Integrand size = 21, antiderivative size = 97 \[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 x \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (a+b x^2\right )}{3 b}-\frac {4 a^{3/2} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 b^{3/2}} \] Output:
2/3*x*(c/(b*x^2+a)^(1/2))^(3/2)*(b*x^2+a)/b-4/3*a^(3/2)*(c/(b*x^2+a)^(1/2) )^(3/2)*(1+b*x^2/a)^(3/4)*InverseJacobiAM(1/2*arctan(b^(1/2)*x/a^(1/2)),2^ (1/2))/b^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.91 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70 \[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 x \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (a+b x^2-a \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {b x^2}{a}\right )\right )}{3 b} \] Input:
Integrate[x^2*(c/Sqrt[a + b*x^2])^(3/2),x]
Output:
(2*x*(c/Sqrt[a + b*x^2])^(3/2)*(a + b*x^2 - a*(1 + (b*x^2)/a)^(3/4)*Hyperg eometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)]))/(3*b)
Time = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2045, 262, 229}
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.
\(\displaystyle \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \int \frac {x^2}{\left (\frac {b x^2}{a}+1\right )^{3/4}}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (\frac {2 a x \sqrt [4]{\frac {b x^2}{a}+1}}{3 b}-\frac {2 a \int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{3/4}}dx}{3 b}\right )\) |
\(\Big \downarrow \) 229 |
\(\displaystyle \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \left (\frac {2 a x \sqrt [4]{\frac {b x^2}{a}+1}}{3 b}-\frac {4 a^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 b^{3/2}}\right )\) |
Input:
Int[x^2*(c/Sqrt[a + b*x^2])^(3/2),x]
Output:
(c/Sqrt[a + b*x^2])^(3/2)*(1 + (b*x^2)/a)^(3/4)*((2*a*x*(1 + (b*x^2)/a)^(1 /4))/(3*b) - (4*a^(3/2)*EllipticF[ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(3*b^ (3/2)))
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) )*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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]
\[\int x^{2} {\left (\frac {c}{\sqrt {b \,x^{2}+a}}\right )}^{\frac {3}{2}}d x\]
Input:
int(x^2*(c/(b*x^2+a)^(1/2))^(3/2),x)
Output:
int(x^2*(c/(b*x^2+a)^(1/2))^(3/2),x)
\[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\int { x^{2} \left (\frac {c}{\sqrt {b x^{2} + a}}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^2*(c/(b*x^2+a)^(1/2))^(3/2),x, algorithm="fricas")
Output:
integral(c*x^2*sqrt(c/sqrt(b*x^2 + a))/sqrt(b*x^2 + a), x)
\[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\int x^{2} \left (\frac {c}{\sqrt {a + b x^{2}}}\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x**2*(c/(b*x**2+a)**(1/2))**(3/2),x)
Output:
Integral(x**2*(c/sqrt(a + b*x**2))**(3/2), x)
\[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\int { x^{2} \left (\frac {c}{\sqrt {b x^{2} + a}}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^2*(c/(b*x^2+a)^(1/2))^(3/2),x, algorithm="maxima")
Output:
integrate(x^2*(c/sqrt(b*x^2 + a))^(3/2), x)
\[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\int { x^{2} \left (\frac {c}{\sqrt {b x^{2} + a}}\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(x^2*(c/(b*x^2+a)^(1/2))^(3/2),x, algorithm="giac")
Output:
integrate(x^2*(c/sqrt(b*x^2 + a))^(3/2), x)
Timed out. \[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\int x^2\,{\left (\frac {c}{\sqrt {b\,x^2+a}}\right )}^{3/2} \,d x \] Input:
int(x^2*(c/(a + b*x^2)^(1/2))^(3/2),x)
Output:
int(x^2*(c/(a + b*x^2)^(1/2))^(3/2), x)
\[ \int x^2 \left (\frac {c}{\sqrt {a+b x^2}}\right )^{3/2} \, dx=\frac {2 \sqrt {c}\, c \left (\left (b \,x^{2}+a \right )^{\frac {1}{4}} x -\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x \right ) a \right )}{3 b} \] Input:
int(x^2*(c/(b*x^2+a)^(1/2))^(3/2),x)
Output:
(2*sqrt(c)*c*((a + b*x**2)**(1/4)*x - int((a + b*x**2)**(1/4)/(a + b*x**2) ,x)*a))/(3*b)