3.862 \(\int \frac{\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=442 \[ -\frac{(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt{c+d x^2}} \]

[Out]

(-2*a^2)/(c*e*Sqrt[e*x]*(c + d*x^2)^(3/2)) - ((b^2*c^2 - 2*a*b*c*d + 7*a^2*d^2)*
(e*x)^(3/2))/(3*c^2*d*e^3*(c + d*x^2)^(3/2)) + ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*
(e*x)^(3/2))/(2*c^3*d*e^3*Sqrt[c + d*x^2]) - ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*Sq
rt[e*x]*Sqrt[c + d*x^2])/(2*c^3*d^(3/2)*e^2*(Sqrt[c] + Sqrt[d]*x)) + ((b^2*c^2 +
a*d*(2*b*c - 7*a*d))*(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])/(2*c^(11/
4)*d^(7/4)*e^(3/2)*Sqrt[c + d*x^2]) - ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*(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])/(4*c^(11/4)*d^(7/4)*e^(3/2)*Sqrt[c + d*x
^2])

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Rubi [A]  time = 1.01156, 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 \[ -\frac{(e x)^{3/2} \left (7 a^2 d^2-2 a b c d+b^2 c^2\right )}{3 c^2 d e^3 \left (c+d x^2\right )^{3/2}}-\frac{2 a^2}{c e \sqrt{e x} \left (c+d x^2\right )^{3/2}}-\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (2 b c-7 a d)+b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 c^{11/4} d^{7/4} e^{3/2} \sqrt{c+d x^2}}-\frac{\sqrt{e x} \sqrt{c+d x^2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d^{3/2} e^2 \left (\sqrt{c}+\sqrt{d} x\right )}+\frac{(e x)^{3/2} \left (a d (2 b c-7 a d)+b^2 c^2\right )}{2 c^3 d e^3 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(5/2)),x]

[Out]

(-2*a^2)/(c*e*Sqrt[e*x]*(c + d*x^2)^(3/2)) - ((b^2*c^2 - 2*a*b*c*d + 7*a^2*d^2)*
(e*x)^(3/2))/(3*c^2*d*e^3*(c + d*x^2)^(3/2)) + ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*
(e*x)^(3/2))/(2*c^3*d*e^3*Sqrt[c + d*x^2]) - ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*Sq
rt[e*x]*Sqrt[c + d*x^2])/(2*c^3*d^(3/2)*e^2*(Sqrt[c] + Sqrt[d]*x)) + ((b^2*c^2 +
a*d*(2*b*c - 7*a*d))*(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])/(2*c^(11/
4)*d^(7/4)*e^(3/2)*Sqrt[c + d*x^2]) - ((b^2*c^2 + a*d*(2*b*c - 7*a*d))*(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])/(4*c^(11/4)*d^(7/4)*e^(3/2)*Sqrt[c + d*x
^2])

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Rubi in Sympy [A]  time = 100.651, size = 406, normalized size = 0.92 \[ - \frac{2 a^{2}}{c e \sqrt{e x} \left (c + d x^{2}\right )^{\frac{3}{2}}} - \frac{\left (e x\right )^{\frac{3}{2}} \left (a d \left (7 a d - 2 b c\right ) + b^{2} c^{2}\right )}{3 c^{2} d e^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (- a d \left (7 a d - 2 b c\right ) + b^{2} c^{2}\right )}{2 c^{3} d e^{3} \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \sqrt{c + d x^{2}} \left (- a d \left (7 a d - 2 b c\right ) + b^{2} c^{2}\right )}{2 c^{3} d^{\frac{3}{2}} e^{2} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (7 a d - 2 b c\right ) + b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 c^{\frac{11}{4}} d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (7 a d - 2 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4 c^{\frac{11}{4}} d^{\frac{7}{4}} e^{\frac{3}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(e*x)**(3/2)/(d*x**2+c)**(5/2),x)

[Out]

-2*a**2/(c*e*sqrt(e*x)*(c + d*x**2)**(3/2)) - (e*x)**(3/2)*(a*d*(7*a*d - 2*b*c)
+ b**2*c**2)/(3*c**2*d*e**3*(c + d*x**2)**(3/2)) + (e*x)**(3/2)*(-a*d*(7*a*d - 2
*b*c) + b**2*c**2)/(2*c**3*d*e**3*sqrt(c + d*x**2)) - sqrt(e*x)*sqrt(c + d*x**2)
*(-a*d*(7*a*d - 2*b*c) + b**2*c**2)/(2*c**3*d**(3/2)*e**2*(sqrt(c) + sqrt(d)*x))
+ sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-a*d*(7*a*
d - 2*b*c) + b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e)))
, 1/2)/(2*c**(11/4)*d**(7/4)*e**(3/2)*sqrt(c + d*x**2)) - sqrt((c + d*x**2)/(sqr
t(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-a*d*(7*a*d - 2*b*c) + b**2*c**2)*e
lliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(4*c**(11/4)*d**(7
/4)*e**(3/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.798618, size = 222, normalized size = 0.5 \[ \frac{x \left (\frac{a^2 (-d) \left (12 c^2+35 c d x^2+21 d^2 x^4\right )+2 a b c d x^2 \left (5 c+3 d x^2\right )+b^2 c^2 x^2 \left (c+3 d x^2\right )}{c+d x^2}-\frac{3 i x^2 \sqrt{\frac{d x^2}{c}+1} \left (-7 a^2 d^2+2 a b c d+b^2 c^2\right ) \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{d} x}{\sqrt{c}}\right )^{3/2}}\right )}{6 c^3 d (e x)^{3/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/((e*x)^(3/2)*(c + d*x^2)^(5/2)),x]

[Out]

(x*((b^2*c^2*x^2*(c + 3*d*x^2) + 2*a*b*c*d*x^2*(5*c + 3*d*x^2) - a^2*d*(12*c^2 +
35*c*d*x^2 + 21*d^2*x^4))/(c + d*x^2) - ((3*I)*(b^2*c^2 + 2*a*b*c*d - 7*a^2*d^2
)*x^2*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1]
- EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1]))/((I*Sqrt[d]*x)/Sqrt[c
])^(3/2)))/(6*c^3*d*(e*x)^(3/2)*Sqrt[c + d*x^2])

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Maple [B]  time = 0.039, size = 1187, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(e*x)^(3/2)/(d*x^2+c)^(5/2),x)

[Out]

1/12*(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))*x^2*a^2*c*d^3-12*((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)*E
llipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b*c^2*d^2-6*
((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-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))*x^2*a^2*c*d^3+6*((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+3*((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+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))*a^
2*c^2*d^2-12*((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-6*((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)*El
lipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^4-21*((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))*a^2*c^2*d^2+6*((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+3*((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-
42*x^4*a^2*d^4+12*x^4*a*b*c*d^3+6*x^4*b^2*c^2*d^2-70*x^2*a^2*c*d^3+20*x^2*a*b*c^
2*d^2+2*x^2*b^2*c^3*d-24*a^2*c^2*d^2)/d^2/c^3/e/(e*x)^(1/2)/(d*x^2+c)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

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)^(3/2)),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(3/2)), x)

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

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)^(3/2)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

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)^(3/2)),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2/((d*x^2 + c)^(5/2)*(e*x)^(3/2)), x)