3.2951 \(\int (d x)^m \sqrt{a+\frac{b}{\left (c x^2\right )^{3/2}}} \, dx\)

Optimal. Leaf size=90 \[ \frac{(d x)^{m+1} \sqrt{a+\frac{b}{\left (c x^2\right )^{3/2}}} \, _2F_1\left (-\frac{1}{2},\frac{1}{3} (-m-1);\frac{2-m}{3};-\frac{b}{a \left (c x^2\right )^{3/2}}\right )}{d (m+1) \sqrt{\frac{b}{a \left (c x^2\right )^{3/2}}+1}} \]

[Out]

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

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Rubi [A]  time = 0.216331, antiderivative size = 90, 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 \[ \frac{(d x)^{m+1} \sqrt{a+\frac{b}{\left (c x^2\right )^{3/2}}} \, _2F_1\left (-\frac{1}{2},\frac{1}{3} (-m-1);\frac{2-m}{3};-\frac{b}{a \left (c x^2\right )^{3/2}}\right )}{d (m+1) \sqrt{\frac{b}{a \left (c x^2\right )^{3/2}}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d*x)**m*sqrt(a + b/(c*x**2)**(3/2)), x)

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Mathematica [A]  time = 0.0697227, size = 0, normalized size = 0. \[ \int (d x)^m \sqrt{a+\frac{b}{\left (c x^2\right )^{3/2}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((d*x)^m*(a+b/(c*x^2)^(3/2))^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x)^m*sqrt((sqrt(c*x^2)*a*c*x^2 + b)/(sqrt(c*x^2)*c*x^2)), 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((d*x)**m*(a+b/(c*x**2)**(3/2))**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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