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

Optimal. Leaf size=153 \[ -\frac{2 a^{7/4} c^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{a+b x^2}}+\frac{2 (c x)^{5/2} \sqrt{a+b x^2}}{7 c}+\frac{4 a c \sqrt{c x} \sqrt{a+b x^2}}{21 b} \]

[Out]

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

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Rubi [A]  time = 0.249118, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2 a^{7/4} c^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{21 b^{5/4} \sqrt{a+b x^2}}+\frac{2 (c x)^{5/2} \sqrt{a+b x^2}}{7 c}+\frac{4 a c \sqrt{c x} \sqrt{a+b x^2}}{21 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.866, size = 139, normalized size = 0.91 \[ - \frac{2 a^{\frac{7}{4}} c^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{21 b^{\frac{5}{4}} \sqrt{a + b x^{2}}} + \frac{4 a c \sqrt{c x} \sqrt{a + b x^{2}}}{21 b} + \frac{2 \left (c x\right )^{\frac{5}{2}} \sqrt{a + b x^{2}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(7/4)*c**(3/2)*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)/(21*b**(5/
4)*sqrt(a + b*x**2)) + 4*a*c*sqrt(c*x)*sqrt(a + b*x**2)/(21*b) + 2*(c*x)**(5/2)*
sqrt(a + b*x**2)/(7*c)

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Mathematica [C]  time = 0.148747, size = 142, normalized size = 0.93 \[ \frac{2 c \sqrt{c x} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (2 a^2+5 a b x^2+3 b^2 x^4\right )-2 i a^2 \sqrt{x} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{21 b \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*Sqrt[c*x]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(2*a^2 + 5*a*b*x^2 + 3*b^2*x^4) - (2*I
)*a^2*Sqrt[1 + a/(b*x^2)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/
Sqrt[x]], -1]))/(21*Sqrt[(I*Sqrt[a])/Sqrt[b]]*b*Sqrt[a + b*x^2])

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Maple [A]  time = 0.032, size = 138, normalized size = 0.9 \[ -{\frac{2\,c}{21\,{b}^{2}x}\sqrt{cx} \left ( \sqrt{-ab}\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{2}{a}^{2}-3\,{b}^{3}{x}^{5}-5\,a{b}^{2}{x}^{3}-2\,{a}^{2}bx \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*c/x*(c*x)^(1/2)/(b*x^2+a)^(1/2)*((-a*b)^(1/2)*((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)*E
llipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2-3*b^3*
x^5-5*a*b^2*x^3-2*a^2*b*x)/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.2517, size = 46, normalized size = 0.3 \[ \frac{\sqrt{a} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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