3.852 \(\int \frac{\sqrt{e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=384 \[ -\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right )}{5 c d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{11/4} \sqrt{c+d x^2}}+\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{11/4} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d^2 e} \]

[Out]

((b*c - a*d)^2*(e*x)^(3/2))/(c*d^2*e*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(3/2)*Sqrt[
c + d*x^2])/(5*d^2*e) - ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*Sqrt[e*x]*Sqrt[c
+ d*x^2])/(5*c*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + ((21*b^2*c^2 - 30*a*b*c*d + 5*a^
2*d^2)*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(3/4)*d^(11
/4)*Sqrt[c + d*x^2]) - ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*Sqrt[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])/(10*c^(3/4)*d^(11/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.801673, antiderivative size = 384, 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 \[ -\frac{\sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right )}{5 c d^{5/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{11/4} \sqrt{c+d x^2}}+\frac{\sqrt{e} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 c^{3/4} d^{11/4} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{3/2} \sqrt{c+d x^2}}{5 d^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[e*x]*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

((b*c - a*d)^2*(e*x)^(3/2))/(c*d^2*e*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(3/2)*Sqrt[
c + d*x^2])/(5*d^2*e) - ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*Sqrt[e*x]*Sqrt[c
+ d*x^2])/(5*c*d^(5/2)*(Sqrt[c] + Sqrt[d]*x)) + ((21*b^2*c^2 - 30*a*b*c*d + 5*a^
2*d^2)*Sqrt[e]*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*E
llipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(5*c^(3/4)*d^(11
/4)*Sqrt[c + d*x^2]) - ((21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*Sqrt[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])/(10*c^(3/4)*d^(11/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 91.0302, size = 357, normalized size = 0.93 \[ \frac{2 b^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{5 d^{2} e} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a d - b c\right )^{2}}{c d^{2} e \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (5 a^{2} d^{2} - 30 a b c d + 21 b^{2} c^{2}\right )}{5 c d^{\frac{5}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} - 30 a b c d + 21 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} d^{\frac{11}{4}} \sqrt{c + d x^{2}}} - \frac{\sqrt{e} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} - 30 a b c d + 21 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{3}{4}} d^{\frac{11}{4}} \sqrt{c + d x^{2}}} \]

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)**(3/2),x)

[Out]

2*b**2*(e*x)**(3/2)*sqrt(c + d*x**2)/(5*d**2*e) + (e*x)**(3/2)*(a*d - b*c)**2/(c
*d**2*e*sqrt(c + d*x**2)) - sqrt(e*x)*sqrt(c + d*x**2)*(5*a**2*d**2 - 30*a*b*c*d
+ 21*b**2*c**2)/(5*c*d**(5/2)*(sqrt(c) + sqrt(d)*x)) + sqrt(e)*sqrt((c + d*x**2
)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(5*a**2*d**2 - 30*a*b*c*d + 21
*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(5*c*
*(3/4)*d**(11/4)*sqrt(c + d*x**2)) - sqrt(e)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d
)*x)**2)*(sqrt(c) + sqrt(d)*x)*(5*a**2*d**2 - 30*a*b*c*d + 21*b**2*c**2)*ellipti
c_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(10*c**(3/4)*d**(11/4)*s
qrt(c + d*x**2))

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Mathematica [C]  time = 0.878989, size = 244, normalized size = 0.64 \[ \frac{e \left (d x^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (5 (b c-a d)^2+2 b^2 c \left (c+d x^2\right )\right )-\left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} \left (F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )\right )\right )}{5 c d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[e*x]*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

(e*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d*x^2*(5*(b*c - a*d)^2 + 2*b^2*c*(c + d*x^2)) - (2
1*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(c + d*x^2) + Sqr
t[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*(-EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/
Sqrt[d]]/Sqrt[x]], -1] + EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]],
-1]))))/(5*c*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3*Sqrt[e*x]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.031, size = 597, normalized size = 1.6 \[ -{\frac{1}{10\,{d}^{3}xc}\sqrt{ex} \left ( 10\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-60\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+30\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-4\,{x}^{4}{b}^{2}c{d}^{2}-10\,{x}^{2}{a}^{2}{d}^{3}+20\,{x}^{2}abc{d}^{2}-14\,{x}^{2}{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

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)^(3/2),x)

[Out]

-1/10*(e*x)^(1/2)*(10*((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^2-60*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2
))^(1/2),1/2*2^(1/2))*b^2*c^3-5*((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+30*((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^3-4*x^4*b^2*c*d^2-10*x^2*a^2*d^3+20*x^2*a*
b*c*d^2-14*x^2*b^2*c^2*d)/(d*x^2+c)^(1/2)/d^3/x/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{e x}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(e*x)/(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(e*x)/(d*x^2 + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{e x}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(e*x)/(d*x^2 + c)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(e*x)/(d*x^2 + c)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(sqrt(e*x)*(a + b*x**2)**2/(c + d*x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{e x}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(e*x)/(d*x^2 + c)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(e*x)/(d*x^2 + c)^(3/2), x)