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

Optimal. Leaf size=296 \[ \frac{12 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}-\frac{24 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}+\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}+\frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{5 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}} \]

[Out]

(12*b*(c*x)^(3/2)*Sqrt[a + b*x^2])/(5*c^3) + (24*a*Sqrt[b]*Sqrt[c*x]*Sqrt[a + b*
x^2])/(5*c^2*(Sqrt[a] + Sqrt[b]*x)) - (2*(a + b*x^2)^(3/2))/(c*Sqrt[c*x]) - (24*
a^(5/4)*b^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*
EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(5*c^(3/2)*Sqrt
[a + b*x^2]) + (12*a^(5/4)*b^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[
a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/
2])/(5*c^(3/2)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.575039, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{12 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}-\frac{24 a^{5/4} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{5 c^{3/2} \sqrt{a+b x^2}}+\frac{12 b (c x)^{3/2} \sqrt{a+b x^2}}{5 c^3}+\frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{5 c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2}}{c \sqrt{c x}} \]

Antiderivative was successfully verified.

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

[Out]

(12*b*(c*x)^(3/2)*Sqrt[a + b*x^2])/(5*c^3) + (24*a*Sqrt[b]*Sqrt[c*x]*Sqrt[a + b*
x^2])/(5*c^2*(Sqrt[a] + Sqrt[b]*x)) - (2*(a + b*x^2)^(3/2))/(c*Sqrt[c*x]) - (24*
a^(5/4)*b^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*
EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(5*c^(3/2)*Sqrt
[a + b*x^2]) + (12*a^(5/4)*b^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[
a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/
2])/(5*c^(3/2)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 56.639, size = 274, normalized size = 0.93 \[ - \frac{24 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{12 a^{\frac{5}{4}} \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{2}} \sqrt{a + b x^{2}}} + \frac{24 a \sqrt{b} \sqrt{c x} \sqrt{a + b x^{2}}}{5 c^{2} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{12 b \left (c x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{5 c^{3}} - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{2}}}{c \sqrt{c x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-24*a**(5/4)*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqr
t(b)*x)*elliptic_e(2*atan(b**(1/4)*sqrt(c*x)/(a**(1/4)*sqrt(c))), 1/2)/(5*c**(3/
2)*sqrt(a + b*x**2)) + 12*a**(5/4)*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)
*x)**2)*(sqrt(a) + sqrt(b)*x)*elliptic_f(2*atan(b**(1/4)*sqrt(c*x)/(a**(1/4)*sqr
t(c))), 1/2)/(5*c**(3/2)*sqrt(a + b*x**2)) + 24*a*sqrt(b)*sqrt(c*x)*sqrt(a + b*x
**2)/(5*c**2*(sqrt(a) + sqrt(b)*x)) + 12*b*(c*x)**(3/2)*sqrt(a + b*x**2)/(5*c**3
) - 2*(a + b*x**2)**(3/2)/(c*sqrt(c*x))

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Mathematica [C]  time = 0.367438, size = 190, normalized size = 0.64 \[ \frac{x \left (-24 a^{3/2} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+24 a^{3/2} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+2 \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (-5 a^2-4 a b x^2+b^2 x^4\right )\right )}{5 (c x)^{3/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(-5*a^2 - 4*a*b*x^2 + b^2*x^4) + 24*a^(3/2)*Sq
rt[b]*x*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1
] - 24*a^(3/2)*Sqrt[b]*x*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b]
*x)/Sqrt[a]]], -1]))/(5*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(c*x)^(3/2)*Sqrt[a + b*x^2])

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Maple [A]  time = 0.023, size = 208, normalized size = 0.7 \[{\frac{2}{5\,c} \left ( 12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{2}-6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{2}+{b}^{2}{x}^{4}-4\,ab{x}^{2}-5\,{a}^{2} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(12*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2
))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(
1/2),1/2*2^(1/2))*2^(1/2)*a^2-6*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(
-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b
)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2+b^2*x^4-4*a*b*x^2-5*a^2)/(
b*x^2+a)^(1/2)/c/(c*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(3/2)/(sqrt(c*x)*c*x), x)

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Sympy [A]  time = 12.5809, size = 49, normalized size = 0.17 \[ \frac{a^{\frac{3}{2}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(3/2)/(c*x)**(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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