3.2913 \(\int (d x)^m \sqrt{a+b \sqrt{c x}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{4 a (d x)^m \left (a+b \sqrt{c x}\right )^{3/2} \left (-\frac{b \sqrt{c x}}{a}\right )^{-2 m} \, _2F_1\left (\frac{3}{2},-2 m-1;\frac{5}{2};\frac{\sqrt{c x} b}{a}+1\right )}{3 b^2 c} \]

[Out]

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

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Rubi [A]  time = 0.161698, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{4 a (d x)^m \left (a+b \sqrt{c x}\right )^{3/2} \left (-\frac{b \sqrt{c x}}{a}\right )^{-2 m} \, _2F_1\left (\frac{3}{2},-2 m-1;\frac{5}{2};\frac{\sqrt{c x} b}{a}+1\right )}{3 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0396, size = 66, normalized size = 0.89 \[ - \frac{4 a \left (d x\right )^{m} \left (- \frac{b \sqrt{c x}}{a}\right )^{- 2 m} \left (a + b \sqrt{c x}\right )^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} - 2 m - 1, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{1 + \frac{b \sqrt{c x}}{a}} \right )}}{3 b^{2} c} \]

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

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Mathematica [A]  time = 0.0720285, size = 72, normalized size = 0.97 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{c x}} \, _2F_1\left (-\frac{1}{2},2 m+2;2 m+3;-\frac{b \sqrt{c x}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{c x}}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{cx}}\, dx \]

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. \[ \int \sqrt{\sqrt{c x} b + a} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

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. \[ \int \sqrt{\sqrt{c x} b + a} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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