\(\int \frac {\sqrt {a+b x^2} (A+B x^2)}{\sqrt {e x}} \, dx\) [787]

Optimal result

Integrand size = 26, antiderivative size = 176 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {2 (7 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{21 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 b e}+\frac {2 a^{3/4} (7 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 b^{5/4} \sqrt {e} \sqrt {a+b x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 285, 335, 226} \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {2 a^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{21 b^{5/4} \sqrt {e} \sqrt {a+b x^2}}+\frac {2 \sqrt {e x} \sqrt {a+b x^2} (7 A b-a B)}{21 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 b e} \]

[In]

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

[Out]

(2*(7*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b*e) + (2*B*Sqrt[e*x]*(a + b*x^2)^(3/2))/(7*b*e) + (2*a^(3/4)*
(7*A*b - a*B)*(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])/(21*b^(5/4)*Sqrt[e]*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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 B \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 b e}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {a B}{2}\right )\right ) \int \frac {\sqrt {a+b x^2}}{\sqrt {e x}} \, dx}{7 b} \\ & = \frac {2 (7 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{21 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 b e}+\frac {(2 a (7 A b-a B)) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{21 b} \\ & = \frac {2 (7 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{21 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 b e}+\frac {(4 a (7 A b-a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b e} \\ & = \frac {2 (7 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{21 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{3/2}}{7 b e}+\frac {2 a^{3/4} (7 A b-a B) \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 )}{21 b^{5/4} \sqrt {e} \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 = 10.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (B \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}}+(7 A b-a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{7 b \sqrt {e x} \sqrt {1+\frac {b x^2}{a}}} \]

[In]

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

[Out]

(2*x*Sqrt[a + b*x^2]*(B*(a + b*x^2)*Sqrt[1 + (b*x^2)/a] + (7*A*b - a*B)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b
*x^2)/a)]))/(7*b*Sqrt[e*x]*Sqrt[1 + (b*x^2)/a])

Maple [A] (verified)

Time = 3.01 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.05

method	result	size
risch	\(\frac {2 \left (3 b B \,x^{2}+7 A b +2 B a \right ) x \sqrt {b \,x^{2}+a}}{21 b \sqrt {e x}}+\frac {2 a \left (7 A b -B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{21 b^{2} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(184\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,x^{2} \sqrt {b e \,x^{3}+a e x}}{7 e}+\frac {2 \left (A b +\frac {2 B a}{7}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}+\frac {\left (A a -\frac {a \left (A b +\frac {2 B a}{7}\right )}{3 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(207\)
default	\(\frac {\frac {2 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b}{3}-\frac {2 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}}{21}+\frac {2 b^{3} B \,x^{5}}{7}+\frac {2 A \,b^{3} x^{3}}{3}+\frac {10 B a \,b^{2} x^{3}}{21}+\frac {2 a \,b^{2} A x}{3}+\frac {4 a^{2} b B x}{21}}{\sqrt {b \,x^{2}+a}\, \sqrt {e x}\, b^{2}}\)	\(246\)

[In]

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

[Out]

2/21*(3*B*b*x^2+7*A*b+2*B*a)*x*(b*x^2+a)^(1/2)/b/(e*x)^(1/2)+2/21*a*(7*A*b-B*a)/b^2*(-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)*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))*((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 = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=-\frac {2 \, {\left (2 \, {\left (B a^{2} - 7 \, A a b\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (3 \, B b^{2} x^{2} + 2 \, B a b + 7 \, A b^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{21 \, b^{2} e} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {A \sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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