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

Optimal. Leaf size=245 \[ \frac{e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt{c+d x^2}}-\frac{e \sqrt{e x} \sqrt{c+d x^2} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right )}{21 c d^3}+\frac{(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d^2 e} \]

[Out]

((b*c - a*d)^2*(e*x)^(5/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((45*b^2*c^2 - 70*a*b*c*
d + 21*a^2*d^2)*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*c*d^3) + (2*b^2*(e*x)^(5/2)*Sqr
t[c + d*x^2])/(7*d^2*e) + ((45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*e^(3/2)*(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])/(42*c^(1/4)*d^(13/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.52173, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{e^{3/2} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt{c+d x^2}}-\frac{e \sqrt{e x} \sqrt{c+d x^2} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right )}{21 c d^3}+\frac{(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt{c+d x^2}}+\frac{2 b^2 (e x)^{5/2} \sqrt{c+d x^2}}{7 d^2 e} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*(e*x)^(5/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((45*b^2*c^2 - 70*a*b*c*
d + 21*a^2*d^2)*e*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*c*d^3) + (2*b^2*(e*x)^(5/2)*Sqr
t[c + d*x^2])/(7*d^2*e) + ((45*b^2*c^2 - 70*a*b*c*d + 21*a^2*d^2)*e^(3/2)*(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])/(42*c^(1/4)*d^(13/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 60.1852, size = 228, normalized size = 0.93 \[ \frac{2 b^{2} \left (e x\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}{7 d^{2} e} + \frac{\left (e x\right )^{\frac{5}{2}} \left (a d - b c\right )^{2}}{c d^{2} e \sqrt{c + d x^{2}}} - \frac{e \sqrt{e x} \sqrt{c + d x^{2}} \left (21 a^{2} d^{2} - 70 a b c d + 45 b^{2} c^{2}\right )}{21 c d^{3}} + \frac{e^{\frac{3}{2}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (21 a^{2} d^{2} - 70 a b c d + 45 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{42 \sqrt [4]{c} d^{\frac{13}{4}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b**2*(e*x)**(5/2)*sqrt(c + d*x**2)/(7*d**2*e) + (e*x)**(5/2)*(a*d - b*c)**2/(c
*d**2*e*sqrt(c + d*x**2)) - e*sqrt(e*x)*sqrt(c + d*x**2)*(21*a**2*d**2 - 70*a*b*
c*d + 45*b**2*c**2)/(21*c*d**3) + e**(3/2)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*
x)**2)*(sqrt(c) + sqrt(d)*x)*(21*a**2*d**2 - 70*a*b*c*d + 45*b**2*c**2)*elliptic
_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(42*c**(1/4)*d**(13/4)*sq
rt(c + d*x**2))

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Mathematica [C]  time = 0.328986, size = 191, normalized size = 0.78 \[ \frac{e \sqrt{e x} \left (i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \left (-21 a^2 d^2+14 a b d \left (5 c+2 d x^2\right )-3 b^2 \left (15 c^2+6 c d x^2-2 d^2 x^4\right )\right )\right )}{21 d^3 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(e*Sqrt[e*x]*(Sqrt[(I*Sqrt[c])/Sqrt[d]]*(-21*a^2*d^2 + 14*a*b*d*(5*c + 2*d*x^2)
- 3*b^2*(15*c^2 + 6*c*d*x^2 - 2*d^2*x^4)) + I*(45*b^2*c^2 - 70*a*b*c*d + 21*a^2*
d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/S
qrt[x]], -1]))/(21*Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3*Sqrt[c + d*x^2])

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Maple [A]  time = 0.032, size = 363, normalized size = 1.5 \[{\frac{e}{42\,x{d}^{4}}\sqrt{ex} \left ( 21\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}{d}^{2}-70\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) abcd+45\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{2}+12\,{x}^{5}{b}^{2}{d}^{3}+56\,{x}^{3}ab{d}^{3}-36\,{x}^{3}{b}^{2}c{d}^{2}-42\,x{a}^{2}{d}^{3}+140\,xabc{d}^{2}-90\,x{b}^{2}{c}^{2}d \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*e/x*(e*x)^(1/2)*(21*(-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)*Ellipti
cF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*d^2-70*(-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*d+45*(-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)*Ell
ipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2+12*x^5*b^2*d
^3+56*x^3*a*b*d^3-36*x^3*b^2*c*d^2-42*x*a^2*d^3+140*x*a*b*c*d^2-90*x*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. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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