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

Optimal. Leaf size=184 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3} \]

[Out]

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

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Rubi [A]  time = 0.390037, antiderivative size = 184, 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 \[ -\frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{3 c^{5/4} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 a^2 \sqrt{c+d x^2}}{3 c e (e x)^{3/2}}+\frac{2 b^2 \sqrt{e x} \sqrt{c+d x^2}}{3 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.289, size = 165, normalized size = 0.9 \[ - \frac{2 a^{2} \sqrt{c + d x^{2}}}{3 c e \left (e x\right )^{\frac{3}{2}}} + \frac{2 b^{2} \sqrt{e x} \sqrt{c + d x^{2}}}{3 d e^{3}} - \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (a d - 6 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{3 c^{\frac{5}{4}} d^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{c + d x^{2}}} \]

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)**(1/2),x)

[Out]

-2*a**2*sqrt(c + d*x**2)/(3*c*e*(e*x)**(3/2)) + 2*b**2*sqrt(e*x)*sqrt(c + d*x**2
)/(3*d*e**3) - sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)
*(a*d*(a*d - 6*b*c) + b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*
sqrt(e))), 1/2)/(3*c**(5/4)*d**(5/4)*e**(5/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.288025, size = 165, normalized size = 0.9 \[ \frac{x \left (2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right ) \left (b^2 c x^2-a^2 d\right )-2 i x^{5/2} \sqrt{\frac{c}{d x^2}+1} \left (a^2 d^2-6 a b c d+b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 c d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} (e x)^{5/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.029, size = 352, normalized size = 1.9 \[ -{\frac{1}{3\,cx{e}^{2}{d}^{2}} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-cd}x{a}^{2}{d}^{2}-6\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}xabcd+\sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-cd}x{b}^{2}{c}^{2}-2\,{x}^{4}{b}^{2}c{d}^{2}+2\,{x}^{2}{a}^{2}{d}^{3}-2\,{x}^{2}{b}^{2}{c}^{2}d+2\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

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)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/(sqrt(d*x^2 + c)*sqrt(e*x)*e^2*x^2), 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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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