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

Optimal. Leaf size=234 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}-\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (b^2 c^2-7 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 \sqrt [4]{c} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (b^2 c^2-7 a d (a d+2 b c)\right )}{21 c d e^3}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{3/2}}{7 d e^3} \]

[Out]

(-2*(b^2*c^2 - 7*a*d*(2*b*c + a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*c*d*e^3) - (2
*a^2*(c + d*x^2)^(3/2))/(3*c*e*(e*x)^(3/2)) + (2*b^2*Sqrt[e*x]*(c + d*x^2)^(3/2)
)/(7*d*e^3) - (2*(b^2*c^2 - 7*a*d*(2*b*c + a*d))*(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])/(21*c^(1/4)*d^(5/4)*e^(5/2)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.506539, antiderivative size = 234, 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 \[ -\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{3 c e (e x)^{3/2}}-\frac{2 \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (b^2 c^2-7 a d (a d+2 b c)\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )|\frac{1}{2}\right )}{21 \sqrt [4]{c} d^{5/4} e^{5/2} \sqrt{c+d x^2}}-\frac{2 \sqrt{e x} \sqrt{c+d x^2} \left (b^2 c^2-7 a d (a d+2 b c)\right )}{21 c d e^3}+\frac{2 b^2 \sqrt{e x} \left (c+d x^2\right )^{3/2}}{7 d e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b^2*c^2 - 7*a*d*(2*b*c + a*d))*Sqrt[e*x]*Sqrt[c + d*x^2])/(21*c*d*e^3) - (2
*a^2*(c + d*x^2)^(3/2))/(3*c*e*(e*x)^(3/2)) + (2*b^2*Sqrt[e*x]*(c + d*x^2)^(3/2)
)/(7*d*e^3) - (2*(b^2*c^2 - 7*a*d*(2*b*c + a*d))*(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])/(21*c^(1/4)*d^(5/4)*e^(5/2)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 51.0397, size = 216, normalized size = 0.92 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 c e \left (e x\right )^{\frac{3}{2}}} + \frac{2 b^{2} \sqrt{e x} \left (c + d x^{2}\right )^{\frac{3}{2}}}{7 d e^{3}} - \frac{2 \sqrt{e x} \sqrt{c + d x^{2}} \left (- 7 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right )}{21 c d e^{3}} - \frac{2 \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (- 7 a d \left (a d + 2 b c\right ) + b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{21 \sqrt [4]{c} d^{\frac{5}{4}} e^{\frac{5}{2}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2*(c + d*x**2)**(3/2)/(3*c*e*(e*x)**(3/2)) + 2*b**2*sqrt(e*x)*(c + d*x**2)
**(3/2)/(7*d*e**3) - 2*sqrt(e*x)*sqrt(c + d*x**2)*(-7*a*d*(a*d + 2*b*c) + b**2*c
**2)/(21*c*d*e**3) - 2*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sq
rt(d)*x)*(-7*a*d*(a*d + 2*b*c) + b**2*c**2)*elliptic_f(2*atan(d**(1/4)*sqrt(e*x)
/(c**(1/4)*sqrt(e))), 1/2)/(21*c**(1/4)*d**(5/4)*e**(5/2)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.294712, size = 171, normalized size = 0.73 \[ \frac{x^{5/2} \left (\frac{2 \left (c+d x^2\right ) \left (-7 a^2 d+14 a b d x^2+b^2 x^2 \left (2 c+3 d x^2\right )\right )}{d x^{3/2}}+\frac{4 i x \sqrt{\frac{c}{d x^2}+1} \left (7 a^2 d^2+14 a b c d-b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{d \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}\right )}{21 (e x)^{5/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(5/2)*((2*(c + d*x^2)*(-7*a^2*d + 14*a*b*d*x^2 + b^2*x^2*(2*c + 3*d*x^2)))/(d
*x^(3/2)) + ((4*I)*(-(b^2*c^2) + 14*a*b*c*d + 7*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*E
llipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sq
rt[d]]*d)))/(21*(e*x)^(5/2)*Sqrt[c + d*x^2])

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Maple [A]  time = 0.046, size = 383, normalized size = 1.6 \[{\frac{2}{21\,x{e}^{2}{d}^{2}} \left ( 7\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}x{a}^{2}{d}^{2}+14\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}xabcd-\sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{2}\sqrt{{1 \left ( -dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}}\sqrt{-{dx{\frac{1}{\sqrt{-cd}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( dx+\sqrt{-cd} \right ){\frac{1}{\sqrt{-cd}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-cd}x{b}^{2}{c}^{2}+3\,{x}^{6}{b}^{2}{d}^{3}+14\,{x}^{4}ab{d}^{3}+5\,{x}^{4}{b}^{2}c{d}^{2}-7\,{x}^{2}{a}^{2}{d}^{3}+14\,{x}^{2}abc{d}^{2}+2\,{x}^{2}{b}^{2}{c}^{2}d-7\,{a}^{2}c{d}^{2} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21/(d*x^2+c)^(1/2)/x*(7*((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+14*((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+3*x^6*b^
2*d^3+14*x^4*a*b*d^3+5*x^4*b^2*c*d^2-7*x^2*a^2*d^3+14*x^2*a*b*c*d^2+2*x^2*b^2*c^
2*d-7*a^2*c*d^2)/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. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{\sqrt{e x} e^{2} x^{2}}, x\right ) \]

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 [A]  time = 83.4005, size = 153, normalized size = 0.65 \[ \frac{a^{2} \sqrt{c} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{4}\right )} + \frac{a b \sqrt{c} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} + \frac{b^{2} \sqrt{c} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac{5}{2}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c)*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), d*x**2*exp_polar(I*pi)/c)/(
2*e**(5/2)*x**(3/2)*gamma(1/4)) + a*b*sqrt(c)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/
4), (5/4,), d*x**2*exp_polar(I*pi)/c)/(e**(5/2)*gamma(5/4)) + b**2*sqrt(c)*x**(5
/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(5/2)*
gamma(9/4))

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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)