3.2914 \(\int (d x)^m \sqrt{a+\frac{b}{\sqrt{c x}}} \, dx\)

Optimal. Leaf size=76 \[ \frac{4 b^2 (d x)^m \left (a+\frac{b}{\sqrt{c x}}\right )^{3/2} \left (-\frac{b}{a \sqrt{c x}}\right )^{2 m} \, _2F_1\left (\frac{3}{2},2 m+3;\frac{5}{2};\frac{b}{a \sqrt{c x}}+1\right )}{3 a^3 c} \]

[Out]

(4*b^2*(d*x)^m*(-(b/(a*Sqrt[c*x])))^(2*m)*(a + b/Sqrt[c*x])^(3/2)*Hypergeometric
2F1[3/2, 3 + 2*m, 5/2, 1 + b/(a*Sqrt[c*x])])/(3*a^3*c)

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Rubi [A]  time = 0.251199, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{4 b^2 (d x)^m \left (a+\frac{b}{\sqrt{c x}}\right )^{3/2} \left (-\frac{b}{a \sqrt{c x}}\right )^{2 m} \, _2F_1\left (\frac{3}{2},2 m+3;\frac{5}{2};\frac{b}{a \sqrt{c x}}+1\right )}{3 a^3 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*b^2*(d*x)^m*(-(b/(a*Sqrt[c*x])))^(2*m)*(a + b/Sqrt[c*x])^(3/2)*Hypergeometric
2F1[3/2, 3 + 2*m, 5/2, 1 + b/(a*Sqrt[c*x])])/(3*a^3*c)

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Rubi in Sympy [A]  time = 19.3112, size = 83, normalized size = 1.09 \[ \frac{4 b^{2} \left (c x\right )^{- m - \frac{1}{2}} \left (c x\right )^{m + \frac{1}{2}} \left (d x\right )^{m} \left (- \frac{b}{a \sqrt{c x}}\right )^{2 m} \left (a + \frac{b}{\sqrt{c x}}\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} 2 m + 3, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{b}{a \sqrt{c x}}} \right )}}{3 a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*b**2*(c*x)**(-m - 1/2)*(c*x)**(m + 1/2)*(d*x)**m*(-b/(a*sqrt(c*x)))**(2*m)*(a
+ b/sqrt(c*x))**(3/2)*hyper((2*m + 3, 3/2), (5/2,), 1 + b/(a*sqrt(c*x)))/(3*a**3
*c)

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Mathematica [A]  time = 0.097326, size = 79, normalized size = 1.04 \[ \frac{4 x (d x)^m \sqrt{a+\frac{b}{\sqrt{c x}}} \, _2F_1\left (-\frac{1}{2},2 m+\frac{3}{2};2 m+\frac{5}{2};-\frac{a \sqrt{c x}}{b}\right )}{(4 m+3) \sqrt{\frac{a \sqrt{c x}}{b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(4*x*(d*x)^m*Sqrt[a + b/Sqrt[c*x]]*Hypergeometric2F1[-1/2, 3/2 + 2*m, 5/2 + 2*m,
-((a*Sqrt[c*x])/b)])/((3 + 4*m)*Sqrt[1 + (a*Sqrt[c*x])/b])

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b{\frac{1}{\sqrt{cx}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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