3.2915 \(\int (d x)^m \sqrt{a+\frac{b}{(c x)^{3/2}}} \, dx\)

Optimal. Leaf size=78 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{(c x)^{3/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a (c x)^{3/2}}\right )}{(m+1) \sqrt{\frac{b}{a (c x)^{3/2}}+1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.286943, antiderivative size = 78, 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 \[ \frac{x (d x)^m \sqrt{a+\frac{b}{(c x)^{3/2}}} \, _2F_1\left (-\frac{1}{2},-\frac{2}{3} (m+1);\frac{1}{3} (1-2 m);-\frac{b}{a (c x)^{3/2}}\right )}{(m+1) \sqrt{\frac{b}{a (c x)^{3/2}}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.3218, size = 100, normalized size = 1.28 \[ \frac{\left (c x\right )^{- m} \left (c x\right )^{- m - \frac{1}{2}} \left (c x\right )^{m + \frac{1}{2}} \left (c x\right )^{m + 1} \left (d x\right )^{m} \sqrt{a + \frac{b}{\left (c x\right )^{\frac{3}{2}}}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{2 m}{3} - \frac{2}{3} \\ - \frac{2 m}{3} + \frac{1}{3} \end{matrix}\middle |{- \frac{b}{a \left (c x\right )^{\frac{3}{2}}}} \right )}}{c \sqrt{1 + \frac{b}{a \left (c x\right )^{\frac{3}{2}}}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.113655, size = 86, normalized size = 1.1 \[ \frac{4 x (d x)^m \sqrt{a+\frac{b}{(c x)^{3/2}}} \, _2F_1\left (-\frac{1}{2},\frac{1}{6} (4 m+1);\frac{1}{6} (4 m+1)+1;-\frac{a (c x)^{3/2}}{b}\right )}{(4 m+1) \sqrt{\frac{a (c x)^{3/2}+b}{b}}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+{b \left ( cx \right ) ^{-{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^(3/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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