3.605 \(\int \frac{\left (a+b x^2\right )^{3/2}}{(c x)^{9/2}} \, dx\)

Optimal. Leaf size=152 \[ \frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}} \]

[Out]

(-4*b*Sqrt[a + b*x^2])/(7*c^3*(c*x)^(3/2)) - (2*(a + b*x^2)^(3/2))/(7*c*(c*x)^(7
/2)) + (4*b^(7/4)*(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])/(7*a^(1/4)*c^
(9/2)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.256562, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{4 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{7 \sqrt [4]{a} c^{9/2} \sqrt{a+b x^2}}-\frac{4 b \sqrt{a+b x^2}}{7 c^3 (c x)^{3/2}}-\frac{2 \left (a+b x^2\right )^{3/2}}{7 c (c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/2)/(c*x)^(9/2),x]

[Out]

(-4*b*Sqrt[a + b*x^2])/(7*c^3*(c*x)^(3/2)) - (2*(a + b*x^2)^(3/2))/(7*c*(c*x)^(7
/2)) + (4*b^(7/4)*(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])/(7*a^(1/4)*c^
(9/2)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 23.6911, size = 139, normalized size = 0.91 \[ - \frac{4 b \sqrt{a + b x^{2}}}{7 c^{3} \left (c x\right )^{\frac{3}{2}}} - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{2}}}{7 c \left (c x\right )^{\frac{7}{2}}} + \frac{4 b^{\frac{7}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}} \right )}\middle | \frac{1}{2}\right )}{7 \sqrt [4]{a} c^{\frac{9}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)/(c*x)**(9/2),x)

[Out]

-4*b*sqrt(a + b*x**2)/(7*c**3*(c*x)**(3/2)) - 2*(a + b*x**2)**(3/2)/(7*c*(c*x)**
(7/2)) + 4*b**(7/4)*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)*sqrt(c))), 1/2)/(7*a**(1/4)
*c**(9/2)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.242044, size = 121, normalized size = 0.8 \[ \frac{x^{9/2} \left (-\frac{2 \left (a+b x^2\right ) \left (a+3 b x^2\right )}{x^{7/2}}+\frac{8 i b^2 x \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{7 (c x)^{9/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(3/2)/(c*x)^(9/2),x]

[Out]

(x^(9/2)*((-2*(a + b*x^2)*(a + 3*b*x^2))/x^(7/2) + ((8*I)*b^2*Sqrt[1 + a/(b*x^2)
]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[a]
)/Sqrt[b]]))/(7*(c*x)^(9/2)*Sqrt[a + b*x^2])

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Maple [A]  time = 0.04, size = 135, normalized size = 0.9 \[{\frac{2}{7\,{x}^{3}{c}^{4}} \left ( 2\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}\sqrt{2}{x}^{3}b-3\,{b}^{2}{x}^{4}-4\,ab{x}^{2}-{a}^{2} \right ){\frac{1}{\sqrt{cx}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/(c*x)^(9/2),x)

[Out]

2/7/(b*x^2+a)^(1/2)/x^3*(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)*EllipticF(((b*x+(-a*b)^(1/
2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*2^(1/2)*x^3*b-3*b^2*x^4-4*a*b*
x^2-a^2)/c^4/(c*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\sqrt{c x} c^{4} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(3/2)/(sqrt(c*x)*c^4*x^4), 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)**(3/2)/(c*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (c x\right )^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(3/2)/(c*x)^(9/2), x)