Integrand size = 19, antiderivative size = 185 \[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}+\frac {3}{2 a^2 c (c x)^{3/2} \sqrt {a+b x^2}}-\frac {5 \sqrt {a+b x^2}}{2 a^3 c (c x)^{3/2}}-\frac {5 b^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{4 a^{13/4} c^{5/2} \sqrt {a+b x^2}} \]
1/3/a/c/(c*x)^(3/2)/(b*x^2+a)^(3/2)+3/2/a^2/c/(c*x)^(3/2)/(b*x^2+a)^(1/2)- 5/2*(b*x^2+a)^(1/2)/a^3/c/(c*x)^(3/2)-5/4*b^(3/4)*(cos(2*arctan(b^(1/4)*(c *x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1 /4)/c^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))) ,1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/ a^(13/4)/c^(5/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{2},\frac {1}{4},-\frac {b x^2}{a}\right )}{3 a^2 (c x)^{5/2} \sqrt {a+b x^2}} \]
(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-3/4, 5/2, 1/4, -((b*x^2)/a)]) /(3*a^2*(c*x)^(5/2)*Sqrt[a + b*x^2])
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {253, 253, 264, 266, 761}
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.
| \(\displaystyle \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {3 \int \frac {1}{(c x)^{5/2} \left (b x^2+a\right )^{3/2}}dx}{2 a}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {3 \left (\frac {5 \int \frac {1}{(c x)^{5/2} \sqrt {b x^2+a}}dx}{2 a}+\frac {1}{a c (c x)^{3/2} \sqrt {a+b x^2}}\right )}{2 a}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {3 \left (\frac {5 \left (-\frac {b \int \frac {1}{\sqrt {c x} \sqrt {b x^2+a}}dx}{3 a c^2}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{2 a}+\frac {1}{a c (c x)^{3/2} \sqrt {a+b x^2}}\right )}{2 a}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \left (\frac {5 \left (-\frac {2 b \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{3 a c^3}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{2 a}+\frac {1}{a c (c x)^{3/2} \sqrt {a+b x^2}}\right )}{2 a}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {3 \left (\frac {5 \left (-\frac {b^{3/4} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{3 a^{5/4} c^{7/2} \sqrt {a+b x^2}}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{2 a}+\frac {1}{a c (c x)^{3/2} \sqrt {a+b x^2}}\right )}{2 a}+\frac {1}{3 a c (c x)^{3/2} \left (a+b x^2\right )^{3/2}}\) |
1/(3*a*c*(c*x)^(3/2)*(a + b*x^2)^(3/2)) + (3*(1/(a*c*(c*x)^(3/2)*Sqrt[a + b*x^2]) + (5*((-2*Sqrt[a + b*x^2])/(3*a*c*(c*x)^(3/2)) - (b^(3/4)*(Sqrt[a] *c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x)^2]*El lipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(3*a^(5/4)* c^(7/2)*Sqrt[a + b*x^2])))/(2*a)))/(2*a)
3.7.35.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Time = 2.85 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {\sqrt {b c \,x^{3}+a c x}}{3 a^{2} c^{3} b \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {11 b x}{6 c^{2} a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {2 \sqrt {b c \,x^{3}+a c x}}{3 a^{3} c^{3} x^{2}}-\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{4 a^{3} c^{2} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(226\) |
| default | \(-\frac {15 \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b \,x^{3}+15 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a x +30 b^{2} x^{4}+42 a b \,x^{2}+8 a^{2}}{12 x \,c^{2} \sqrt {c x}\, a^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(227\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}}{3 a^{3} x \,c^{2} \sqrt {c x}}-\frac {b \left (\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b c \,x^{3}+a c x}}+3 a^{2} \left (\frac {\sqrt {b c \,x^{3}+a c x}}{3 a c \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 x}{6 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}+\frac {5 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} b \sqrt {b c \,x^{3}+a c x}}\right )+3 a \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b c \,x^{3}+a c x}}\right )\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{3 a^{3} c^{2} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(484\) |
(c*x*(b*x^2+a))^(1/2)/(c*x)^(1/2)/(b*x^2+a)^(1/2)*(-1/3/a^2/c^3/b*(b*c*x^3 +a*c*x)^(1/2)/(x^2+a/b)^2-11/6*b/c^2*x/a^3/((x^2+a/b)*b*c*x)^(1/2)-2/3/a^3 /c^3*(b*c*x^3+a*c*x)^(1/2)/x^2-5/4/a^3/c^2*(-a*b)^(1/2)*((x+(-a*b)^(1/2)/b )/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-x/( -a*b)^(1/2)*b)^(1/2)/(b*c*x^3+a*c*x)^(1/2)*EllipticF(((x+(-a*b)^(1/2)/b)/( -a*b)^(1/2)*b)^(1/2),1/2*2^(1/2)))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=-\frac {15 \, {\left (b^{2} x^{6} + 2 \, a b x^{4} + a^{2} x^{2}\right )} \sqrt {b c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (15 \, b^{2} x^{4} + 21 \, a b x^{2} + 4 \, a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{6 \, {\left (a^{3} b^{2} c^{3} x^{6} + 2 \, a^{4} b c^{3} x^{4} + a^{5} c^{3} x^{2}\right )}} \]
-1/6*(15*(b^2*x^6 + 2*a*b*x^4 + a^2*x^2)*sqrt(b*c)*weierstrassPInverse(-4* a/b, 0, x) + (15*b^2*x^4 + 21*a*b*x^2 + 4*a^2)*sqrt(b*x^2 + a)*sqrt(c*x))/ (a^3*b^2*c^3*x^6 + 2*a^4*b*c^3*x^4 + a^5*c^3*x^2)
Result contains complex when optimal does not.
Time = 8.00 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.26 \[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {5}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} c^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
gamma(-3/4)*hyper((-3/4, 5/2), (1/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/ 2)*c**(5/2)*x**(3/2)*gamma(1/4))
\[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (c x\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (c x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(c x)^{5/2} \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \]