3.839 \(\int \frac{(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=430 \[ -\frac{c^{5/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{15/4} \sqrt{c+d x^2}}+\frac{2 c^{5/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{15/4} \sqrt{c+d x^2}}-\frac{2 c e^2 \sqrt{e x} \sqrt{c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{195 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 e (e x)^{3/2} \sqrt{c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{585 d^3}-\frac{2 b (e x)^{7/2} \sqrt{c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{11/2} \sqrt{c+d x^2}}{13 d e^3} \]

[Out]

(2*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(585*d
^3) - (2*b*(11*b*c - 26*a*d)*(e*x)^(7/2)*Sqrt[c + d*x^2])/(117*d^2*e) + (2*b^2*(
e*x)^(11/2)*Sqrt[c + d*x^2])/(13*d*e^3) - (2*c*(117*a^2*d^2 + 7*b*c*(11*b*c - 26
*a*d))*e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*c
^(5/4)*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqr
t[(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])/(195*d^(15/4)*Sqrt[c + d*x^2]) - (c^(5/4)*(117*a^2*d^2 +
7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(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
])/(195*d^(15/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.990731, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{c^{5/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{15/4} \sqrt{c+d x^2}}+\frac{2 c^{5/4} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{195 d^{15/4} \sqrt{c+d x^2}}-\frac{2 c e^2 \sqrt{e x} \sqrt{c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{195 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{2 e (e x)^{3/2} \sqrt{c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{585 d^3}-\frac{2 b (e x)^{7/2} \sqrt{c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac{2 b^2 (e x)^{11/2} \sqrt{c+d x^2}}{13 d e^3} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(5/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]

[Out]

(2*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(585*d
^3) - (2*b*(11*b*c - 26*a*d)*(e*x)^(7/2)*Sqrt[c + d*x^2])/(117*d^2*e) + (2*b^2*(
e*x)^(11/2)*Sqrt[c + d*x^2])/(13*d*e^3) - (2*c*(117*a^2*d^2 + 7*b*c*(11*b*c - 26
*a*d))*e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(195*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + (2*c
^(5/4)*(117*a^2*d^2 + 7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqr
t[(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])/(195*d^(15/4)*Sqrt[c + d*x^2]) - (c^(5/4)*(117*a^2*d^2 +
7*b*c*(11*b*c - 26*a*d))*e^(5/2)*(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
])/(195*d^(15/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 92.2903, size = 413, normalized size = 0.96 \[ \frac{2 b^{2} \left (e x\right )^{\frac{11}{2}} \sqrt{c + d x^{2}}}{13 d e^{3}} + \frac{2 b \left (e x\right )^{\frac{7}{2}} \sqrt{c + d x^{2}} \left (26 a d - 11 b c\right )}{117 d^{2} e} + \frac{2 c^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (117 a^{2} d^{2} - 7 b c \left (26 a d - 11 b c\right )\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 d^{\frac{15}{4}} \sqrt{c + d x^{2}}} - \frac{c^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (117 a^{2} d^{2} - 7 b c \left (26 a d - 11 b c\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{195 d^{\frac{15}{4}} \sqrt{c + d x^{2}}} - \frac{2 c e^{2} \sqrt{e x} \sqrt{c + d x^{2}} \left (117 a^{2} d^{2} - 7 b c \left (26 a d - 11 b c\right )\right )}{195 d^{\frac{7}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{2 e \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (117 a^{2} d^{2} - 7 b c \left (26 a d - 11 b c\right )\right )}{585 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(1/2),x)

[Out]

2*b**2*(e*x)**(11/2)*sqrt(c + d*x**2)/(13*d*e**3) + 2*b*(e*x)**(7/2)*sqrt(c + d*
x**2)*(26*a*d - 11*b*c)/(117*d**2*e) + 2*c**(5/4)*e**(5/2)*sqrt((c + d*x**2)/(sq
rt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(117*a**2*d**2 - 7*b*c*(26*a*d - 11
*b*c))*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(195*d**(1
5/4)*sqrt(c + d*x**2)) - c**(5/4)*e**(5/2)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*
x)**2)*(sqrt(c) + sqrt(d)*x)*(117*a**2*d**2 - 7*b*c*(26*a*d - 11*b*c))*elliptic_
f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(195*d**(15/4)*sqrt(c + d*
x**2)) - 2*c*e**2*sqrt(e*x)*sqrt(c + d*x**2)*(117*a**2*d**2 - 7*b*c*(26*a*d - 11
*b*c))/(195*d**(7/2)*(sqrt(c) + sqrt(d)*x)) + 2*e*(e*x)**(3/2)*sqrt(c + d*x**2)*
(117*a**2*d**2 - 7*b*c*(26*a*d - 11*b*c))/(585*d**3)

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Mathematica [C]  time = 1.74581, size = 237, normalized size = 0.55 \[ \frac{2 (e x)^{5/2} \left (d \sqrt{x} \left (c+d x^2\right ) \left (117 a^2 d^2+26 a b d \left (5 d x^2-7 c\right )+b^2 \left (77 c^2-55 c d x^2+45 d^2 x^4\right )\right )+3 c \left (117 a^2 d^2-182 a b c d+77 b^2 c^2\right ) \left (-\sqrt{x} \left (\frac{c}{x^2}+d\right )+\frac{i c \sqrt{\frac{c}{d x^2}+1} \left (E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{d}}\right )^{3/2}}\right )\right )}{585 d^4 x^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(5/2)*(a + b*x^2)^2)/Sqrt[c + d*x^2],x]

[Out]

(2*(e*x)^(5/2)*(d*Sqrt[x]*(c + d*x^2)*(117*a^2*d^2 + 26*a*b*d*(-7*c + 5*d*x^2) +
b^2*(77*c^2 - 55*c*d*x^2 + 45*d^2*x^4)) + 3*c*(77*b^2*c^2 - 182*a*b*c*d + 117*a
^2*d^2)*(-((d + c/x^2)*Sqrt[x]) + (I*c*Sqrt[1 + c/(d*x^2)]*(EllipticE[I*ArcSinh[
Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/S
qrt[d]]/Sqrt[x]], -1]))/((I*Sqrt[c])/Sqrt[d])^(3/2))))/(585*d^4*x^(3/2)*Sqrt[c +
d*x^2])

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Maple [A]  time = 0.044, size = 661, normalized size = 1.5 \[ -{\frac{{e}^{2}}{585\,x{d}^{4}}\sqrt{ex} \left ( -90\,{x}^{8}{b}^{2}{d}^{4}-260\,{x}^{6}ab{d}^{4}+20\,{x}^{6}{b}^{2}c{d}^{3}+702\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}-1092\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d+462\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}-351\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{c}^{2}{d}^{2}+546\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{3}d-231\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{4}-234\,{x}^{4}{a}^{2}{d}^{4}+104\,{x}^{4}abc{d}^{3}-44\,{x}^{4}{b}^{2}{c}^{2}{d}^{2}-234\,{x}^{2}{a}^{2}c{d}^{3}+364\,{x}^{2}ab{c}^{2}{d}^{2}-154\,{x}^{2}{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(b*x^2+a)^2/(d*x^2+c)^(1/2),x)

[Out]

-1/585/x*e^2*(e*x)^(1/2)/(d*x^2+c)^(1/2)/d^4*(-90*x^8*b^2*d^4-260*x^6*a*b*d^4+20
*x^6*b^2*c*d^3+702*((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^2*d^2-1092*((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^3*
d+462*((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^4-351*((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^2*d^2+546*((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^3*d-231*((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^4-234*x^4*a^2*d^4+104*x^4*a*b*c*d^
3-44*x^4*b^2*c^2*d^2-234*x^2*a^2*c*d^3+364*x^2*a*b*c^2*d^2-154*x^2*b^2*c^3*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(e*x)^(5/2)/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(5/2)/sqrt(d*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}\right )} \sqrt{e x}}{\sqrt{d x^{2} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(e*x)^(5/2)/sqrt(d*x^2 + c),x, algorithm="fricas")

[Out]

integral((b^2*e^2*x^6 + 2*a*b*e^2*x^4 + a^2*e^2*x^2)*sqrt(e*x)/sqrt(d*x^2 + c),
x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(5/2)*(b*x**2+a)**2/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}{\sqrt{d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(e*x)^(5/2)/sqrt(d*x^2 + c),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(5/2)/sqrt(d*x^2 + c), x)