3.2950 \(\int (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^2}}} \, dx\)

Optimal. Leaf size=87 \[ -\frac{2 b (d x)^{m+1} \left (a+\frac{b}{\sqrt{c x^2}}\right )^{3/2} \left (-\frac{b}{a \sqrt{c x^2}}\right )^m \, _2F_1\left (\frac{3}{2},m+2;\frac{5}{2};\frac{b}{a \sqrt{c x^2}}+1\right )}{3 a^2 d \sqrt{c x^2}} \]

[Out]

(-2*b*(d*x)^(1 + m)*(-(b/(a*Sqrt[c*x^2])))^m*(a + b/Sqrt[c*x^2])^(3/2)*Hypergeom
etric2F1[3/2, 2 + m, 5/2, 1 + b/(a*Sqrt[c*x^2])])/(3*a^2*d*Sqrt[c*x^2])

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Rubi [A]  time = 0.183341, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{2 b (d x)^{m+1} \left (a+\frac{b}{\sqrt{c x^2}}\right )^{3/2} \left (-\frac{b}{a \sqrt{c x^2}}\right )^m \, _2F_1\left (\frac{3}{2},m+2;\frac{5}{2};\frac{b}{a \sqrt{c x^2}}+1\right )}{3 a^2 d \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*b*(d*x)^(1 + m)*(-(b/(a*Sqrt[c*x^2])))^m*(a + b/Sqrt[c*x^2])^(3/2)*Hypergeom
etric2F1[3/2, 2 + m, 5/2, 1 + b/(a*Sqrt[c*x^2])])/(3*a^2*d*Sqrt[c*x^2])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m*sqrt(a + b/sqrt(c*x**2)), x)

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Mathematica [A]  time = 0.100161, size = 81, normalized size = 0.93 \[ \frac{2 x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^2}}} \, _2F_1\left (-\frac{1}{2},m+\frac{1}{2};m+\frac{3}{2};-\frac{a \sqrt{c x^2}}{b}\right )}{(2 m+1) \sqrt{\frac{a \sqrt{c x^2}}{b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(d*x)^m*Sqrt[a + b/Sqrt[c*x^2]]*Hypergeometric2F1[-1/2, 1/2 + m, 3/2 + m, -
((a*Sqrt[c*x^2])/b)])/((1 + 2*m)*Sqrt[1 + (a*Sqrt[c*x^2])/b])

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b{\frac{1}{\sqrt{c{x}^{2}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((d*x)^m*(a+b/(c*x^2)^(1/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m*sqrt(a + b/sqrt(c*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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