3.2973 \(\int (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, dx\)

Optimal. Leaf size=84 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(m+1) \sqrt{\frac{b}{a \sqrt{c x^3}}+1}} \]

[Out]

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

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Rubi [A]  time = 0.300232, antiderivative size = 84, 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 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a \sqrt{c x^3}}\right )}{(m+1) \sqrt{\frac{b}{a \sqrt{c x^3}}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107233, size = 89, normalized size = 1.06 \[ \frac{4 x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x^3}}} \, _2F_1\left (-\frac{1}{2},\frac{2 m}{3}+\frac{1}{6};\frac{2 m}{3}+\frac{7}{6};-\frac{a \sqrt{c x^3}}{b}\right )}{(4 m+1) \sqrt{\frac{a \sqrt{c x^3}}{b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(4*x*(d*x)^m*Sqrt[a + b/Sqrt[c*x^3]]*Hypergeometric2F1[-1/2, 1/6 + (2*m)/3, 7/6
+ (2*m)/3, -((a*Sqrt[c*x^3])/b)])/((1 + 4*m)*Sqrt[1 + (a*Sqrt[c*x^3])/b])

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b{\frac{1}{\sqrt{c{x}^{3}}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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