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

Optimal. Leaf size=207 \[ -\frac{\sqrt{e x} \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{3 c^2 d e^3 \sqrt{c+d x^2}}-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-5 a d)+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 c^{9/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}} \]

[Out]

(-2*a^2)/(3*c*e*(e*x)^(3/2)*Sqrt[c + d*x^2]) - ((3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d
^2)*Sqrt[e*x])/(3*c^2*d*e^3*Sqrt[c + d*x^2]) + ((3*b^2*c^2 + a*d*(6*b*c - 5*a*d)
)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(6*c^(9/4)*d^(5/4)*e^(5/2)*Sq
rt[c + d*x^2])

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Rubi [A]  time = 0.472997, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\sqrt{e x} \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{3 c^2 d e^3 \sqrt{c+d x^2}}-\frac{2 a^2}{3 c e (e x)^{3/2} \sqrt{c+d x^2}}+\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a d (6 b c-5 a d)+3 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 c^{9/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2)/(3*c*e*(e*x)^(3/2)*Sqrt[c + d*x^2]) - ((3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d
^2)*Sqrt[e*x])/(3*c^2*d*e^3*Sqrt[c + d*x^2]) + ((3*b^2*c^2 + a*d*(6*b*c - 5*a*d)
)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*Ar
cTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(6*c^(9/4)*d^(5/4)*e^(5/2)*Sq
rt[c + d*x^2])

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Rubi in Sympy [A]  time = 45.4871, size = 189, normalized size = 0.91 \[ - \frac{2 a^{2}}{3 c e \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}} - \frac{\sqrt{e x} \left (a d \left (5 a d - 6 b c\right ) + 3 b^{2} c^{2}\right )}{3 c^{2} d e^{3} \sqrt{c + d x^{2}}} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- a d \left (5 a d - 6 b c\right ) + 3 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{6 c^{\frac{9}{4}} d^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/(3*c*e*(e*x)**(3/2)*sqrt(c + d*x**2)) - sqrt(e*x)*(a*d*(5*a*d - 6*b*c) +
3*b**2*c**2)/(3*c**2*d*e**3*sqrt(c + d*x**2)) + sqrt((c + d*x**2)/(sqrt(c) + sq
rt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(-a*d*(5*a*d - 6*b*c) + 3*b**2*c**2)*elliptic
_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(6*c**(9/4)*d**(5/4)*e**(
5/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.297353, size = 181, normalized size = 0.87 \[ \frac{x \left (-i x^{5/2} \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2-6 a b c d-3 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (a^2 d \left (2 c+5 d x^2\right )-6 a b c d x^2+3 b^2 c^2 x^2\right )\right )}{3 c^2 d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} (e x)^{5/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(3*b^2*c^2*x^2 - 6*a*b*c*d*x^2 + a^2*d*(2*c + 5*
d*x^2))) - I*(-3*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x^(5/2)*El
lipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(3*c^2*Sqrt[(I*Sqrt[
c])/Sqrt[d]]*d*(e*x)^(5/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.036, size = 353, normalized size = 1.7 \[ -{\frac{1}{6\,x{c}^{2}{e}^{2}{d}^{2}} \left ( 5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}x{a}^{2}{d}^{2}-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}xabcd-3\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}x{b}^{2}{c}^{2}+10\,{x}^{2}{a}^{2}{d}^{3}-12\,{x}^{2}abc{d}^{2}+6\,{x}^{2}{b}^{2}{c}^{2}d+4\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/x*(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))*(-c*d)^(1/2)*x*a^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))*(-c*d)^(1
/2)*x*a*b*c*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))*(-c*d)^(1/2)*x*b^2*c^2+10*x^2*a^2*d^3-12*x^2*a
*b*c*d^2+6*x^2*b^2*c^2*d+4*a^2*c*d^2)/(d*x^2+c)^(1/2)/c^2/e^2/(e*x)^(1/2)/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

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