Integrand size = 19, antiderivative size = 158 \[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}+\frac {10 b \sqrt {a+b x^2}}{21 a^2 c^3 (c x)^{3/2}}+\frac {5 b^{7/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{21 a^{9/4} c^{9/2} \sqrt {a+b x^2}} \] Output:
-2/7*(b*x^2+a)^(1/2)/a/c/(c*x)^(7/2)+10/21*b*(b*x^2+a)^(1/2)/a^2/c^3/(c*x) ^(3/2)+5/21*b^(7/4)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^ (1/2)*InverseJacobiAM(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)),1/2*2^ (1/2))/a^(9/4)/c^(9/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.35 \[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {1}{2},-\frac {3}{4},-\frac {b x^2}{a}\right )}{7 (c x)^{9/2} \sqrt {a+b x^2}} \] Input:
Integrate[1/((c*x)^(9/2)*Sqrt[a + b*x^2]),x]
Output:
(-2*x*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[-7/4, 1/2, -3/4, -((b*x^2)/a)] )/(7*(c*x)^(9/2)*Sqrt[a + b*x^2])
Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {264, 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.
\(\displaystyle \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {5 b \int \frac {1}{(c x)^{5/2} \sqrt {b x^2+a}}dx}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {5 b \left (-\frac {b \int \frac {1}{\sqrt {c x} \sqrt {b x^2+a}}dx}{3 a c^2}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {5 b \left (-\frac {2 b \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{3 a c^3}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle -\frac {5 b \left (-\frac {b^{3/4} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{3 a^{5/4} c^{7/2} \sqrt {a+b x^2}}-\frac {2 \sqrt {a+b x^2}}{3 a c (c x)^{3/2}}\right )}{7 a c^2}-\frac {2 \sqrt {a+b x^2}}{7 a c (c x)^{7/2}}\) |
Input:
Int[1/((c*x)^(9/2)*Sqrt[a + b*x^2]),x]
Output:
(-2*Sqrt[a + b*x^2])/(7*a*c*(c*x)^(7/2)) - (5*b*((-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]*EllipticF[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])))/(7*a*c^2)
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 = 1.87 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {5 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, b \,x^{3}+10 b^{2} x^{4}+4 a b \,x^{2}-6 a^{2}}{21 \sqrt {b \,x^{2}+a}\, x^{3} a^{2} c^{4} \sqrt {c x}}\) | \(138\) |
risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-5 b \,x^{2}+3 a \right )}{21 a^{2} x^{3} c^{4} \sqrt {c x}}+\frac {5 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{21 a^{2} \sqrt {b c \,x^{3}+a c x}\, c^{4} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(177\) |
elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {b c \,x^{3}+a c x}}{7 a \,c^{5} x^{4}}+\frac {10 b \sqrt {b c \,x^{3}+a c x}}{21 a^{2} c^{5} x^{2}}+\frac {5 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 a^{2} c^{4} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(192\) |
Input:
int(1/(c*x)^(9/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/21/(b*x^2+a)^(1/2)/x^3*(5*((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)*(-b/(-a*b)^(1/2)*x)^(1/2)*Ellip ticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*b*x ^3+10*b^2*x^4+4*a*b*x^2-6*a^2)/a^2/c^4/(c*x)^(1/2)
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=\frac {2 \, {\left (5 \, \sqrt {b c} b x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (5 \, b x^{2} - 3 \, a\right )} \sqrt {b x^{2} + a} \sqrt {c x}\right )}}{21 \, a^{2} c^{5} x^{4}} \] Input:
integrate(1/(c*x)^(9/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/21*(5*sqrt(b*c)*b*x^4*weierstrassPInverse(-4*a/b, 0, x) + (5*b*x^2 - 3*a )*sqrt(b*x^2 + a)*sqrt(c*x))/(a^2*c^5*x^4)
Result contains complex when optimal does not.
Time = 21.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=\frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} c^{\frac {9}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} \] Input:
integrate(1/(c*x)**(9/2)/(b*x**2+a)**(1/2),x)
Output:
gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt( a)*c**(9/2)*x**(7/2)*gamma(-3/4))
\[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \left (c x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(9/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^2 + a)*(c*x)^(9/2)), x)
\[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \left (c x\right )^{\frac {9}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(9/2)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^2 + a)*(c*x)^(9/2)), x)
Timed out. \[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{9/2}\,\sqrt {b\,x^2+a}} \,d x \] Input:
int(1/((c*x)^(9/2)*(a + b*x^2)^(1/2)),x)
Output:
int(1/((c*x)^(9/2)*(a + b*x^2)^(1/2)), x)
\[ \int \frac {1}{(c x)^{9/2} \sqrt {a+b x^2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{7}+a \,x^{5}}d x \right )}{c^{5}} \] Input:
int(1/(c*x)^(9/2)/(b*x^2+a)^(1/2),x)
Output:
(sqrt(c)*int((sqrt(x)*sqrt(a + b*x**2))/(a*x**5 + b*x**7),x))/c**5