Integrand size = 21, antiderivative size = 154 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{3 x^3}-\frac {b \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a x}+\frac {b^2 x \left (c \sqrt {a+b x^2}\right )^{3/2}}{2 a \left (a+b x^2\right )}-\frac {b^{3/2} \left (c \sqrt {a+b x^2}\right )^{3/2} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 a^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
-1/3*(c*(b*x^2+a)^(1/2))^(3/2)/x^3-1/2*b*(c*(b*x^2+a)^(1/2))^(3/2)/a/x+1/2 *b^2*x*(c*(b*x^2+a)^(1/2))^(3/2)/a/(b*x^2+a)-1/2*b^(3/2)*(cos(1/2*arctan(x *b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arctan(x*b^(1/2)/a^(1/2)))*EllipticE(s in(1/2*arctan(x*b^(1/2)/a^(1/2))),2^(1/2))*(c*(b*x^2+a)^(1/2))^(3/2)/a^(3/ 2)/(1+b*x^2/a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.37 \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=-\frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {1}{2},-\frac {b x^2}{a}\right )}{3 x^3 \left (1+\frac {b x^2}{a}\right )^{3/4}} \]
-1/3*((c*Sqrt[a + b*x^2])^(3/2)*Hypergeometric2F1[-3/2, -3/4, -1/2, -((b*x ^2)/a)])/(x^3*(1 + (b*x^2)/a)^(3/4))
Time = 0.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2045, 247, 264, 225, 212}
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.
\(\displaystyle \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \int \frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x^4}dx}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {b \int \frac {1}{x^2 \sqrt [4]{\frac {b x^2}{a}+1}}dx}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{3 x^3}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt [4]{\frac {b x^2}{a}+1}}dx}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{3 x^3}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {b \left (\frac {b \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{3 x^3}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {\left (c \sqrt {a+b x^2}\right )^{3/2} \left (\frac {b \left (\frac {b \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{x}\right )}{2 a}-\frac {\left (\frac {b x^2}{a}+1\right )^{3/4}}{3 x^3}\right )}{\left (\frac {b x^2}{a}+1\right )^{3/4}}\) |
((c*Sqrt[a + b*x^2])^(3/2)*(-1/3*(1 + (b*x^2)/a)^(3/4)/x^3 + (b*(-((1 + (b *x^2)/a)^(3/4)/x) + (b*((2*x)/(1 + (b*x^2)/a)^(1/4) - (2*Sqrt[a]*EllipticE [ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/Sqrt[b]))/(2*a)))/(2*a)))/(1 + (b*x^2) /a)^(3/4)
3.3.58.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a , 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && 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]
\[\int \frac {\left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x^{4}}d x\]
Exception generated. \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=\int \frac {\left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{4}}\, dx \]
\[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=\int { \frac {\left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
\[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=\int { \frac {\left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (c \sqrt {a+b x^2}\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (c\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^4} \,d x \]