\(\int (d x)^m \sqrt {a+b (c x^3)^{3/2}} \, dx\) [76]

Optimal result

Integrand size = 23, antiderivative size = 88 \[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {(d x)^{1+m} \sqrt {a+b \left (c x^3\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2 (1+m)}{9},1+\frac {2 (1+m)}{9},-\frac {b \left (c x^3\right )^{3/2}}{a}\right )}{d (1+m) \sqrt {1+\frac {b \left (c x^3\right )^{3/2}}{a}}} \] Output:

(d*x)^(1+m)*(a+b*(c*x^3)^(3/2))^(1/2)*hypergeom([-1/2, 2/9+2/9*m],[11/9+2/ 
9*m],-b*(c*x^3)^(3/2)/a)/d/(1+m)/(1+b*(c*x^3)^(3/2)/a)^(1/2)
 

Mathematica [F]

\[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.48, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {893, 866, 864, 889, 888}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 893

\(\displaystyle \int (d x)^m \sqrt {a+b c^{3/2} x^{9/2}}dx\)

\(\Big \downarrow \) 866

\(\displaystyle x^{-m} (d x)^m \int x^m \sqrt {b c^{3/2} x^{9/2}+a}dx\)

\(\Big \downarrow \) 864

\(\displaystyle 2 x^{-m} (d x)^m \int \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m+1} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}+a}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {2 x^{-m} (d x)^m \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}} \int \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m+1} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1}}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {x^{-m} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 (m+1)} (d x)^m \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^3 x^9}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2 (m+1)}{9},\frac {1}{9} (2 m+11),-\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}\right )}{(m+1) \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^3 x^9}+1}}\)

Input:

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

Output:

((d*x)^m*(Sqrt[c*x^3]/(Sqrt[c]*x))^(2*(1 + m))*Sqrt[a + (b*(c*x^3)^(9/2))/ 
(c^3*x^9)]*Hypergeometric2F1[-1/2, (2*(1 + m))/9, (11 + 2*m)/9, -((b*(c*x^ 
3)^(9/2))/(a*c^3*x^9))])/((1 + m)*x^m*Sqrt[1 + (b*(c*x^3)^(9/2))/(a*c^3*x^ 
9)])
 

Defintions of rubi rules used

rule 864

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi 
nator[n]}, Simp[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 866

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

rule 888

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

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[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 893

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> With[{k = Denominator[n]}, Subst[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]
 

Maple [F]

\[\int \left (d x \right )^{m} \sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   algl 
ogextint: unimplemented
 

Sympy [F]

\[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int \left (d x\right )^{m} \sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \left (d x\right )^{m} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int { \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \left (d x\right )^{m} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (d x)^m \sqrt {a+b \left (c x^3\right )^{3/2}} \, dx=\frac {d^{m} \left (4 x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b c \,x^{4}+a}\, x -36 \sqrt {c}\, \left (\int \frac {x^{m +\frac {1}{2}} \sqrt {\sqrt {x}\, \sqrt {c}\, b c \,x^{4}+a}\, x^{4}}{-4 b^{2} c^{3} m \,x^{9}-13 b^{2} c^{3} x^{9}+4 a^{2} m +13 a^{2}}d x \right ) a b c m -117 \sqrt {c}\, \left (\int \frac {x^{m +\frac {1}{2}} \sqrt {\sqrt {x}\, \sqrt {c}\, b c \,x^{4}+a}\, x^{4}}{-4 b^{2} c^{3} m \,x^{9}-13 b^{2} c^{3} x^{9}+4 a^{2} m +13 a^{2}}d x \right ) a b c +36 \left (\int \frac {x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b c \,x^{4}+a}}{-4 b^{2} c^{3} m \,x^{9}-13 b^{2} c^{3} x^{9}+4 a^{2} m +13 a^{2}}d x \right ) a^{2} m +117 \left (\int \frac {x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b c \,x^{4}+a}}{-4 b^{2} c^{3} m \,x^{9}-13 b^{2} c^{3} x^{9}+4 a^{2} m +13 a^{2}}d x \right ) a^{2}\right )}{4 m +13} \] Input:

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

Output:

(d**m*(4*x**m*sqrt(sqrt(x)*sqrt(c)*b*c*x**4 + a)*x - 36*sqrt(c)*int((x**(( 
2*m + 1)/2)*sqrt(sqrt(x)*sqrt(c)*b*c*x**4 + a)*x**4)/(4*a**2*m + 13*a**2 - 
 4*b**2*c**3*m*x**9 - 13*b**2*c**3*x**9),x)*a*b*c*m - 117*sqrt(c)*int((x** 
((2*m + 1)/2)*sqrt(sqrt(x)*sqrt(c)*b*c*x**4 + a)*x**4)/(4*a**2*m + 13*a**2 
 - 4*b**2*c**3*m*x**9 - 13*b**2*c**3*x**9),x)*a*b*c + 36*int((x**m*sqrt(sq 
rt(x)*sqrt(c)*b*c*x**4 + a))/(4*a**2*m + 13*a**2 - 4*b**2*c**3*m*x**9 - 13 
*b**2*c**3*x**9),x)*a**2*m + 117*int((x**m*sqrt(sqrt(x)*sqrt(c)*b*c*x**4 + 
 a))/(4*a**2*m + 13*a**2 - 4*b**2*c**3*m*x**9 - 13*b**2*c**3*x**9),x)*a**2 
))/(4*m + 13)