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

Optimal. Leaf size=213 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2+2 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 c^{9/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{\sqrt{e x} (5 a d+7 b c) (b c-a d)}{6 c^2 d^2 e \sqrt{c+d x^2}}+\frac{\sqrt{e x} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \]

[Out]

((b*c - a*d)^2*Sqrt[e*x])/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(7*b*c +
5*a*d)*Sqrt[e*x])/(6*c^2*d^2*e*Sqrt[c + d*x^2]) + ((5*b^2*c^2 + 2*a*b*c*d + 5*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)*Sqrt[e])], 1/2])/(12*c^(9/4)*d^(9/4)*Sqrt
[e]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.437618, antiderivative size = 213, 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 \[ \frac{\left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2+2 a b c d+5 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 c^{9/4} d^{9/4} \sqrt{e} \sqrt{c+d x^2}}-\frac{\sqrt{e x} (5 a d+7 b c) (b c-a d)}{6 c^2 d^2 e \sqrt{c+d x^2}}+\frac{\sqrt{e x} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^2*Sqrt[e*x])/(3*c*d^2*e*(c + d*x^2)^(3/2)) - ((b*c - a*d)*(7*b*c +
5*a*d)*Sqrt[e*x])/(6*c^2*d^2*e*Sqrt[c + d*x^2]) + ((5*b^2*c^2 + 2*a*b*c*d + 5*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)*Sqrt[e])], 1/2])/(12*c^(9/4)*d^(9/4)*Sqrt
[e]*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 53.821, size = 194, normalized size = 0.91 \[ \frac{\sqrt{e x} \left (a d - b c\right )^{2}}{3 c d^{2} e \left (c + d x^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{e x} \left (a d - b c\right ) \left (5 a d + 7 b c\right )}{6 c^{2} d^{2} e \sqrt{c + d x^{2}}} + \frac{\sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (5 a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{12 c^{\frac{9}{4}} d^{\frac{9}{4}} \sqrt{e} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/(d*x**2+c)**(5/2)/(e*x)**(1/2),x)

[Out]

sqrt(e*x)*(a*d - b*c)**2/(3*c*d**2*e*(c + d*x**2)**(3/2)) + sqrt(e*x)*(a*d - b*c
)*(5*a*d + 7*b*c)/(6*c**2*d**2*e*sqrt(c + d*x**2)) + sqrt((c + d*x**2)/(sqrt(c)
+ sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(5*a**2*d**2 + 2*a*b*c*d + 5*b**2*c**2)*e
lliptic_f(2*atan(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), 1/2)/(12*c**(9/4)*d**(9
/4)*sqrt(e)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.449948, size = 169, normalized size = 0.79 \[ \frac{x \left (\frac{i \sqrt{x} \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2+2 a b c d+5 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}+5 a^2 d^2+\frac{2 c (b c-a d)^2}{c+d x^2}+2 a b c d-7 b^2 c^2\right )}{6 c^2 d^2 \sqrt{e x} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-7*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2 + (2*c*(b*c - a*d)^2)/(c + d*x^2) + (I*(5
*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*Sqrt[x]*EllipticF[I*ArcSin
h[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sqrt[d]]))/(6*c^2*d^
2*Sqrt[e*x]*Sqrt[c + d*x^2])

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Maple [B]  time = 0.035, size = 660, normalized size = 3.1 \[{\frac{1}{12\,{c}^{2}{d}^{3}} \left ( 5\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}\sqrt{-cd}{x}^{2}{a}^{2}{d}^{3}+2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}\sqrt{-cd}{x}^{2}abc{d}^{2}+5\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}\sqrt{-cd}{x}^{2}{b}^{2}{c}^{2}d+5\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){a}^{2}c{d}^{2}+2\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) ab{c}^{2}d+5\,\sqrt{-cd}\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{3}+10\,{x}^{3}{a}^{2}{d}^{4}+4\,{x}^{3}abc{d}^{3}-14\,{x}^{3}{b}^{2}{c}^{2}{d}^{2}+14\,x{a}^{2}c{d}^{3}-4\,xab{c}^{2}{d}^{2}-10\,x{b}^{2}{c}^{3}d \right ){\frac{1}{\sqrt{ex}}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(d*x^2+c)^(5/2)/(e*x)^(1/2),x)

[Out]

1/12*(5*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(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)*(-c*d)^(1/2)*x^2*a^2*d^3+2*EllipticF(((d*x+(-c*d)^(1/2
))/(-c*d)^(1/2))^(1/2),1/2*2^(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)*(-c*d)^(1
/2)*x^2*a*b*c*d^2+5*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(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)*(-c*d)^(1/2)*x^2*b^2*c^2*d+5*(-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+2*(-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)*El
lipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*d+5*(-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+10*x^3*a^2*d^4+4*x^3*a*b*c*d^3-14*x^3*b^2*c^2*d^
2+14*x*a^2*c*d^3-4*x*a*b*c^2*d^2-10*x*b^2*c^3*d)/(e*x)^(1/2)/c^2/d^3/(d*x^2+c)^(
3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)/((d^2*x^4 + 2*c*d*x^2 + c^2)*sqrt(d*x^2 + c
)*sqrt(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/(d*x**2+c)**(5/2)/(e*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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