3.6.92 \(\int x^m \sqrt {a+b x^3} \, dx\) [592]

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

[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 = 0.01, 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 = {372, 371} \begin {gather*} \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}} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 = 0.08, size = 65, normalized size = 1.03 \begin {gather*} \frac {x^{1+m} \sqrt {a+b x^3} \, _2F_1\left (-\frac {1}{2},\frac {1+m}{3};1+\frac {1+m}{3};-\frac {b x^3}{a}\right )}{(1+m) \sqrt {1+\frac {b x^3}{a}}} \end {gather*}

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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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{m} \sqrt {b \,x^{3}+a}\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.36, size = 15, normalized size = 0.24 \begin {gather*} {\rm integral}\left (\sqrt {b x^{3} + a} x^{m}, x\right ) \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.57, size = 54, normalized size = 0.86 \begin {gather*} \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 )} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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