3.849 \(\int \frac{(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=296 \[ \frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{231 d^4}-\frac{e (e x)^{5/2} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{77 c d^3}-\frac{5 c^{3/4} e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{462 d^{17/4} \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e} \]

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(c*d^2*e*Sqrt[c + d*x^2]) + (5*(117*b^2*c^2 - 198*a*
b*c*d + 77*a^2*d^2)*e^3*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^4) - ((117*b^2*c^2 - 1
98*a*b*c*d + 77*a^2*d^2)*e*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*c*d^3) + (2*b^2*(e*x
)^(9/2)*Sqrt[c + d*x^2])/(11*d^2*e) - (5*c^(3/4)*(117*b^2*c^2 - 198*a*b*c*d + 77
*a^2*d^2)*e^(7/2)*(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)*Sqrt[e])], 1/2])/(462*d^(17/4)
*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.617181, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 e^3 \sqrt{e x} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{231 d^4}-\frac{e (e x)^{5/2} \sqrt{c+d x^2} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right )}{77 c d^3}-\frac{5 c^{3/4} e^{7/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{462 d^{17/4} \sqrt{c+d x^2}}+\frac{(e x)^{9/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{9/2} \sqrt{c+d x^2}}{11 d^2 e} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

((b*c - a*d)^2*(e*x)^(9/2))/(c*d^2*e*Sqrt[c + d*x^2]) + (5*(117*b^2*c^2 - 198*a*
b*c*d + 77*a^2*d^2)*e^3*Sqrt[e*x]*Sqrt[c + d*x^2])/(231*d^4) - ((117*b^2*c^2 - 1
98*a*b*c*d + 77*a^2*d^2)*e*(e*x)^(5/2)*Sqrt[c + d*x^2])/(77*c*d^3) + (2*b^2*(e*x
)^(9/2)*Sqrt[c + d*x^2])/(11*d^2*e) - (5*c^(3/4)*(117*b^2*c^2 - 198*a*b*c*d + 77
*a^2*d^2)*e^(7/2)*(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)*Sqrt[e])], 1/2])/(462*d^(17/4)
*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 69.6021, size = 282, normalized size = 0.95 \[ \frac{2 b^{2} \left (e x\right )^{\frac{9}{2}} \sqrt{c + d x^{2}}}{11 d^{2} e} - \frac{5 c^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (77 a^{2} d^{2} - 198 a b c d + 117 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{462 d^{\frac{17}{4}} \sqrt{c + d x^{2}}} + \frac{5 e^{3} \sqrt{e x} \sqrt{c + d x^{2}} \left (77 a^{2} d^{2} - 198 a b c d + 117 b^{2} c^{2}\right )}{231 d^{4}} + \frac{\left (e x\right )^{\frac{9}{2}} \left (a d - b c\right )^{2}}{c d^{2} e \sqrt{c + d x^{2}}} - \frac{e \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (77 a^{2} d^{2} - 198 a b c d + 117 b^{2} c^{2}\right )}{77 c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

2*b**2*(e*x)**(9/2)*sqrt(c + d*x**2)/(11*d**2*e) - 5*c**(3/4)*e**(7/2)*sqrt((c +
d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(77*a**2*d**2 - 198*a*b
*c*d + 117*b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))),
1/2)/(462*d**(17/4)*sqrt(c + d*x**2)) + 5*e**3*sqrt(e*x)*sqrt(c + d*x**2)*(77*a*
*2*d**2 - 198*a*b*c*d + 117*b**2*c**2)/(231*d**4) + (e*x)**(9/2)*(a*d - b*c)**2/
(c*d**2*e*sqrt(c + d*x**2)) - e*(e*x)**(5/2)*sqrt(c + d*x**2)*(77*a**2*d**2 - 19
8*a*b*c*d + 117*b**2*c**2)/(77*c*d**3)

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Mathematica [C]  time = 0.434476, size = 226, normalized size = 0.76 \[ \frac{e^3 \sqrt{e x} \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (77 a^2 d^2 \left (5 c+2 d x^2\right )+66 a b d \left (-15 c^2-6 c d x^2+2 d^2 x^4\right )+3 b^2 \left (195 c^3+78 c^2 d x^2-26 c d^2 x^4+14 d^3 x^6\right )\right )-5 i c \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left (77 a^2 d^2-198 a b c d+117 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{231 d^4 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^(3/2),x]

[Out]

(e^3*Sqrt[e*x]*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(77*a^2*d^2*(5*c + 2*d*x^2) + 66*a*b*d
*(-15*c^2 - 6*c*d*x^2 + 2*d^2*x^4) + 3*b^2*(195*c^3 + 78*c^2*d*x^2 - 26*c*d^2*x^
4 + 14*d^3*x^6)) - (5*I)*c*(117*b^2*c^2 - 198*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(
d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))/(2
31*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^4*Sqrt[c + d*x^2])

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Maple [A]  time = 0.056, size = 407, normalized size = 1.4 \[ -{\frac{{e}^{3}}{462\,x{d}^{5}}\sqrt{ex} \left ( -84\,{x}^{7}{b}^{2}{d}^{4}+385\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-990\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+585\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-264\,{x}^{5}ab{d}^{4}+156\,{x}^{5}{b}^{2}c{d}^{3}-308\,{x}^{3}{a}^{2}{d}^{4}+792\,{x}^{3}abc{d}^{3}-468\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}-770\,x{a}^{2}c{d}^{3}+1980\,xab{c}^{2}{d}^{2}-1170\,x{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x)

[Out]

-1/462*e^3/x*(e*x)^(1/2)*(-84*x^7*b^2*d^4+385*(-c*d)^(1/2)*((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*d^2-990*(-c*d)^(1/2)*((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^2*d+585*(-c*d)^(1/2)*((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^3-264*x^5*a*b*d^4+156*x^5*b^2*c*d^3-308*x^3*a^2*d^4+792*x^3*a*b*c*d
^3-468*x^3*b^2*c^2*d^2-770*x*a^2*c*d^3+1980*x*a*b*c^2*d^2-1170*x*b^2*c^3*d)/(d*x
^2+c)^(1/2)/d^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(7/2)/(d*x^2 + c)^(3/2), x)

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

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)^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*e^3*x^7 + 2*a*b*e^3*x^5 + a^2*e^3*x^3)*sqrt(e*x)/(d*x^2 + c)^(3/2)
, 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((e*x)**(7/2)*(b*x**2+a)**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. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{7}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*(e*x)^(7/2)/(d*x^2 + c)^(3/2), x)