\(\int \frac {(a-b x^2)^{3/2} (a+b x^2)^{3/2}}{(e x)^{3/2}} \, dx\) [1366]

Optimal result

Integrand size = 31, antiderivative size = 156 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=-\frac {24 b^2 (e x)^{7/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{11 e^5}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}-\frac {96 a^2 b^2 (e x)^{7/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{8},\frac {15}{8},\frac {b^2 x^4}{a^2}\right )}{77 e^5 \sqrt {a-b x^2} \sqrt {a+b x^2}} \] Output:

-24/11*b^2*(e*x)^(7/2)*(-b*x^2+a)^(1/2)*(b*x^2+a)^(1/2)/e^5-2*(-b*x^2+a)^( 
3/2)*(b*x^2+a)^(3/2)/e/(e*x)^(1/2)-96/77*a^2*b^2*(e*x)^(7/2)*(1-b^2*x^4/a^ 
2)^(1/2)*hypergeom([1/2, 7/8],[15/8],b^2*x^4/a^2)/e^5/(-b*x^2+a)^(1/2)/(b* 
x^2+a)^(1/2)
 

Mathematica [A] (verified)

Time = 8.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=-\frac {2 a^2 x \sqrt {a-b x^2} \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {1}{8};\frac {7}{8};\frac {b^2 x^4}{a^2}\right )}{(e x)^{3/2} \sqrt {1-\frac {b^2 x^4}{a^2}}} \] Input:

Integrate[((a - b*x^2)^(3/2)*(a + b*x^2)^(3/2))/(e*x)^(3/2),x]
 

Output:

(-2*a^2*x*Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*HypergeometricPFQ[{-3/2, -1/8}, 
{7/8}, (b^2*x^4)/a^2])/((e*x)^(3/2)*Sqrt[1 - (b^2*x^4)/a^2])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {339, 340, 344, 851, 889, 888}

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.

\(\displaystyle \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 339

\(\displaystyle -\frac {12 b^2 \int (e x)^{5/2} \sqrt {a-b x^2} \sqrt {b x^2+a}dx}{e^4}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}\)

\(\Big \downarrow \) 340

\(\displaystyle -\frac {12 b^2 \left (\frac {4}{11} a^2 \int \frac {(e x)^{5/2}}{\sqrt {a-b x^2} \sqrt {b x^2+a}}dx+\frac {2 (e x)^{7/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{11 e}\right )}{e^4}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}\)

\(\Big \downarrow \) 344

\(\displaystyle -\frac {12 b^2 \left (\frac {4 a^2 \sqrt {a^2-b^2 x^4} \int \frac {(e x)^{5/2}}{\sqrt {a^2-b^2 x^4}}dx}{11 \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {2 (e x)^{7/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{11 e}\right )}{e^4}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}\)

\(\Big \downarrow \) 851

\(\displaystyle -\frac {12 b^2 \left (\frac {8 a^2 \sqrt {a^2-b^2 x^4} \int \frac {e^3 x^3}{\sqrt {a^2-b^2 x^4}}d\sqrt {e x}}{11 e \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {2 (e x)^{7/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{11 e}\right )}{e^4}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}\)

\(\Big \downarrow \) 889

\(\displaystyle -\frac {12 b^2 \left (\frac {8 a^2 \sqrt {1-\frac {b^2 x^4}{a^2}} \int \frac {e^3 x^3}{\sqrt {1-\frac {b^2 x^4}{a^2}}}d\sqrt {e x}}{11 e \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {2 (e x)^{7/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{11 e}\right )}{e^4}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}\)

\(\Big \downarrow \) 888

\(\displaystyle -\frac {12 b^2 \left (\frac {8 a^2 (e x)^{7/2} \sqrt {1-\frac {b^2 x^4}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{8},\frac {15}{8},\frac {b^2 x^4}{a^2}\right )}{77 e \sqrt {a-b x^2} \sqrt {a+b x^2}}+\frac {2 (e x)^{7/2} \sqrt {a-b x^2} \sqrt {a+b x^2}}{11 e}\right )}{e^4}-\frac {2 \left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{e \sqrt {e x}}\)

Input:

Int[((a - b*x^2)^(3/2)*(a + b*x^2)^(3/2))/(e*x)^(3/2),x]
 

Output:

(-2*(a - b*x^2)^(3/2)*(a + b*x^2)^(3/2))/(e*Sqrt[e*x]) - (12*b^2*((2*(e*x) 
^(7/2)*Sqrt[a - b*x^2]*Sqrt[a + b*x^2])/(11*e) + (8*a^2*(e*x)^(7/2)*Sqrt[1 
 - (b^2*x^4)/a^2]*Hypergeometric2F1[1/2, 7/8, 15/8, (b^2*x^4)/a^2])/(77*e* 
Sqrt[a - b*x^2]*Sqrt[a + b*x^2])))/e^4
 

Defintions of rubi rules used

rule 339

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^p*((c + d*x^2)^p/(e*(m + 1))) 
, x] - Simp[4*b*d*(p/(e^4*(m + 1)))   Int[(e*x)^(m + 4)*(a + b*x^2)^(p - 1) 
*(c + d*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c + a 
*d, 0] && GtQ[p, 0] && LtQ[m, -1]
 

rule 340

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^p*((c + d*x^2)^p/(e*(m + 4*p 
 + 1))), x] + Simp[4*a*c*(p/(m + 4*p + 1))   Int[(e*x)^m*(a + b*x^2)^(p - 1 
)*(c + d*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c + 
a*d, 0] && GtQ[p, 0] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*m]
 

rule 344

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c 
+ b*d*x^4)^FracPart[p])   Int[(e*x)^m*(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a 
, b, c, d, e, m, p}, x] && EqQ[b*c + a*d, 0] &&  !IntegerQ[p]
 

rule 851

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]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

Maple [F]

\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (e x \right )^{\frac {3}{2}}}d x\]

Input:

int((-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/(e*x)^(3/2),x)
 

Output:

int((-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/(e*x)^(3/2),x)
 

Fricas [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/(e*x)^(3/2),x, algorithm="frica 
s")
 

Output:

integral(-(b^2*x^4 - a^2)*sqrt(b*x^2 + a)*sqrt(-b*x^2 + a)*sqrt(e*x)/(e^2* 
x^2), x)
 

Sympy [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((-b*x**2+a)**(3/2)*(b*x**2+a)**(3/2)/(e*x)**(3/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(a + b*x**2)**(3/2)/(e*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/(e*x)^(3/2),x, algorithm="maxim 
a")
 

Output:

integrate((b*x^2 + a)^(3/2)*(-b*x^2 + a)^(3/2)/(e*x)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/(e*x)^(3/2),x, algorithm="giac" 
)
 

Output:

integrate((b*x^2 + a)^(3/2)*(-b*x^2 + a)^(3/2)/(e*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (a-b\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(a - b*x^2)^(3/2))/(e*x)^(3/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(a - b*x^2)^(3/2))/(e*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (a-b x^2\right )^{3/2} \left (a+b x^2\right )^{3/2}}{(e x)^{3/2}} \, dx=\frac {2 \sqrt {e}\, \left (5 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, a^{2}-\sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, b^{2} x^{4}+8 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}}{-b^{2} x^{6}+a^{2} x^{2}}d x \right ) a^{4}\right )}{11 \sqrt {x}\, e^{2}} \] Input:

int((-b*x^2+a)^(3/2)*(b*x^2+a)^(3/2)/(e*x)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(2*sqrt(e)*(5*sqrt(a + b*x**2)*sqrt(a - b*x**2)*a**2 - sqrt(a + b*x**2)*sq 
rt(a - b*x**2)*b**2*x**4 + 8*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2)*sqrt(a 
- b*x**2))/(a**2*x**2 - b**2*x**6),x)*a**4))/(11*sqrt(x)*e**2)