3.864 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{7/2} \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=489 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{15/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{15/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right )}{10 c^4 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right )}{10 c^4 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{(e x)^{3/2} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right )}{15 c^3 e^5 \left (c+d x^2\right )^{3/2}}-\frac{2 a^2}{5 c e (e x)^{5/2} \left (c+d x^2\right )^{3/2}}-\frac{2 a (10 b c-11 a d)}{5 c^2 e^3 \sqrt{e x} \left (c+d x^2\right )^{3/2}} \]

[Out]

(-2*a^2)/(5*c*e*(e*x)^(5/2)*(c + d*x^2)^(3/2)) - (2*a*(10*b*c - 11*a*d))/(5*c^2*
e^3*Sqrt[e*x]*(c + d*x^2)^(3/2)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(e*x)^
(3/2))/(15*c^3*e^5*(c + d*x^2)^(3/2)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(
e*x)^(3/2))/(10*c^4*e^5*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2
)*Sqrt[e*x]*Sqrt[c + d*x^2])/(10*c^4*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + ((5*b^
2*c^2 - 70*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]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2]
)/(10*c^(15/4)*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 70*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]*Ellipti
cF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(20*c^(15/4)*d^(3/4)*e
^(7/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 1.08812, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{20 c^{15/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 c^{15/4} d^{3/4} e^{7/2} \sqrt{c+d x^2}}+\frac{(e x)^{3/2} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right )}{10 c^4 e^5 \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right )}{10 c^4 \sqrt{d} e^4 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{(e x)^{3/2} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right )}{15 c^3 e^5 \left (c+d x^2\right )^{3/2}}-\frac{2 a^2}{5 c e (e x)^{5/2} \left (c+d x^2\right )^{3/2}}-\frac{2 a (10 b c-11 a d)}{5 c^2 e^3 \sqrt{e x} \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(5/2)),x]

[Out]

(-2*a^2)/(5*c*e*(e*x)^(5/2)*(c + d*x^2)^(3/2)) - (2*a*(10*b*c - 11*a*d))/(5*c^2*
e^3*Sqrt[e*x]*(c + d*x^2)^(3/2)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(e*x)^
(3/2))/(15*c^3*e^5*(c + d*x^2)^(3/2)) + ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*(
e*x)^(3/2))/(10*c^4*e^5*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2
)*Sqrt[e*x]*Sqrt[c + d*x^2])/(10*c^4*Sqrt[d]*e^4*(Sqrt[c] + Sqrt[d]*x)) + ((5*b^
2*c^2 - 70*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]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2]
)/(10*c^(15/4)*d^(3/4)*e^(7/2)*Sqrt[c + d*x^2]) - ((5*b^2*c^2 - 70*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]*Ellipti
cF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(20*c^(15/4)*d^(3/4)*e
^(7/2)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 110.417, size = 461, normalized size = 0.94 \[ - \frac{2 a^{2}}{5 c e \left (e x\right )^{\frac{5}{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{2 a \left (11 a d - 10 b c\right )}{5 c^{2} e^{3} \sqrt{e x} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (7 a d \left (11 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{15 c^{3} e^{5} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (7 a d \left (11 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{10 c^{4} e^{5} \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (7 a d \left (11 a d - 10 b c\right ) + 5 b^{2} c^{2}\right )}{10 c^{4} \sqrt{d} e^{4} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (7 a d \left (11 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 c^{\frac{15}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (7 a d \left (11 a d - 10 b c\right ) + 5 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{20 c^{\frac{15}{4}} d^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(5/2),x)

[Out]

-2*a**2/(5*c*e*(e*x)**(5/2)*(c + d*x**2)**(3/2)) + 2*a*(11*a*d - 10*b*c)/(5*c**2
*e**3*sqrt(e*x)*(c + d*x**2)**(3/2)) + (e*x)**(3/2)*(7*a*d*(11*a*d - 10*b*c) + 5
*b**2*c**2)/(15*c**3*e**5*(c + d*x**2)**(3/2)) + (e*x)**(3/2)*(7*a*d*(11*a*d - 1
0*b*c) + 5*b**2*c**2)/(10*c**4*e**5*sqrt(c + d*x**2)) - sqrt(e*x)*sqrt(c + d*x**
2)*(7*a*d*(11*a*d - 10*b*c) + 5*b**2*c**2)/(10*c**4*sqrt(d)*e**4*(sqrt(c) + sqrt
(d)*x)) + sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(7*a
*d*(11*a*d - 10*b*c) + 5*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/
4)*sqrt(e))), 1/2)/(10*c**(15/4)*d**(3/4)*e**(7/2)*sqrt(c + d*x**2)) - sqrt((c +
d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(7*a*d*(11*a*d - 10*b*c
) + 5*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/
(20*c**(15/4)*d**(3/4)*e**(7/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.824256, size = 246, normalized size = 0.5 \[ \frac{x \left (\frac{a^2 \left (-12 c^3+132 c^2 d x^2+385 c d^2 x^4+231 d^3 x^6\right )-10 a b c x^2 \left (12 c^2+35 c d x^2+21 d^2 x^4\right )+5 b^2 c^2 x^4 \left (5 c+3 d x^2\right )}{c+d x^2}+\frac{3 i c x^2 \sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}} \sqrt{\frac{d x^2}{c}+1} \left (77 a^2 d^2-70 a b c d+5 b^2 c^2\right ) \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )\right )}{d}\right )}{30 c^4 (e x)^{7/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/((e*x)^(7/2)*(c + d*x^2)^(5/2)),x]

[Out]

(x*((5*b^2*c^2*x^4*(5*c + 3*d*x^2) - 10*a*b*c*x^2*(12*c^2 + 35*c*d*x^2 + 21*d^2*
x^4) + a^2*(-12*c^3 + 132*c^2*d*x^2 + 385*c*d^2*x^4 + 231*d^3*x^6))/(c + d*x^2)
+ ((3*I)*c*(5*b^2*c^2 - 70*a*b*c*d + 77*a^2*d^2)*x^2*Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]
*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1] - El
lipticF[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1]))/d))/(30*c^4*(e*x)^(7/2)*Sq
rt[c + d*x^2])

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Maple [B]  time = 0.04, size = 1231, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(e*x)^(7/2)/(d*x^2+c)^(5/2),x)

[Out]

-1/60*(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^4*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^4*a*b*c^2*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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2
))^(1/2),1/2*2^(1/2))*x^4*b^2*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)*Ellip
ticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^4*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^4*a*b*c^2*d^2-15*((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))*x^4*b^2*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))*x^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)*EllipticE(((d*x+(-
c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b*c^3*d+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*b^2*c^4-231*((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*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))*x^2*a*b*
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)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)
^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^2*c^4-462*x^6*a^2*d^4+420*x^6*a*b*c*d^3-30*x^6*
b^2*c^2*d^2-770*x^4*a^2*c*d^3+700*x^4*a*b*c^2*d^2-50*x^4*b^2*c^3*d-264*x^2*a^2*c
^2*d^2+240*x^2*a*b*c^3*d+24*a^2*c^3*d)/x^2/d/c^4/e^3/(e*x)^(1/2)/(d*x^2+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(7/2)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}{{\left (d^{2} e^{3} x^{7} + 2 \, c d e^{3} x^{5} + c^{2} e^{3} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/((d^2*e^3*x^7 + 2*c*d*e^3*x^5 + c^2*e^3*x^3
)*sqrt(d*x^2 + c)*sqrt(e*x)), 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((b*x**2+a)**2/(e*x)**(7/2)/(d*x**2+c)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(7/2)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(7/2)), x)