\(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{(e x)^{5/2}} \, dx\) [301]

Optimal result

Integrand size = 26, antiderivative size = 210 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\frac {4 (7 A b+3 a B) \sqrt {e x} \sqrt {a+b x^2}}{21 e^3}+\frac {2 (7 A b+3 a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{21 a e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}+\frac {4 a^{3/4} (7 A b+3 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 \sqrt [4]{b} e^{5/2} \sqrt {a+b x^2}} \] Output:

4/21*(7*A*b+3*B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/e^3+2/21*(7*A*b+3*B*a)*(e*x 
)^(1/2)*(b*x^2+a)^(3/2)/a/e^3-2/3*A*(b*x^2+a)^(5/2)/a/e/(e*x)^(3/2)+4/21*a 
^(3/4)*(7*A*b+3*B*a)*(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)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2 
^(1/2))/b^(1/4)/e^(5/2)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.40 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (-\frac {A \left (a+b x^2\right )^2}{a}+\frac {(7 A b+3 a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}}\right )}{3 (e x)^{5/2}} \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(5/2),x]
 

Output:

(2*x*Sqrt[a + b*x^2]*(-((A*(a + b*x^2)^2)/a) + ((7*A*b + 3*a*B)*x^2*Hyperg 
eometric2F1[-3/2, 1/4, 5/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(3*(e*x)^ 
(5/2))
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {359, 248, 248, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 359

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

\(\Big \downarrow \) 248

\(\displaystyle \frac {(3 a B+7 A b) \left (\frac {6}{7} a \int \frac {\sqrt {b x^2+a}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 e}\right )}{3 a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {(3 a B+7 A b) \left (\frac {6}{7} a \left (\frac {2}{3} a \int \frac {1}{\sqrt {e x} \sqrt {b x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+b x^2}}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 e}\right )}{3 a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {(3 a B+7 A b) \left (\frac {6}{7} a \left (\frac {4 a \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{3 e}+\frac {2 \sqrt {e x} \sqrt {a+b x^2}}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 e}\right )}{3 a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {(3 a B+7 A b) \left (\frac {6}{7} a \left (\frac {2 a^{3/4} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} e^{3/2} \sqrt {a+b x^2}}+\frac {2 \sqrt {e x} \sqrt {a+b x^2}}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 e}\right )}{3 a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{3 a e (e x)^{3/2}}\)

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(5/2),x]
 

Output:

(-2*A*(a + b*x^2)^(5/2))/(3*a*e*(e*x)^(3/2)) + ((7*A*b + 3*a*B)*((2*Sqrt[e 
*x]*(a + b*x^2)^(3/2))/(7*e) + (6*a*((2*Sqrt[e*x]*Sqrt[a + b*x^2])/(3*e) + 
 (2*a^(3/4)*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e 
+ Sqrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e]) 
], 1/2])/(3*b^(1/4)*e^(3/2)*Sqrt[a + b*x^2])))/7))/(3*a*e^2)
 

Defintions of rubi rules used

rule 248

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

rule 266

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]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 761

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]
 

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-3 b B \,x^{4}-7 A b \,x^{2}-9 B a \,x^{2}+7 A a \right )}{21 x \,e^{2} \sqrt {e x}}+\frac {4 a \left (7 A b +3 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{21 b \sqrt {b e \,x^{3}+a e x}\, e^{2} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(199\)
default	\(\frac {\frac {4 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a b}\, a b x}{3}+\frac {4 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a b}\, a^{2} x}{7}+\frac {2 b^{3} B \,x^{6}}{7}+\frac {2 A \,x^{4} b^{3}}{3}+\frac {8 B \,x^{4} a \,b^{2}}{7}+\frac {6 B \,a^{2} b \,x^{2}}{7}-\frac {2 a^{2} b A}{3}}{\sqrt {b \,x^{2}+a}\, x b \,e^{2} \sqrt {e x}}\)	\(255\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 a A \sqrt {b e \,x^{3}+a e x}}{3 e^{3} x^{2}}+\frac {2 B b \,x^{2} \sqrt {b e \,x^{3}+a e x}}{7 e^{3}}+\frac {2 \left (\frac {b \left (A b +2 B a \right )}{e^{2}}-\frac {5 B b a}{7 e^{2}}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}+\frac {\left (\frac {a \left (2 A b +B a \right )}{e^{2}}-\frac {b a A}{3 e^{2}}-\frac {\left (\frac {b \left (A b +2 B a \right )}{e^{2}}-\frac {5 B b a}{7 e^{2}}\right ) a}{3 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(277\)

Input:

int((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/21*(b*x^2+a)^(1/2)*(-3*B*b*x^4-7*A*b*x^2-9*B*a*x^2+7*A*a)/x/e^2/(e*x)^( 
1/2)+4/21*a*(7*A*b+3*B*a)/b*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1 
/2))^(1/2)*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-b/(-a*b)^(1/2) 
*x)^(1/2)/(b*e*x^3+a*e*x)^(1/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^( 
1/2))^(1/2),1/2*2^(1/2))/e^2*((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^( 
1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\frac {2 \, {\left (4 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} \sqrt {b e} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, B b^{2} x^{4} - 7 \, A a b + {\left (9 \, B a b + 7 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{21 \, b e^{3} x^{2}} \] Input:

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(5/2),x, algorithm="fricas")
 

Output:

2/21*(4*(3*B*a^2 + 7*A*a*b)*sqrt(b*e)*x^2*weierstrassPInverse(-4*a/b, 0, x 
) + (3*B*b^2*x^4 - 7*A*a*b + (9*B*a*b + 7*A*b^2)*x^2)*sqrt(b*x^2 + a)*sqrt 
(e*x))/(b*e^3*x^2)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {A \sqrt {a} b \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((b*x**2+a)**(3/2)*(B*x**2+A)/(e*x)**(5/2),x)
 

Output:

A*a**(3/2)*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), b*x**2*exp_polar(I*pi)/ 
a)/(2*e**(5/2)*x**(3/2)*gamma(1/4)) + A*sqrt(a)*b*sqrt(x)*gamma(1/4)*hyper 
((-1/2, 1/4), (5/4,), b*x**2*exp_polar(I*pi)/a)/(2*e**(5/2)*gamma(5/4)) + 
B*a**(3/2)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**2*exp_polar( 
I*pi)/a)/(2*e**(5/2)*gamma(5/4)) + B*sqrt(a)*b*x**(5/2)*gamma(5/4)*hyper(( 
-1/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*e**(5/2)*gamma(9/4))
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/(e*x)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/(e*x)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}} \,d x \] Input:

int(((A + B*x^2)*(a + b*x^2)^(3/2))/(e*x)^(5/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2)^(3/2))/(e*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{5/2}} \, dx=\frac {2 \sqrt {e}\, \left (-47 \sqrt {b \,x^{2}+a}\, a^{2}+16 \sqrt {b \,x^{2}+a}\, a b \,x^{2}+3 \sqrt {b \,x^{2}+a}\, b^{2} x^{4}-60 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{5}+a \,x^{3}}d x \right ) a^{3} x \right )}{21 \sqrt {x}\, e^{3} x} \] Input:

int((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(5/2),x)
 

Output:

(2*sqrt(e)*( - 47*sqrt(a + b*x**2)*a**2 + 16*sqrt(a + b*x**2)*a*b*x**2 + 3 
*sqrt(a + b*x**2)*b**2*x**4 - 60*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2))/(a 
*x**3 + b*x**5),x)*a**3*x))/(21*sqrt(x)*e**3*x)