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

Optimal. Leaf size=436 \[ \frac{\sqrt [4]{c} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{30 d^{15/4} \sqrt{c+d x^2}}-\frac{\sqrt [4]{c} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{15/4} \sqrt{c+d x^2}}+\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{15 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{e (e x)^{3/2} \sqrt{c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{45 c d^3}+\frac{(e x)^{7/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2} \sqrt{c+d x^2}}{9 d^2 e} \]

[Out]

((b*c - a*d)^2*(e*x)^(7/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 126*a*b*c
*d + 45*a^2*d^2)*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(45*c*d^3) + (2*b^2*(e*x)^(7/2)*
Sqrt[c + d*x^2])/(9*d^2*e) + ((77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e^2*Sqrt[e
*x]*Sqrt[c + d*x^2])/(15*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) - (c^(1/4)*(77*b^2*c^2 -
126*a*b*c*d + 45*a^2*d^2)*e^(5/2)*(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])/(15*d^(15/4)*Sqrt[c + d*x^2]) + (c^(1/4)*(77*b^2*c^2 - 126*a*b*c*d + 45*a^2*
d^2)*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ell
ipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(30*d^(15/4)*Sqrt[
c + d*x^2])

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Rubi [A]  time = 0.911696, antiderivative size = 436, 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 \[ \frac{\sqrt [4]{c} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{30 d^{15/4} \sqrt{c+d x^2}}-\frac{\sqrt [4]{c} e^{5/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 d^{15/4} \sqrt{c+d x^2}}+\frac{e^2 \sqrt{e x} \sqrt{c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{15 d^{7/2} \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{e (e x)^{3/2} \sqrt{c+d x^2} \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right )}{45 c d^3}+\frac{(e x)^{7/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{7/2} \sqrt{c+d x^2}}{9 d^2 e} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(e*x)^(7/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((77*b^2*c^2 - 126*a*b*c
*d + 45*a^2*d^2)*e*(e*x)^(3/2)*Sqrt[c + d*x^2])/(45*c*d^3) + (2*b^2*(e*x)^(7/2)*
Sqrt[c + d*x^2])/(9*d^2*e) + ((77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*e^2*Sqrt[e
*x]*Sqrt[c + d*x^2])/(15*d^(7/2)*(Sqrt[c] + Sqrt[d]*x)) - (c^(1/4)*(77*b^2*c^2 -
126*a*b*c*d + 45*a^2*d^2)*e^(5/2)*(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])/(15*d^(15/4)*Sqrt[c + d*x^2]) + (c^(1/4)*(77*b^2*c^2 - 126*a*b*c*d + 45*a^2*
d^2)*e^(5/2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*Ell
ipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(30*d^(15/4)*Sqrt[
c + d*x^2])

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Rubi in Sympy [A]  time = 104.712, size = 410, normalized size = 0.94 \[ \frac{2 b^{2} \left (e x\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}{9 d^{2} e} - \frac{\sqrt [4]{c} e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (45 a^{2} d^{2} - 126 a b c d + 77 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 d^{\frac{15}{4}} \sqrt{c + d x^{2}}} + \frac{\sqrt [4]{c} e^{\frac{5}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (45 a^{2} d^{2} - 126 a b c d + 77 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{30 d^{\frac{15}{4}} \sqrt{c + d x^{2}}} + \frac{e^{2} \sqrt{e x} \sqrt{c + d x^{2}} \left (45 a^{2} d^{2} - 126 a b c d + 77 b^{2} c^{2}\right )}{15 d^{\frac{7}{2}} \left (\sqrt{c} + \sqrt{d} x\right )} + \frac{\left (e x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}}{c d^{2} e \sqrt{c + d x^{2}}} - \frac{e \left (e x\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (45 a^{2} d^{2} - 126 a b c d + 77 b^{2} c^{2}\right )}{45 c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(7/2)*sqrt(c + d*x**2)/(9*d**2*e) - c**(1/4)*e**(5/2)*sqrt((c + d*
x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(45*a**2*d**2 - 126*a*b*c*
d + 77*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)
/(15*d**(15/4)*sqrt(c + d*x**2)) + c**(1/4)*e**(5/2)*sqrt((c + d*x**2)/(sqrt(c)
+ sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(45*a**2*d**2 - 126*a*b*c*d + 77*b**2*c**
2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(30*d**(15/4)*
sqrt(c + d*x**2)) + e**2*sqrt(e*x)*sqrt(c + d*x**2)*(45*a**2*d**2 - 126*a*b*c*d
+ 77*b**2*c**2)/(15*d**(7/2)*(sqrt(c) + sqrt(d)*x)) + (e*x)**(7/2)*(a*d - b*c)**
2/(c*d**2*e*sqrt(c + d*x**2)) - e*(e*x)**(3/2)*sqrt(c + d*x**2)*(45*a**2*d**2 -
126*a*b*c*d + 77*b**2*c**2)/(45*c*d**3)

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Mathematica [C]  time = 1.16269, size = 276, normalized size = 0.63 \[ \frac{(e x)^{5/2} \left (d x^2 \left (-45 a^2 d^2+18 a b d \left (7 c+2 d x^2\right )+b^2 \left (-77 c^2-22 c d x^2+10 d^2 x^4\right )\right )+\frac{3 \left (45 a^2 d^2-126 a b c d+77 b^2 c^2\right ) \left (\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (c+d x^2\right )+\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{c} \sqrt{d} x^{3/2} \sqrt{\frac{c}{d x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{45 d^4 x^3 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((e*x)^(5/2)*(d*x^2*(-45*a^2*d^2 + 18*a*b*d*(7*c + 2*d*x^2) + b^2*(-77*c^2 - 22*
c*d*x^2 + 10*d^2*x^4)) + (3*(77*b^2*c^2 - 126*a*b*c*d + 45*a^2*d^2)*(Sqrt[(I*Sqr
t[c])/Sqrt[d]]*(c + d*x^2) - Sqrt[c]*Sqrt[d]*Sqrt[1 + c/(d*x^2)]*x^(3/2)*Ellipti
cE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1] + Sqrt[c]*Sqrt[d]*Sqrt[1 +
c/(d*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1]))
/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(45*d^4*x^3*Sqrt[c + d*x^2])

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Maple [A]  time = 0.056, size = 618, normalized size = 1.4 \[{\frac{{e}^{2}}{90\,x{d}^{4}}\sqrt{ex} \left ( 20\,{x}^{6}{b}^{2}{d}^{3}+270\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}-756\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+462\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}-135\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+378\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d-231\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+72\,{x}^{4}ab{d}^{3}-44\,{x}^{4}{b}^{2}c{d}^{2}-90\,{x}^{2}{a}^{2}{d}^{3}+252\,{x}^{2}abc{d}^{2}-154\,{x}^{2}{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90/x*e^2*(e*x)^(1/2)*(20*x^6*b^2*d^3+270*((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))*a^2*c*d^2-756*((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^2*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))*b^2*c^3-135*((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+378*((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-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)*Elli
pticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^3+72*x^4*a*b*d^
3-44*x^4*b^2*c*d^2-90*x^2*a^2*d^3+252*x^2*a*b*c*d^2-154*x^2*b^2*c^2*d)/(d*x^2+c)
^(1/2)/d^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} e^{2} x^{6} + 2 \, a b e^{2} x^{4} + a^{2} e^{2} x^{2}\right )} \sqrt{e x}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*e^2*x^6 + 2*a*b*e^2*x^4 + a^2*e^2*x^2)*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)**(5/2)*(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. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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