\(\int \sqrt {e x} (a+b x^2)^{3/2} (A+B x^2) \, dx\) [795]

Optimal result

Integrand size = 26, antiderivative size = 377 \[ \int \sqrt {e x} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {4 a (13 A b-3 a B) (e x)^{3/2} \sqrt {a+b x^2}}{195 b e}+\frac {8 a^2 (13 A b-3 a B) \sqrt {e x} \sqrt {a+b x^2}}{195 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}-\frac {8 a^{9/4} (13 A b-3 a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 b^{7/4} \sqrt {a+b x^2}}+\frac {4 a^{9/4} (13 A b-3 a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 b^{7/4} \sqrt {a+b x^2}} \]

[Out]

2/117*(13*A*b-3*B*a)*(e*x)^(3/2)*(b*x^2+a)^(3/2)/b/e+2/13*B*(e*x)^(3/2)*(b*x^2+a)^(5/2)/b/e+4/195*a*(13*A*b-3*
B*a)*(e*x)^(3/2)*(b*x^2+a)^(1/2)/b/e+8/195*a^2*(13*A*b-3*B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^(3/2)/(a^(1/2)+x*b
^(1/2))-8/195*a^(9/4)*(13*A*b-3*B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan
(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2
))*(a^(1/2)+x*b^(1/2))*e^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(7/4)/(b*x^2+a)^(1/2)+4/195*a^(9/4)*(
13*A*b-3*B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^
(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*
e^(1/2)*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(7/4)/(b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {470, 285, 335, 311, 226, 1210} \[ \int \sqrt {e x} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {4 a^{9/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (13 A b-3 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 b^{7/4} \sqrt {a+b x^2}}-\frac {8 a^{9/4} \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (13 A b-3 a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 b^{7/4} \sqrt {a+b x^2}}+\frac {8 a^2 \sqrt {e x} \sqrt {a+b x^2} (13 A b-3 a B)}{195 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2} (13 A b-3 a B)}{117 b e}+\frac {4 a (e x)^{3/2} \sqrt {a+b x^2} (13 A b-3 a B)}{195 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e} \]

[In]

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

[Out]

(4*a*(13*A*b - 3*a*B)*(e*x)^(3/2)*Sqrt[a + b*x^2])/(195*b*e) + (8*a^2*(13*A*b - 3*a*B)*Sqrt[e*x]*Sqrt[a + b*x^
2])/(195*b^(3/2)*(Sqrt[a] + Sqrt[b]*x)) + (2*(13*A*b - 3*a*B)*(e*x)^(3/2)*(a + b*x^2)^(3/2))/(117*b*e) + (2*B*
(e*x)^(3/2)*(a + b*x^2)^(5/2))/(13*b*e) - (8*a^(9/4)*(13*A*b - 3*a*B)*Sqrt[e]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a +
b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(195*b^(7/4)*
Sqrt[a + b*x^2]) + (4*a^(9/4)*(13*A*b - 3*a*B)*Sqrt[e]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[
b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(195*b^(7/4)*Sqrt[a + b*x^2])

Rule 226

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]

Rule 285

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] + Dist[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]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[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 470

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

Rule 1210

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
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}-\frac {\left (2 \left (-\frac {13 A b}{2}+\frac {3 a B}{2}\right )\right ) \int \sqrt {e x} \left (a+b x^2\right )^{3/2} \, dx}{13 b} \\ & = \frac {2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac {(2 a (13 A b-3 a B)) \int \sqrt {e x} \sqrt {a+b x^2} \, dx}{39 b} \\ & = \frac {4 a (13 A b-3 a B) (e x)^{3/2} \sqrt {a+b x^2}}{195 b e}+\frac {2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac {\left (4 a^2 (13 A b-3 a B)\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{195 b} \\ & = \frac {4 a (13 A b-3 a B) (e x)^{3/2} \sqrt {a+b x^2}}{195 b e}+\frac {2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac {\left (8 a^2 (13 A b-3 a B)\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 b e} \\ & = \frac {4 a (13 A b-3 a B) (e x)^{3/2} \sqrt {a+b x^2}}{195 b e}+\frac {2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}+\frac {\left (8 a^{5/2} (13 A b-3 a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 b^{3/2}}-\frac {\left (8 a^{5/2} (13 A b-3 a B)\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} e}}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 b^{3/2}} \\ & = \frac {4 a (13 A b-3 a B) (e x)^{3/2} \sqrt {a+b x^2}}{195 b e}+\frac {8 a^2 (13 A b-3 a B) \sqrt {e x} \sqrt {a+b x^2}}{195 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (13 A b-3 a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{117 b e}+\frac {2 B (e x)^{3/2} \left (a+b x^2\right )^{5/2}}{13 b e}-\frac {8 a^{9/4} (13 A b-3 a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 b^{7/4} \sqrt {a+b x^2}}+\frac {4 a^{9/4} (13 A b-3 a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 b^{7/4} \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.26 \[ \int \sqrt {e x} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {2 x \sqrt {e x} \sqrt {a+b x^2} \left (3 B \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}+a (13 A b-3 a B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{39 b \sqrt {1+\frac {b x^2}{a}}} \]

[In]

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

[Out]

(2*x*Sqrt[e*x]*Sqrt[a + b*x^2]*(3*B*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] + a*(13*A*b - 3*a*B)*Hypergeometric2F1[-
3/2, 3/4, 7/4, -((b*x^2)/a)]))/(39*b*Sqrt[1 + (b*x^2)/a])

Maple [A] (verified)

Time = 3.06 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {2 x^{2} \left (45 b^{2} B \,x^{4}+65 A \,b^{2} x^{2}+75 B a b \,x^{2}+143 a b A +12 a^{2} B \right ) \sqrt {b \,x^{2}+a}\, e}{585 b \sqrt {e x}}+\frac {4 a^{2} \left (13 A b -3 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) e \sqrt {\left (b \,x^{2}+a \right ) e x}}{195 b^{2} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(262\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B b \,x^{5} \sqrt {b e \,x^{3}+a e x}}{13}+\frac {2 \left (b \left (A b +2 B a \right ) e -\frac {11 B a b e}{13}\right ) x^{3} \sqrt {b e \,x^{3}+a e x}}{9 b e}+\frac {2 \left (a \left (2 A b +B a \right ) e -\frac {7 \left (b \left (A b +2 B a \right ) e -\frac {11 B a b e}{13}\right ) a}{9 b}\right ) x \sqrt {b e \,x^{3}+a e x}}{5 b e}+\frac {\left (a^{2} A e -\frac {3 \left (a \left (2 A b +B a \right ) e -\frac {7 \left (b \left (A b +2 B a \right ) e -\frac {11 B a b e}{13}\right ) a}{9 b}\right ) a}{5 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\)	\(363\)
default	\(\frac {2 \sqrt {e x}\, \left (45 B \,x^{8} b^{4}+65 A \,x^{6} b^{4}+120 B \,x^{6} a \,b^{3}+156 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3} b -78 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3} b -36 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{4}+18 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{4}+208 A a \,b^{3} x^{4}+87 B \,a^{2} b^{2} x^{4}+143 A \,a^{2} b^{2} x^{2}+12 B \,a^{3} b \,x^{2}\right )}{585 \sqrt {b \,x^{2}+a}\, b^{2} x}\)	\(438\)

[In]

int((b*x^2+a)^(3/2)*(B*x^2+A)*(e*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/585/b*x^2*(45*B*b^2*x^4+65*A*b^2*x^2+75*B*a*b*x^2+143*A*a*b+12*B*a^2)*(b*x^2+a)^(1/2)*e/(e*x)^(1/2)+4/195*a^
2/b^2*(13*A*b-3*B*a)*(-a*b)^(1/2)*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2
)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*e*x^3+a*e*x)^(1/2)*(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-
a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^(1/2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2
)))*e*((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.28 \[ \int \sqrt {e x} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {2 \, {\left (12 \, {\left (3 \, B a^{3} - 13 \, A a^{2} b\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (45 \, B b^{3} x^{5} + 5 \, {\left (15 \, B a b^{2} + 13 \, A b^{3}\right )} x^{3} + {\left (12 \, B a^{2} b + 143 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{585 \, b^{2}} \]

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)*(e*x)^(1/2),x, algorithm="fricas")

[Out]

2/585*(12*(3*B*a^3 - 13*A*a^2*b)*sqrt(b*e)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + (45
*B*b^3*x^5 + 5*(15*B*a*b^2 + 13*A*b^3)*x^3 + (12*B*a^2*b + 143*A*a*b^2)*x)*sqrt(b*x^2 + a)*sqrt(e*x))/b^2

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.74 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.53 \[ \int \sqrt {e x} \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx=\frac {A a^{\frac {3}{2}} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {A \sqrt {a} b \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {B a^{\frac {3}{2}} \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {B \sqrt {a} b \sqrt {e} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} \]

[In]

integrate((b*x**2+a)**(3/2)*(B*x**2+A)*(e*x)**(1/2),x)

[Out]

A*a**(3/2)*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(7/4)) + A
*sqrt(a)*b*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(11/4)) +
B*a**(3/2)*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(11/4))
+ B*sqrt(a)*b*sqrt(e)*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(15
/4))

Maxima [F]

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

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)*(e*x)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((b*x^2+a)^(3/2)*(B*x^2+A)*(e*x)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((A + B*x^2)*(e*x)^(1/2)*(a + b*x^2)^(3/2),x)

[Out]

int((A + B*x^2)*(e*x)^(1/2)*(a + b*x^2)^(3/2), x)