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

Optimal. Leaf size=297 \[ \frac{4 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{c x} \sqrt{a+b x^2}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \]

[Out]

(4*a*(c*x)^(3/2)*Sqrt[a + b*x^2])/(15*c) + (8*a^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(15
*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x)) + (2*(c*x)^(3/2)*(a + b*x^2)^(3/2))/(9*c) - (8*a
^(9/4)*Sqrt[c]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*E
llipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(15*b^(3/4)*Sqrt
[a + b*x^2]) + (4*a^(9/4)*Sqrt[c]*(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
])/(15*b^(3/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.579623, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{4 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}-\frac{8 a^{9/4} \sqrt{c} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{15 b^{3/4} \sqrt{a+b x^2}}+\frac{8 a^2 \sqrt{c x} \sqrt{a+b x^2}}{15 \sqrt{b} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{4 a (c x)^{3/2} \sqrt{a+b x^2}}{15 c}+\frac{2 (c x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*a*(c*x)^(3/2)*Sqrt[a + b*x^2])/(15*c) + (8*a^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(15
*Sqrt[b]*(Sqrt[a] + Sqrt[b]*x)) + (2*(c*x)^(3/2)*(a + b*x^2)^(3/2))/(9*c) - (8*a
^(9/4)*Sqrt[c]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*E
llipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(15*b^(3/4)*Sqrt
[a + b*x^2]) + (4*a^(9/4)*Sqrt[c]*(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
])/(15*b^(3/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 55.5755, size = 272, normalized size = 0.92 \[ - \frac{8 a^{\frac{9}{4}} \sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + b x^{2}}} + \frac{4 a^{\frac{9}{4}} \sqrt{c} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{3}{4}} \sqrt{a + b x^{2}}} + \frac{8 a^{2} \sqrt{c x} \sqrt{a + b x^{2}}}{15 \sqrt{b} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{4 a \left (c x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}}{15 c} + \frac{2 \left (c x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}}}{9 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(1/2)*(b*x**2+a)**(3/2),x)

[Out]

-8*a**(9/4)*sqrt(c)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(
b)*x)*elliptic_e(2*atan(b**(1/4)*sqrt(c*x)/(a**(1/4)*sqrt(c))), 1/2)/(15*b**(3/4
)*sqrt(a + b*x**2)) + 4*a**(9/4)*sqrt(c)*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)*sqrt(c
))), 1/2)/(15*b**(3/4)*sqrt(a + b*x**2)) + 8*a**2*sqrt(c*x)*sqrt(a + b*x**2)/(15
*sqrt(b)*(sqrt(a) + sqrt(b)*x)) + 4*a*(c*x)**(3/2)*sqrt(a + b*x**2)/(15*c) + 2*(
c*x)**(3/2)*(a + b*x**2)**(3/2)/(9*c)

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Mathematica [C]  time = 0.266469, size = 188, normalized size = 0.63 \[ \frac{2 \sqrt{c x} \left (-12 a^{5/2} \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+12 a^{5/2} \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+\sqrt{b} x \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (11 a^2+16 a b x^2+5 b^2 x^4\right )\right )}{45 \sqrt{b} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c*x]*(Sqrt[b]*x*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(11*a^2 + 16*a*b*x^2 + 5*b^2
*x^4) + 12*a^(5/2)*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sq
rt[a]]], -1] - 12*a^(5/2)*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b
]*x)/Sqrt[a]]], -1]))/(45*Sqrt[b]*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.017, size = 218, normalized size = 0.7 \[{\frac{2}{45\,bx}\sqrt{cx} \left ( 5\,{b}^{3}{x}^{6}+12\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{3}-6\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{2}{a}^{3}+16\,a{b}^{2}{x}^{4}+11\,{a}^{2}b{x}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(1/2)*(b*x^2+a)^(3/2),x)

[Out]

2/45*(c*x)^(1/2)/(b*x^2+a)^(1/2)/b*(5*b^3*x^6+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)*Elli
pticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3-6*((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^3+16*a*b^2*x^4+11*a^2*b*x^2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.4824, size = 46, normalized size = 0.15 \[ \frac{a^{\frac{3}{2}} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)**(1/2)*(b*x**2+a)**(3/2),x)

[Out]

a**(3/2)*sqrt(c)*x**(3/2)*gamma(3/4)*hyper((-3/2, 3/4), (7/4,), b*x**2*exp_polar
(I*pi)/a)/(2*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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