3.592 \(\int x^m \sqrt {a+b x^3} \, dx\)

Optimal. Leaf size=63 \[ \frac {x^{m+1} \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{(m+1) \sqrt {\frac {b x^3}{a}+1}} \]

[Out]

x^(1+m)*hypergeom([-1/2, 1/3+1/3*m],[4/3+1/3*m],-b*x^3/a)*(b*x^3+a)^(1/2)/(1+m)/(1+b*x^3/a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac {x^{m+1} \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{(m+1) \sqrt {\frac {b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sqrt[a + b*x^3],x]

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int x^m \sqrt {a+b x^3} \, dx &=\frac {\sqrt {a+b x^3} \int x^m \sqrt {1+\frac {b x^3}{a}} \, dx}{\sqrt {1+\frac {b x^3}{a}}}\\ &=\frac {x^{1+m} \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{(1+m) \sqrt {1+\frac {b x^3}{a}}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 65, normalized size = 1.03 \[ \frac {x^{m+1} \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{3};\frac {m+1}{3}+1;-\frac {b x^3}{a}\right )}{(m+1) \sqrt {\frac {b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Sqrt[a + b*x^3],x]

[Out]

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

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fricas [F]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x^{3} + a} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a)*x^m, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{3} + a} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^3 + a)*x^m, x)

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,x^{3}+a}\, x^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{3} + a} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^3 + a)*x^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,\sqrt {b\,x^3+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^3)^(1/2),x)

[Out]

int(x^m*(a + b*x^3)^(1/2), x)

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sympy [C]  time = 1.84, size = 54, normalized size = 0.86 \[ \frac {\sqrt {a} x x^{m} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**3+a)**(1/2),x)

[Out]

sqrt(a)*x*x**m*gamma(m/3 + 1/3)*hyper((-1/2, m/3 + 1/3), (m/3 + 4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(m/3
+ 4/3))

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