\(\int (e x)^{3/2} \sqrt {a+b x^2} (A+B x^2) \, dx\) [785]

Optimal result

Integrand size = 26, antiderivative size = 212 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {2 a^{7/4} (11 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}} \]

[Out]

2/11*B*(e*x)^(5/2)*(b*x^2+a)^(3/2)/b/e+2/77*(11*A*b-5*B*a)*(e*x)^(5/2)*(b*x^2+a)^(1/2)/b/e+4/231*a*(11*A*b-5*B
*a)*e*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^2-2/231*a^(7/4)*(11*A*b-5*B*a)*e^(3/2)*(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^(9/4)/(
b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {470, 285, 327, 335, 226} \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=-\frac {2 a^{7/4} e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (11 A b-5 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}}+\frac {4 a e \sqrt {e x} \sqrt {a+b x^2} (11 A b-5 a B)}{231 b^2}+\frac {2 (e x)^{5/2} \sqrt {a+b x^2} (11 A b-5 a B)}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e} \]

[In]

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

[Out]

(4*a*(11*A*b - 5*a*B)*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(231*b^2) + (2*(11*A*b - 5*a*B)*(e*x)^(5/2)*Sqrt[a + b*x^2]
)/(77*b*e) + (2*B*(e*x)^(5/2)*(a + b*x^2)^(3/2))/(11*b*e) - (2*a^(7/4)*(11*A*b - 5*a*B)*e^(3/2)*(Sqrt[a] + Sqr
t[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])/(231*b^(9/4)*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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && 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 (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {\left (2 \left (-\frac {11 A b}{2}+\frac {5 a B}{2}\right )\right ) \int (e x)^{3/2} \sqrt {a+b x^2} \, dx}{11 b} \\ & = \frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}+\frac {(2 a (11 A b-5 a B)) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{77 b} \\ & = \frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {\left (2 a^2 (11 A b-5 a B) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{231 b^2} \\ & = \frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {\left (4 a^2 (11 A b-5 a B) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{231 b^2} \\ & = \frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {2 a^{7/4} (11 A b-5 a B) e^{3/2} \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 )}{231 b^{9/4} \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 = 10.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.52 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 e \sqrt {e x} \sqrt {a+b x^2} \left (-\left (\left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (-11 A b+5 a B-7 b B x^2\right )\right )+a (-11 A b+5 a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{77 b^2 \sqrt {1+\frac {b x^2}{a}}} \]

[In]

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

[Out]

(2*e*Sqrt[e*x]*Sqrt[a + b*x^2]*(-((a + b*x^2)*Sqrt[1 + (b*x^2)/a]*(-11*A*b + 5*a*B - 7*b*B*x^2)) + a*(-11*A*b
+ 5*a*B)*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)]))/(77*b^2*Sqrt[1 + (b*x^2)/a])

Maple [A] (verified)

Time = 3.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.01

method	result	size
risch	\(\frac {2 \left (21 b^{2} B \,x^{4}+33 A \,b^{2} x^{2}+6 B a b \,x^{2}+22 a b A -10 a^{2} B \right ) x \sqrt {b \,x^{2}+a}\, e^{2}}{231 b^{2} \sqrt {e x}}-\frac {2 a^{2} \left (11 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{231 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(214\)
default	\(-\frac {2 e \sqrt {e x}\, \left (-21 x^{7} B \,b^{4}+11 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2} b -33 A \,b^{4} x^{5}-5 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{3}-27 x^{5} B a \,b^{3}-55 A a \,b^{3} x^{3}+4 B \,a^{2} b^{2} x^{3}-22 A x \,a^{2} b^{2}+10 B x \,a^{3} b \right )}{231 x \sqrt {b \,x^{2}+a}\, b^{3}}\)	\(276\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B e \,x^{4} \sqrt {b e \,x^{3}+a e x}}{11}+\frac {2 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b e}+\frac {2 \left (A a \,e^{2}-\frac {5 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) a}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}-\frac {\left (A a \,e^{2}-\frac {5 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) a}{7 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\)	\(294\)

[In]

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

[Out]

2/231*(21*B*b^2*x^4+33*A*b^2*x^2+6*B*a*b*x^2+22*A*a*b-10*B*a^2)*x*(b*x^2+a)^(1/2)/b^2*e^2/(e*x)^(1/2)-2/231*a^
2*(11*A*b-5*B*a)/b^3*(-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))*e^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 = 101, normalized size of antiderivative = 0.48 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 \, {\left (2 \, {\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} \sqrt {b e} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (21 \, B b^{3} e x^{4} + 3 \, {\left (2 \, B a b^{2} + 11 \, A b^{3}\right )} e x^{2} - 2 \, {\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{231 \, b^{3}} \]

[In]

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

[Out]

2/231*(2*(5*B*a^3 - 11*A*a^2*b)*sqrt(b*e)*e*weierstrassPInverse(-4*a/b, 0, x) + (21*B*b^3*e*x^4 + 3*(2*B*a*b^2
+ 11*A*b^3)*e*x^2 - 2*(5*B*a^2*b - 11*A*a*b^2)*e)*sqrt(b*x^2 + a)*sqrt(e*x))/b^3

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.56 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.46 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {A \sqrt {a} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} \]

[In]

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

[Out]

A*sqrt(a)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(9/4)) + B
*sqrt(a)*e**(3/2)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(13/4))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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