3.2948 \(\int (d x)^m \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{(d x)^{m+1} \sqrt{a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )}{d (m+1) \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.143826, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{(d x)^{m+1} \sqrt{a+b \left (c x^2\right )^{3/2}} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b \left (c x^2\right )^{3/2}}{a}\right )}{d (m+1) \sqrt{\frac{b \left (c x^2\right )^{3/2}}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0485507, size = 0, normalized size = 0. \[ \int (d x)^m \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\sqrt{c x^{2}} b c x^{2} + a} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(sqrt(c*x^2)*b*c*x^2 + a)*(d*x)^m, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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