Integrand size = 19, antiderivative size = 296 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}-\frac {3 \sqrt {a+b x^2}}{a^2 c \sqrt {c x}}+\frac {3 \sqrt {b} \sqrt {c x} \sqrt {a+b x^2}}{a^2 c^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{a^{7/4} c^{3/2} \sqrt {a+b x^2}}+\frac {3 \sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 a^{7/4} c^{3/2} \sqrt {a+b x^2}} \]
1/a/c/(c*x)^(1/2)/(b*x^2+a)^(1/2)-3*(b*x^2+a)^(1/2)/a^2/c/(c*x)^(1/2)+3*b^ (1/2)*(c*x)^(1/2)*(b*x^2+a)^(1/2)/a^2/c^2/(a^(1/2)+x*b^(1/2))-3*b^(1/4)*(c os(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)))*EllipticE(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^(7/4)/c^(3/2)/(b*x^2+a)^(1/2)+3/2*b^(1/4)*(cos(2*a rctan(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^(7/4)/c^(3/2)/(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 = 57, normalized size of antiderivative = 0.19 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},-\frac {b x^2}{a}\right )}{a (c x)^{3/2} \sqrt {a+b x^2}} \]
(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^2)/a)]) /(a*(c*x)^(3/2)*Sqrt[a + b*x^2])
Time = 0.40 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {253, 264, 266, 834, 27, 761, 1510}
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.
\(\displaystyle \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {3 \int \frac {1}{(c x)^{3/2} \sqrt {b x^2+a}}dx}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {3 \left (\frac {b \int \frac {\sqrt {c x}}{\sqrt {b x^2+a}}dx}{a c^2}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {3 \left (\frac {2 b \int \frac {c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {3 \left (\frac {2 b \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\sqrt {a} c \int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {a} c \sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \left (\frac {2 b \left (\frac {\sqrt {a} c \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {3 \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} c-\sqrt {b} c x}{\sqrt {b x^2+a}}d\sqrt {c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {3 \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {c} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {c^2 \sqrt {c x} \sqrt {a+b x^2}}{\sqrt {a} c+\sqrt {b} c x}}{\sqrt {b}}\right )}{a c^3}-\frac {2 \sqrt {a+b x^2}}{a c \sqrt {c x}}\right )}{2 a}+\frac {1}{a c \sqrt {c x} \sqrt {a+b x^2}}\) |
1/(a*c*Sqrt[c*x]*Sqrt[a + b*x^2]) + (3*((-2*Sqrt[a + b*x^2])/(a*c*Sqrt[c*x ]) + (2*b*(-((-((c^2*Sqrt[c*x]*Sqrt[a + b*x^2])/(Sqrt[a]*c + Sqrt[b]*c*x)) + (a^(1/4)*Sqrt[c]*(Sqrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sq rt[a]*c + Sqrt[b]*c*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)* Sqrt[c])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b]) + (a^(1/4)*Sqrt[c]*(S qrt[a]*c + Sqrt[b]*c*x)*Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x) ^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(2*b^ (3/4)*Sqrt[a + b*x^2])))/(a*c^3)))/(2*a)
3.7.26.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Time = 2.40 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {3 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a -6 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a +6 b \,x^{2}+4 a}{2 \sqrt {b \,x^{2}+a}\, c \sqrt {c x}\, a^{2}}\) | \(197\) |
elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 \left (c b \,x^{2}+a c \right )}{a^{2} c^{2} \sqrt {x \left (c b \,x^{2}+a c \right )}}-\frac {b \,x^{2}}{c \,a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}+\frac {3 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a^{2} c \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(251\) |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}}{a^{2} c \sqrt {c x}}+\frac {b^{2} \left (\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{2} \sqrt {b c \,x^{3}+a c x}}-\frac {a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b c \,x^{3}+a c x}}\right )}{b}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{a^{2} c \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(412\) |
-1/2*(3*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2 ))/(-a*b)^(1/2))^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)*EllipticF(((b*x+(-a*b)^(1 /2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a-6*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2) )^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x/(-a*b)^(1/2)* b)^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a+ 6*b*x^2+4*a)/(b*x^2+a)^(1/2)/c/(c*x)^(1/2)/a^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.28 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \, {\left (b x^{3} + a x\right )} \sqrt {b c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, b x^{2} + 2 \, a\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{a^{2} b c^{2} x^{3} + a^{3} c^{2} x} \]
-(3*(b*x^3 + a*x)*sqrt(b*c)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse (-4*a/b, 0, x)) + (3*b*x^2 + 2*a)*sqrt(b*x^2 + a)*sqrt(c*x))/(a^2*b*c^2*x^ 3 + a^3*c^2*x)
Result contains complex when optimal does not.
Time = 1.65 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.16 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/ 2)*c**(3/2)*sqrt(x)*gamma(3/4))
\[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]