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

Optimal. Leaf size=375 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (15 a^2 d^2+b c (7 b c-18 a d)\right )}{15 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\sqrt [4]{c} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{11/4} \sqrt{c+d x^2}}-\frac{2 \sqrt [4]{c} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{11/4} \sqrt{c+d x^2}}-\frac{2 b (e x)^{3/2} \sqrt{c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac{2 b^2 (e x)^{7/2} \sqrt{c+d x^2}}{9 d e^3} \]

[Out]

(-2*b*(7*b*c - 18*a*d)*(e*x)^(3/2)*Sqrt[c + d*x^2])/(45*d^2*e) + (2*b^2*(e*x)^(7
/2)*Sqrt[c + d*x^2])/(9*d*e^3) + (2*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqrt[e*x
]*Sqrt[c + d*x^2])/(15*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) - (2*c^(1/4)*(15*a^2*d^2 +
b*c*(7*b*c - 18*a*d))*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] +
Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/
(15*d^(11/4)*Sqrt[c + d*x^2]) + (c^(1/4)*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqr
t[e]*(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])/(15*d^(11/4)*Sqrt[c + d*x^
2])

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Rubi [A]  time = 0.863554, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (15 a^2 d^2+b c (7 b c-18 a d)\right )}{15 d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{\sqrt [4]{c} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{11/4} \sqrt{c+d x^2}}-\frac{2 \sqrt [4]{c} \sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (15 a^2 d^2+b c (7 b c-18 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{11/4} \sqrt{c+d x^2}}-\frac{2 b (e x)^{3/2} \sqrt{c+d x^2} (7 b c-18 a d)}{45 d^2 e}+\frac{2 b^2 (e x)^{7/2} \sqrt{c+d x^2}}{9 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*(7*b*c - 18*a*d)*(e*x)^(3/2)*Sqrt[c + d*x^2])/(45*d^2*e) + (2*b^2*(e*x)^(7
/2)*Sqrt[c + d*x^2])/(9*d*e^3) + (2*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqrt[e*x
]*Sqrt[c + d*x^2])/(15*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) - (2*c^(1/4)*(15*a^2*d^2 +
b*c*(7*b*c - 18*a*d))*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] +
Sqrt[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/
(15*d^(11/4)*Sqrt[c + d*x^2]) + (c^(1/4)*(15*a^2*d^2 + b*c*(7*b*c - 18*a*d))*Sqr
t[e]*(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])/(15*d^(11/4)*Sqrt[c + d*x^
2])

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Rubi in Sympy [A]  time = 78.5934, size = 354, normalized size = 0.94 \[ \frac{2 b^{2} \left (e x\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}{9 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (18 a d - 7 b c\right )}{45 d^{2} e} - \frac{2 \sqrt [4]{c} \sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (15 a^{2} d^{2} - b c \left (18 a d - 7 b c\right )\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{\sqrt [4]{c} \sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (15 a^{2} d^{2} - b c \left (18 a d - 7 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{2 \sqrt{e x} \sqrt{c + d x^{2}} \left (15 a^{2} d^{2} - b c \left (18 a d - 7 b c\right )\right )}{15 d^{\frac{5}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(7/2)*sqrt(c + d*x**2)/(9*d*e**3) + 2*b*(e*x)**(3/2)*sqrt(c + d*x*
*2)*(18*a*d - 7*b*c)/(45*d**2*e) - 2*c**(1/4)*sqrt(e)*sqrt((c + d*x**2)/(sqrt(c)
+ sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(15*a**2*d**2 - b*c*(18*a*d - 7*b*c))*el
liptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(15*d**(11/4)*sqrt(
c + d*x**2)) + c**(1/4)*sqrt(e)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqr
t(c) + sqrt(d)*x)*(15*a**2*d**2 - b*c*(18*a*d - 7*b*c))*elliptic_f(2*atan(d**(1/
4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(15*d**(11/4)*sqrt(c + d*x**2)) + 2*sqrt(
e*x)*sqrt(c + d*x**2)*(15*a**2*d**2 - b*c*(18*a*d - 7*b*c))/(15*d**(5/2)*(sqrt(c
) + sqrt(d)*x))

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Mathematica [C]  time = 1.0749, size = 249, normalized size = 0.66 \[ \frac{2 e \left (b d x^2 \left (c+d x^2\right ) \left (18 a d-7 b c+5 b d x^2\right )+\frac{3 \left (15 a^2 d^2-18 a b c d+7 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{45 d^3 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(b*d*x^2*(c + d*x^2)*(-7*b*c + 18*a*d + 5*b*d*x^2) + (3*(7*b^2*c^2 - 18*a*b
*c*d + 15*a^2*d^2)*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(c + d*x^2) - Sqrt[c]*Sqrt[d]*Sqrt
[1 + c/(d*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]],
-1] + Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sq
rt[c])/Sqrt[d]]/Sqrt[x]], -1]))/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(45*d^3*Sqrt[e*x]*Sq
rt[c + d*x^2])

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Maple [A]  time = 0.027, size = 604, normalized size = 1.6 \[{\frac{1}{45\,{d}^{3}x}\sqrt{ex} \left ( 10\,{x}^{6}{b}^{2}{d}^{3}+90\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-108\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-45\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+54\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+36\,{x}^{4}ab{d}^{3}-4\,{x}^{4}{b}^{2}c{d}^{2}+36\,{x}^{2}abc{d}^{2}-14\,{x}^{2}{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(e*x)^(1/2)/(d*x^2+c)^(1/2)/d^3*(10*x^6*b^2*d^3+90*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c*
d^2-108*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^
(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*d+42*((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)*Ellipt
icE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3-45*((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+54*((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-21*((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+36*x
^4*a*b*d^3-4*x^4*b^2*c*d^2+36*x^2*a*b*c*d^2-14*x^2*b^2*c^2*d)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.9715, size = 143, normalized size = 0.38 \[ \frac{a^{2} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e \Gamma \left (\frac{7}{4}\right )} + \frac{a b \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt{c} e^{3} \Gamma \left (\frac{11}{4}\right )} + \frac{b^{2} \left (e x\right )^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt{c} e^{5} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(e*x)**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), d*x**2*exp_polar(I*pi)/c)
/(2*sqrt(c)*e*gamma(7/4)) + a*b*(e*x)**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,
), d*x**2*exp_polar(I*pi)/c)/(sqrt(c)*e**3*gamma(11/4)) + b**2*(e*x)**(11/2)*gam
ma(11/4)*hyper((1/2, 11/4), (15/4,), d*x**2*exp_polar(I*pi)/c)/(2*sqrt(c)*e**5*g
amma(15/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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