Integrand size = 31, antiderivative size = 88 \[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {24 \sqrt {e x} \sqrt {1-b^2 x^4}}{65 e}+\frac {2 \sqrt {e x} \left (1-b^2 x^4\right )^{3/2}}{13 e}+\frac {96 \sqrt {e x} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},\frac {1}{2},\frac {9}{8},b^2 x^4\right )}{65 e} \] Output:
24/65*(e*x)^(1/2)*(-b^2*x^4+1)^(1/2)/e+2/13*(e*x)^(1/2)*(-b^2*x^4+1)^(3/2) /e+96/65*(e*x)^(1/2)*hypergeom([1/8, 1/2],[9/8],b^2*x^4)/e
Time = 10.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.31 \[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{8},\frac {9}{8},b^2 x^4\right )}{\sqrt {e x}} \] Input:
Integrate[((1 - b*x^2)^(3/2)*(1 + b*x^2)^(3/2))/Sqrt[e*x],x]
Output:
(2*x*Hypergeometric2F1[-3/2, 1/8, 9/8, b^2*x^4])/Sqrt[e*x]
Leaf count is larger than twice the leaf count of optimal. \(506\) vs. \(2(88)=176\).
Time = 0.62 (sec) , antiderivative size = 506, normalized size of antiderivative = 5.75, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {335, 811, 811, 851, 767, 27, 2422}
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.
\(\displaystyle \int \frac {\left (1-b x^2\right )^{3/2} \left (b x^2+1\right )^{3/2}}{\sqrt {e x}} \, dx\) |
\(\Big \downarrow \) 335 |
\(\displaystyle \int \frac {\left (1-b^2 x^4\right )^{3/2}}{\sqrt {e x}}dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {12}{13} \int \frac {\sqrt {1-b^2 x^4}}{\sqrt {e x}}dx+\frac {2 \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}}{13 e}\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {12}{13} \left (\frac {4}{5} \int \frac {1}{\sqrt {e x} \sqrt {1-b^2 x^4}}dx+\frac {2 \sqrt {1-b^2 x^4} \sqrt {e x}}{5 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}}{13 e}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {12}{13} \left (\frac {8 \int \frac {1}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{5 e}+\frac {2 \sqrt {1-b^2 x^4} \sqrt {e x}}{5 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}}{13 e}\) |
\(\Big \downarrow \) 767 |
\(\displaystyle \frac {12}{13} \left (\frac {8 \left (\frac {1}{2} \int \frac {e-\sqrt [4]{-b^2} e x}{e \sqrt {1-b^2 x^4}}d\sqrt {e x}+\frac {1}{2} \int \frac {\sqrt [4]{-b^2} x e+e}{e \sqrt {1-b^2 x^4}}d\sqrt {e x}\right )}{5 e}+\frac {2 \sqrt {1-b^2 x^4} \sqrt {e x}}{5 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}}{13 e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {12}{13} \left (\frac {8 \left (\frac {\int \frac {e-\sqrt [4]{-b^2} e x}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 e}+\frac {\int \frac {\sqrt [4]{-b^2} x e+e}{\sqrt {1-b^2 x^4}}d\sqrt {e x}}{2 e}\right )}{5 e}+\frac {2 \sqrt {1-b^2 x^4} \sqrt {e x}}{5 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}}{13 e}\) |
\(\Big \downarrow \) 2422 |
\(\displaystyle \frac {12}{13} \left (\frac {8 \left (\frac {\sqrt [4]{-b^2} (e x)^{3/2} \sqrt {\frac {\left (\sqrt [4]{-b^2} e x+e\right )^2}{\sqrt [4]{-b^2} e^2 x}} \sqrt {-\frac {e^4-b^2 e^4 x^4}{\sqrt {-b^2} e^4 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {-\frac {\sqrt {2} \sqrt {-b^2} x^2 e^2-2 \sqrt [4]{-b^2} x e^2+\sqrt {2} e^2}{\sqrt [4]{-b^2} e^2 x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {1-b^2 x^4} \left (\sqrt [4]{-b^2} e x+e\right )}-\frac {\sqrt [4]{-b^2} (e x)^{3/2} \sqrt {-\frac {\left (e-\sqrt [4]{-b^2} e x\right )^2}{\sqrt [4]{-b^2} e^2 x}} \sqrt {-\frac {e^4-b^2 e^4 x^4}{\sqrt {-b^2} e^4 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {\sqrt {2} \sqrt {-b^2} x^2 e^2+2 \sqrt [4]{-b^2} x e^2+\sqrt {2} e^2}{\sqrt [4]{-b^2} e^2 x}}\right ),-2 \left (1-\sqrt {2}\right )\right )}{2 \sqrt {2+\sqrt {2}} \sqrt {1-b^2 x^4} \left (e-\sqrt [4]{-b^2} e x\right )}\right )}{5 e}+\frac {2 \sqrt {1-b^2 x^4} \sqrt {e x}}{5 e}\right )+\frac {2 \left (1-b^2 x^4\right )^{3/2} \sqrt {e x}}{13 e}\) |
Input:
Int[((1 - b*x^2)^(3/2)*(1 + b*x^2)^(3/2))/Sqrt[e*x],x]
Output:
(2*Sqrt[e*x]*(1 - b^2*x^4)^(3/2))/(13*e) + (12*((2*Sqrt[e*x]*Sqrt[1 - b^2* x^4])/(5*e) + (8*(((-b^2)^(1/4)*(e*x)^(3/2)*Sqrt[(e + (-b^2)^(1/4)*e*x)^2/ ((-b^2)^(1/4)*e^2*x)]*Sqrt[-((e^4 - b^2*e^4*x^4)/(Sqrt[-b^2]*e^4*x^2))]*El lipticF[ArcSin[Sqrt[-((Sqrt[2]*e^2 - 2*(-b^2)^(1/4)*e^2*x + Sqrt[2]*Sqrt[- b^2]*e^2*x^2)/((-b^2)^(1/4)*e^2*x))]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sq rt[2]]*(e + (-b^2)^(1/4)*e*x)*Sqrt[1 - b^2*x^4]) - ((-b^2)^(1/4)*(e*x)^(3/ 2)*Sqrt[-((e - (-b^2)^(1/4)*e*x)^2/((-b^2)^(1/4)*e^2*x))]*Sqrt[-((e^4 - b^ 2*e^4*x^4)/(Sqrt[-b^2]*e^4*x^2))]*EllipticF[ArcSin[Sqrt[(Sqrt[2]*e^2 + 2*( -b^2)^(1/4)*e^2*x + Sqrt[2]*Sqrt[-b^2]*e^2*x^2)/((-b^2)^(1/4)*e^2*x)]/2], -2*(1 - Sqrt[2])])/(2*Sqrt[2 + Sqrt[2]]*(e - (-b^2)^(1/4)*e*x)*Sqrt[1 - b^ 2*x^4])))/(5*e)))/13
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p _.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] ))
Int[1/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[1/2 Int[(1 - Rt[b/a, 4 ]*x^2)/Sqrt[a + b*x^8], x], x] + Simp[1/2 Int[(1 + Rt[b/a, 4]*x^2)/Sqrt[a + b*x^8], x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_) + (d_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^8], x_Symbol] :> Simp[(-c) *d*x^3*Sqrt[-(c - d*x^2)^2/(c*d*x^2)]*(Sqrt[(-d^2)*((a + b*x^8)/(b*c^2*x^4) )]/(Sqrt[2 + Sqrt[2]]*(c - d*x^2)*Sqrt[a + b*x^8]))*EllipticF[ArcSin[(1/2)* Sqrt[(Sqrt[2]*c^2 + 2*c*d*x^2 + Sqrt[2]*d^2*x^4)/(c*d*x^2)]], -2*(1 - Sqrt[ 2])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^4 - a*d^4, 0]
\[\int \frac {\left (-b \,x^{2}+1\right )^{\frac {3}{2}} \left (b \,x^{2}+1\right )^{\frac {3}{2}}}{\sqrt {e x}}d x\]
Input:
int((-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2)/(e*x)^(1/2),x)
Output:
int((-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2)/(e*x)^(1/2),x)
\[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + 1\right )}^{\frac {3}{2}} {\left (-b x^{2} + 1\right )}^{\frac {3}{2}}}{\sqrt {e x}} \,d x } \] Input:
integrate((-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2)/(e*x)^(1/2),x, algorithm="frica s")
Output:
integral(-(b^2*x^4 - 1)*sqrt(b*x^2 + 1)*sqrt(-b*x^2 + 1)*sqrt(e*x)/(e*x), x)
\[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\int \frac {\left (- b x^{2} + 1\right )^{\frac {3}{2}} \left (b x^{2} + 1\right )^{\frac {3}{2}}}{\sqrt {e x}}\, dx \] Input:
integrate((-b*x**2+1)**(3/2)*(b*x**2+1)**(3/2)/(e*x)**(1/2),x)
Output:
Integral((-b*x**2 + 1)**(3/2)*(b*x**2 + 1)**(3/2)/sqrt(e*x), x)
\[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + 1\right )}^{\frac {3}{2}} {\left (-b x^{2} + 1\right )}^{\frac {3}{2}}}{\sqrt {e x}} \,d x } \] Input:
integrate((-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2)/(e*x)^(1/2),x, algorithm="maxim a")
Output:
integrate((b*x^2 + 1)^(3/2)*(-b*x^2 + 1)^(3/2)/sqrt(e*x), x)
\[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\int { \frac {{\left (b x^{2} + 1\right )}^{\frac {3}{2}} {\left (-b x^{2} + 1\right )}^{\frac {3}{2}}}{\sqrt {e x}} \,d x } \] Input:
integrate((-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2)/(e*x)^(1/2),x, algorithm="giac" )
Output:
integrate((b*x^2 + 1)^(3/2)*(-b*x^2 + 1)^(3/2)/sqrt(e*x), x)
Timed out. \[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\int \frac {{\left (1-b\,x^2\right )}^{3/2}\,{\left (b\,x^2+1\right )}^{3/2}}{\sqrt {e\,x}} \,d x \] Input:
int(((1 - b*x^2)^(3/2)*(b*x^2 + 1)^(3/2))/(e*x)^(1/2),x)
Output:
int(((1 - b*x^2)^(3/2)*(b*x^2 + 1)^(3/2))/(e*x)^(1/2), x)
\[ \int \frac {\left (1-b x^2\right )^{3/2} \left (1+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx=\frac {2 \sqrt {e}\, \left (-5 \sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}\, b^{2} x^{4}+17 \sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}-24 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+1}\, \sqrt {-b \,x^{2}+1}}{b^{2} x^{5}-x}d x \right )\right )}{65 e} \] Input:
int((-b*x^2+1)^(3/2)*(b*x^2+1)^(3/2)/(e*x)^(1/2),x)
Output:
(2*sqrt(e)*( - 5*sqrt(x)*sqrt(b*x**2 + 1)*sqrt( - b*x**2 + 1)*b**2*x**4 + 17*sqrt(x)*sqrt(b*x**2 + 1)*sqrt( - b*x**2 + 1) - 24*int((sqrt(x)*sqrt(b*x **2 + 1)*sqrt( - b*x**2 + 1))/(b**2*x**5 - x),x)))/(65*e)