\(\int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx\) [2980]

Optimal result

Integrand size = 23, antiderivative size = 84 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{3} (1+m),\frac {1}{3} (1-2 m),-\frac {b}{a \sqrt {c x^3}}\right )}{(1+m) \sqrt {1+\frac {b}{a \sqrt {c x^3}}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {376, 350, 348, 346, 372, 371} \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{3} (m+1),\frac {1}{3} (1-2 m),-\frac {b}{a \sqrt {c x^3}}\right )}{(m+1) \sqrt {\frac {b}{a \sqrt {c x^3}}+1}} \]

[In]

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

[Out]

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

Rule 346

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(-c^(-1))*(c*x)^(m + 1)*(1/x)^(m + 1),
Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m
]

Rule 348

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 350

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[c^IntPart[m]*((c*x)^FracPart[m]/x^FracPa
rt[m]), Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && FractionQ[n]

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])

Rule 376

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} (d x)^m \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (\left (x^{-m} (d x)^m\right ) \int \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} x^m \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \text {Subst}\left (\int \sqrt {a+\frac {b}{\sqrt {c} x^3}} x^{-1+2 (1+m)} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\text {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \text {Subst}\left (\int x^{-1-2 (1+m)} \sqrt {a+\frac {b x^3}{\sqrt {c}}} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = -\text {Subst}\left (\frac {\left (2 \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} x^{-m} (d x)^m\right ) \text {Subst}\left (\int x^{-1-2 (1+m)} \sqrt {1+\frac {b x^3}{a \sqrt {c}}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{\sqrt {1+\frac {b}{a \sqrt {c} x^{3/2}}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, _2F_1\left (-\frac {1}{2},-\frac {2}{3} (1+m);\frac {1}{3} (1-2 m);-\frac {b}{a \sqrt {c x^3}}\right )}{(1+m) \sqrt {1+\frac {b}{a \sqrt {c x^3}}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.57 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.06 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {4 x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6}+\frac {2 m}{3},\frac {7}{6}+\frac {2 m}{3},-\frac {a \sqrt {c x^3}}{b}\right )}{(1+4 m) \sqrt {1+\frac {a \sqrt {c x^3}}{b}}} \]

[In]

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

[Out]

(4*x*(d*x)^m*Sqrt[a + b/Sqrt[c*x^3]]*Hypergeometric2F1[-1/2, 1/6 + (2*m)/3, 7/6 + (2*m)/3, -((a*Sqrt[c*x^3])/b
)])/((1 + 4*m)*Sqrt[1 + (a*Sqrt[c*x^3])/b])

Maple [F]

\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\sqrt {c \,x^{3}}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   alglogextint: unimplemented

Sympy [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {c\,x^3}}} \,d x \]

[In]

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

[Out]

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