Integrand size = 21, antiderivative size = 76 \[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {a^3 c (d x)^{1+m} \sqrt {c \left (a+b x^2\right )^3} \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{d (1+m) \left (1+\frac {b x^2}{a}\right )^{3/2}} \] Output:
a^3*c*(d*x)^(1+m)*(c*(b*x^2+a)^3)^(1/2)*hypergeom([-9/2, 1/2+1/2*m],[3/2+1 /2*m],-b*x^2/a)/d/(1+m)/(1+b*x^2/a)^(3/2)
Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08 \[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\frac {a^4 x (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},\frac {1+m}{2},1+\frac {1+m}{2},-\frac {b x^2}{a}\right )}{(1+m) \left (a+b x^2\right )^4 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[(d*x)^m*(c*(a + b*x^2)^3)^(3/2),x]
Output:
(a^4*x*(d*x)^m*(c*(a + b*x^2)^3)^(3/2)*Hypergeometric2F1[-9/2, (1 + m)/2, 1 + (1 + m)/2, -((b*x^2)/a)])/((1 + m)*(a + b*x^2)^4*Sqrt[1 + (b*x^2)/a])
Time = 0.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2045, 278}
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.
\(\displaystyle \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^3} \int (d x)^m \left (\frac {b x^2}{a}+1\right )^{9/2}dx}{\left (\frac {b x^2}{a}+1\right )^{3/2}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {a^3 c (d x)^{m+1} \sqrt {c \left (a+b x^2\right )^3} \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{d (m+1) \left (\frac {b x^2}{a}+1\right )^{3/2}}\) |
Input:
Int[(d*x)^m*(c*(a + b*x^2)^3)^(3/2),x]
Output:
(a^3*c*(d*x)^(1 + m)*Sqrt[c*(a + b*x^2)^3]*Hypergeometric2F1[-9/2, (1 + m) /2, (3 + m)/2, -((b*x^2)/a)])/(d*(1 + m)*(1 + (b*x^2)/a)^(3/2))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
\[\int \left (x d \right )^{m} {\left (c \left (b \,x^{2}+a \right )^{3}\right )}^{\frac {3}{2}}d x\]
Input:
int((x*d)^m*(c*(b*x^2+a)^3)^(3/2),x)
Output:
int((x*d)^m*(c*(b*x^2+a)^3)^(3/2),x)
\[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int { \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(c*(b*x^2+a)^3)^(3/2),x, algorithm="fricas")
Output:
integral((b^3*c*x^6 + 3*a*b^2*c*x^4 + 3*a^2*b*c*x^2 + a^3*c)^(3/2)*(d*x)^m , x)
Exception generated. \[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((d*x)**m*(c*(b*x**2+a)**3)**(3/2),x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int { \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(c*(b*x^2+a)^3)^(3/2),x, algorithm="maxima")
Output:
integrate(((b*x^2 + a)^3*c)^(3/2)*(d*x)^m, x)
\[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int { \left ({\left (b x^{2} + a\right )}^{3} c\right )^{\frac {3}{2}} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(c*(b*x^2+a)^3)^(3/2),x, algorithm="giac")
Output:
integrate(((b*x^2 + a)^3*c)^(3/2)*(d*x)^m, x)
Timed out. \[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=\int {\left (d\,x\right )}^m\,{\left (c\,{\left (b\,x^2+a\right )}^3\right )}^{3/2} \,d x \] Input:
int((d*x)^m*(c*(a + b*x^2)^3)^(3/2),x)
Output:
int((d*x)^m*(c*(a + b*x^2)^3)^(3/2), x)
\[ \int (d x)^m \left (c \left (a+b x^2\right )^3\right )^{3/2} \, dx=d^{m} \sqrt {c}\, c \left (\left (\int x^{m} \sqrt {b \,x^{2}+a}\, x^{8}d x \right ) b^{4}+4 \left (\int x^{m} \sqrt {b \,x^{2}+a}\, x^{6}d x \right ) a \,b^{3}+6 \left (\int x^{m} \sqrt {b \,x^{2}+a}\, x^{4}d x \right ) a^{2} b^{2}+4 \left (\int x^{m} \sqrt {b \,x^{2}+a}\, x^{2}d x \right ) a^{3} b +\left (\int x^{m} \sqrt {b \,x^{2}+a}d x \right ) a^{4}\right ) \] Input:
int((d*x)^m*(c*(b*x^2+a)^3)^(3/2),x)
Output:
d**m*sqrt(c)*c*(int(x**m*sqrt(a + b*x**2)*x**8,x)*b**4 + 4*int(x**m*sqrt(a + b*x**2)*x**6,x)*a*b**3 + 6*int(x**m*sqrt(a + b*x**2)*x**4,x)*a**2*b**2 + 4*int(x**m*sqrt(a + b*x**2)*x**2,x)*a**3*b + int(x**m*sqrt(a + b*x**2),x )*a**4)