Integrand size = 15, antiderivative size = 287 \[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=-\frac {7 b \sqrt [6]{a+b x^2}}{9 x}-\frac {\left (a+b x^2\right )^{7/6}}{3 x^3}+\frac {7 b \sqrt [6]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}-\left (1-\sqrt {3}\right ) \sqrt [3]{a+b x^2}}{\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{18 \sqrt [4]{3} \sqrt [3]{a} x \sqrt {-\frac {\sqrt [3]{a+b x^2} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\sqrt [3]{a}-\left (1+\sqrt {3}\right ) \sqrt [3]{a+b x^2}\right )^2}}} \] Output:
-7/9*b*(b*x^2+a)^(1/6)/x-1/3*(b*x^2+a)^(7/6)/x^3+7/54*b*(b*x^2+a)^(1/6)*(a ^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3)) /(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))^2)^(1/2)*InverseJacobiAM(arccos((a^ (1/3)-(1-3^(1/2))*(b*x^2+a)^(1/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))), 1/4*6^(1/2)+1/4*2^(1/2))*3^(3/4)/a^(1/3)/x/(-(b*x^2+a)^(1/3)*(a^(1/3)-(b*x ^2+a)^(1/3))/(a^(1/3)-(1+3^(1/2))*(b*x^2+a)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.18 \[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=-\frac {a \sqrt [6]{a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {7}{6},-\frac {1}{2},-\frac {b x^2}{a}\right )}{3 x^3 \sqrt [6]{1+\frac {b x^2}{a}}} \] Input:
Integrate[(a + b*x^2)^(7/6)/x^4,x]
Output:
-1/3*(a*(a + b*x^2)^(1/6)*Hypergeometric2F1[-3/2, -7/6, -1/2, -((b*x^2)/a) ])/(x^3*(1 + (b*x^2)/a)^(1/6))
Time = 0.28 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {247, 247, 236, 234, 760}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {7}{9} b \int \frac {\sqrt [6]{b x^2+a}}{x^2}dx-\frac {\left (a+b x^2\right )^{7/6}}{3 x^3}\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {7}{9} b \left (\frac {1}{3} b \int \frac {1}{\left (b x^2+a\right )^{5/6}}dx-\frac {\sqrt [6]{a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/6}}{3 x^3}\) |
\(\Big \downarrow \) 236 |
\(\displaystyle \frac {7}{9} b \left (\frac {b \int \frac {1}{\left (1-\frac {b x^2}{b x^2+a}\right )^{2/3}}d\frac {x}{\sqrt {b x^2+a}}}{3 \sqrt [3]{\frac {a}{a+b x^2}} \sqrt [3]{a+b x^2}}-\frac {\sqrt [6]{a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/6}}{3 x^3}\) |
\(\Big \downarrow \) 234 |
\(\displaystyle \frac {7}{9} b \left (-\frac {\sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \int \frac {1}{\sqrt {\frac {x^3}{\left (b x^2+a\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {b x^2}{b x^2+a}}}{2 x \sqrt [3]{\frac {a}{a+b x^2}}}-\frac {\sqrt [6]{a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/6}}{3 x^3}\) |
\(\Big \downarrow \) 760 |
\(\displaystyle \frac {7}{9} b \left (\frac {\sqrt {2-\sqrt {3}} \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt [6]{a+b x^2} \left (1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}\right ) \sqrt {\frac {\frac {x^2}{a+b x^2}+\sqrt [3]{1-\frac {b x^2}{a+b x^2}}+1}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {b x^2}{b x^2+a}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} x \sqrt [3]{\frac {a}{a+b x^2}} \sqrt {\frac {x^3}{\left (a+b x^2\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}}{\left (-\sqrt [3]{1-\frac {b x^2}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {\sqrt [6]{a+b x^2}}{x}\right )-\frac {\left (a+b x^2\right )^{7/6}}{3 x^3}\) |
Input:
Int[(a + b*x^2)^(7/6)/x^4,x]
Output:
-1/3*(a + b*x^2)^(7/6)/x^3 + (7*b*(-((a + b*x^2)^(1/6)/x) + (Sqrt[2 - Sqrt [3]]*Sqrt[-((b*x^2)/(a + b*x^2))]*(a + b*x^2)^(1/6)*(1 - (1 - (b*x^2)/(a + b*x^2))^(1/3))*Sqrt[(1 + x^2/(a + b*x^2) + (1 - (b*x^2)/(a + b*x^2))^(1/3 ))/(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))/(1 - Sqrt[3] - (1 - (b*x^2)/( a + b*x^2))^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*x*(a/(a + b*x^2))^(1/3)*Sqr t[-1 + x^3/(a + b*x^2)^(3/2)]*Sqrt[-((1 - (1 - (b*x^2)/(a + b*x^2))^(1/3)) /(1 - Sqrt[3] - (1 - (b*x^2)/(a + b*x^2))^(1/3))^2)])))/9
Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[1/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[((a_) + (b_.)*(x_)^2)^(-5/6), x_Symbol] :> Simp[1/((a/(a + b*x^2))^(1/3 )*(a + b*x^2)^(1/3)) Subst[Int[1/(1 - b*x^2)^(2/3), x], x, x/Sqrt[a + b*x ^2]], x] /; FreeQ[{a, b}, x]
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[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {7}{6}}}{x^{4}}d x\]
Input:
int((b*x^2+a)^(7/6)/x^4,x)
Output:
int((b*x^2+a)^(7/6)/x^4,x)
\[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{6}}}{x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(7/6)/x^4,x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(7/6)/x^4, x)
Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.12 \[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=- \frac {a^{\frac {7}{6}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {7}{6} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 x^{3}} \] Input:
integrate((b*x**2+a)**(7/6)/x**4,x)
Output:
-a**(7/6)*hyper((-3/2, -7/6), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*x**3)
\[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{6}}}{x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(7/6)/x^4,x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(7/6)/x^4, x)
\[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {7}{6}}}{x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(7/6)/x^4,x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(7/6)/x^4, x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{7/6}}{x^4} \,d x \] Input:
int((a + b*x^2)^(7/6)/x^4,x)
Output:
int((a + b*x^2)^(7/6)/x^4, x)
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.13 \[ \int \frac {\left (a+b x^2\right )^{7/6}}{x^4} \, dx=\frac {\left (b \,x^{2}+a \right )^{\frac {1}{6}} \left (-b^{2} x^{4}-2 a b \,x^{2}-a^{2}\right )}{3 a \,x^{3}} \] Input:
int((b*x^2+a)^(7/6)/x^4,x)
Output:
((a + b*x**2)**(5/6)*( - a**2 - 2*a*b*x**2 - b**2*x**4))/(3*(a + b*x**2)** (2/3)*a*x**3)