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

Optimal. Leaf size=442 \[ -\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt{c+d x^2}}-\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}} \]

[Out]

((b*c - a*d)^2*(e*x)^(7/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((77*b^2*c^2 - 70*a*
b*c*d + 5*a^2*d^2)*e*(e*x)^(3/2))/(30*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(7/2
))/(5*d^2*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^2*Sqrt[e
*x]*Sqrt[c + d*x^2])/(10*c*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + ((77*b^2*c^2 - 70*a*
b*c*d + 5*a^2*d^2)*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqr
t[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(10*
c^(3/4)*d^(15/4)*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^(5/
2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*A
rcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(20*c^(3/4)*d^(15/4)*Sqrt[c
+ d*x^2])

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Rubi [A]  time = 0.923458, antiderivative size = 442, 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 \[ -\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt{c+d x^2}}-\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt{c+d x^2}}+\frac{(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac{2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(e*x)^(7/2))/(3*c*d^2*e*(c + d*x^2)^(3/2)) + ((77*b^2*c^2 - 70*a*
b*c*d + 5*a^2*d^2)*e*(e*x)^(3/2))/(30*c*d^3*Sqrt[c + d*x^2]) + (2*b^2*(e*x)^(7/2
))/(5*d^2*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^2*Sqrt[e
*x]*Sqrt[c + d*x^2])/(10*c*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) + ((77*b^2*c^2 - 70*a*
b*c*d + 5*a^2*d^2)*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqr
t[d]*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(10*
c^(3/4)*d^(15/4)*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 70*a*b*c*d + 5*a^2*d^2)*e^(5/
2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*A
rcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(20*c^(3/4)*d^(15/4)*Sqrt[c
+ d*x^2])

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Rubi in Sympy [A]  time = 106.014, size = 413, normalized size = 0.93 \[ \frac{2 b^{2} \left (e x\right )^{\frac{7}{2}}}{5 d^{2} e \sqrt{c + d x^{2}}} + \frac{\left (e x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}}{3 c d^{2} e \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{e \left (e x\right )^{\frac{3}{2}} \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right )}{30 c d^{3} \sqrt{c + d x^{2}}} - \frac{e^{2} \sqrt{e x} \sqrt{c + d x^{2}} \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right )}{10 c d^{\frac{7}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{3}{4}} d^{\frac{15}{4}} \sqrt{c + d x^{2}}} - \frac{e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} - 70 a b c d + 77 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{20 c^{\frac{3}{4}} d^{\frac{15}{4}} \sqrt{c + d x^{2}}} \]

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)**(5/2),x)

[Out]

2*b**2*(e*x)**(7/2)/(5*d**2*e*sqrt(c + d*x**2)) + (e*x)**(7/2)*(a*d - b*c)**2/(3
*c*d**2*e*(c + d*x**2)**(3/2)) + e*(e*x)**(3/2)*(5*a**2*d**2 - 70*a*b*c*d + 77*b
**2*c**2)/(30*c*d**3*sqrt(c + d*x**2)) - e**2*sqrt(e*x)*sqrt(c + d*x**2)*(5*a**2
*d**2 - 70*a*b*c*d + 77*b**2*c**2)/(10*c*d**(7/2)*(sqrt(c) + sqrt(d)*x)) + e**(5
/2)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(5*a**2*d*
*2 - 70*a*b*c*d + 77*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*s
qrt(e))), 1/2)/(10*c**(3/4)*d**(15/4)*sqrt(c + d*x**2)) - e**(5/2)*sqrt((c + d*x
**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(5*a**2*d**2 - 70*a*b*c*d +
77*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(2
0*c**(3/4)*d**(15/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 1.37026, size = 298, normalized size = 0.67 \[ \frac{(e x)^{5/2} \left (-d x^2 \left (-5 a^2 d^2 \left (c+3 d x^2\right )+10 a b c d \left (7 c+9 d x^2\right )+b^2 (-c) \left (77 c^2+99 c d x^2+12 d^2 x^4\right )\right )-\frac{3 \left (c+d x^2\right ) \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{30 c d^4 x^3 \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(5/2)*(-(d*x^2*(-5*a^2*d^2*(c + 3*d*x^2) + 10*a*b*c*d*(7*c + 9*d*x^2) - b
^2*c*(77*c^2 + 99*c*d*x^2 + 12*d^2*x^4))) - (3*(77*b^2*c^2 - 70*a*b*c*d + 5*a^2*
d^2)*(c + d*x^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*Sqrt
[c])/Sqrt[d]]/Sqrt[x]], -1]))/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(30*c*d^4*x^3*(c + d*x
^2)^(3/2))

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Maple [B]  time = 0.059, size = 1191, normalized size = 2.7 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-1/60*(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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d
)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c*d^3-420*((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))*x^2*a*b*c^2*d^2+
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))*x^2*b^2*c^3*d-15*((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
icF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c*d^3+210*((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))*x^2*a*b*c^2*d^2-231*((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))*x^2*b^2*c^3*d-24*x^6*b^2*c*d^
3+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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/
2))^(1/2),1/2*2^(1/2))*a^2*c^2*d^2-420*((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))*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^4-15*((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+210*((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^4-30*x^4*a^2*d^4+180*x^4*a*b*c*d^3-198*x^4*b^2*c^2
*d^2-10*x^2*a^2*c*d^3+140*x^2*a*b*c^2*d^2-154*x^2*b^2*c^3*d)/x*e^2*(e*x)^(1/2)/d
^4/c/(d*x^2+c)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(e*x)^(5/2)/(d*x^2 + c)^(5/2),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)/((d^2*x^4 + 2*c*d
*x^2 + c^2)*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)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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