3.2974 \(\int (d x)^m \sqrt{a+\frac{b}{\left (c x^3\right )^{3/2}}} \, dx\)

Optimal. Leaf size=102 \[ \frac{x (d x)^m \sqrt{a+\frac{b c^3 x^9}{\left (c x^3\right )^{9/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{9} (m+1);\frac{1}{9} (7-2 m);-\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}\right )}{(m+1) \sqrt{\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}+1}} \]

[Out]

(x*(d*x)^m*Sqrt[a + (b*c^3*x^9)/(c*x^3)^(9/2)]*Hypergeometric2F1[-1/2, (-2*(1 +
m))/9, (7 - 2*m)/9, -((b*c^3*x^9)/(a*(c*x^3)^(9/2)))])/((1 + m)*Sqrt[1 + (b*c^3*
x^9)/(a*(c*x^3)^(9/2))])

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Rubi [A]  time = 0.306082, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{x (d x)^m \sqrt{a+\frac{b c^3 x^9}{\left (c x^3\right )^{9/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{9} (m+1);\frac{1}{9} (7-2 m);-\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}\right )}{(m+1) \sqrt{\frac{b c^3 x^9}{a \left (c x^3\right )^{9/2}}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d*x)^m*Sqrt[a + (b*c^3*x^9)/(c*x^3)^(9/2)]*Hypergeometric2F1[-1/2, (-2*(1 +
m))/9, (7 - 2*m)/9, -((b*c^3*x^9)/(a*(c*x^3)^(9/2)))])/((1 + m)*Sqrt[1 + (b*c^3*
x^9)/(a*(c*x^3)^(9/2))])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0721971, size = 0, normalized size = 0. \[ \int (d x)^m \sqrt{a+\frac{b}{\left (c x^3\right )^{3/2}}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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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**3)**(3/2))**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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