^{3}+44\,xab{c}^{2}{d}^{2}-10\,x{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231/(d*x^2+c)^(1/2)*(21*x^7*b^2*d^4+77*(-c*d)^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)
^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*
d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c*d^
2-22*(-c*d)^(1/2)*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^
(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2
))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*d+5*(-c*d)^(1/2)*((d*x+(-c*d)^(1/2))
/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)
^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b
^2*c^3+66*x^5*a*b*d^4+27*x^5*b^2*c*d^3+77*x^3*a^2*d^4+110*x^3*a*b*c*d^3-4*x^3*b^
2*c^2*d^2+77*x*a^2*c*d^3+44*x*a*b*c^2*d^2-10*x*b^2*c^3*d)/(e*x)^(1/2)/d^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/sqrt(e*x), x)

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Sympy [A]  time = 26.9265, size = 150, normalized size = 0.61 \[ \frac{a^{2} \sqrt{c} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{5}{4}\right )} + \frac{a b \sqrt{c} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{e} \Gamma \left (\frac{9}{4}\right )} + \frac{b^{2} \sqrt{c} x^{\frac{9}{2}} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{e} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), d*x**2*exp_polar(I*pi
)/c)/(2*sqrt(e)*gamma(5/4)) + a*b*sqrt(c)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4),
(9/4,), d*x**2*exp_polar(I*pi)/c)/(sqrt(e)*gamma(9/4)) + b**2*sqrt(c)*x**(9/2)*
gamma(9/4)*hyper((-1/2, 9/4), (13/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(e)*gamm
a(13/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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