\(\int \frac {(a+b x^2)^{3/2} (A+B x^2)}{\sqrt {e x}} \, dx\) [796]

Optimal result

Integrand size = 26, antiderivative size = 214 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {4 a (11 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{77 b e}+\frac {2 (11 A b-a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac {4 a^{7/4} (11 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{77 b^{5/4} \sqrt {e} \sqrt {a+b x^2}} \]

[Out]

2/77*(11*A*b-B*a)*(b*x^2+a)^(3/2)*(e*x)^(1/2)/b/e+2/11*B*(b*x^2+a)^(5/2)*(e*x)^(1/2)/b/e+4/77*a*(11*A*b-B*a)*(
e*x)^(1/2)*(b*x^2+a)^(1/2)/b/e+4/77*a^(7/4)*(11*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 = 0.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {470, 285, 335, 226} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {4 a^{7/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (11 A b-a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{77 b^{5/4} \sqrt {e} \sqrt {a+b x^2}}+\frac {2 \sqrt {e x} \left (a+b x^2\right )^{3/2} (11 A b-a B)}{77 b e}+\frac {4 a \sqrt {e x} \sqrt {a+b x^2} (11 A b-a B)}{77 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{5/2}}{11 b e} \]

[In]

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

[Out]

(4*a*(11*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(77*b*e) + (2*(11*A*b - a*B)*Sqrt[e*x]*(a + b*x^2)^(3/2))/(77*b
*e) + (2*B*Sqrt[e*x]*(a + b*x^2)^(5/2))/(11*b*e) + (4*a^(7/4)*(11*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])/(77*b^(5/4)*Sq
rt[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 )^{5/2}}{11 b e}-\frac {\left (2 \left (-\frac {11 A b}{2}+\frac {a B}{2}\right )\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {e x}} \, dx}{11 b} \\ & = \frac {2 (11 A b-a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac {(6 a (11 A b-a B)) \int \frac {\sqrt {a+b x^2}}{\sqrt {e x}} \, dx}{77 b} \\ & = \frac {4 a (11 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{77 b e}+\frac {2 (11 A b-a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac {\left (4 a^2 (11 A b-a B)\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{77 b} \\ & = \frac {4 a (11 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{77 b e}+\frac {2 (11 A b-a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac {\left (8 a^2 (11 A b-a B)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{77 b e} \\ & = \frac {4 a (11 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{77 b e}+\frac {2 (11 A b-a B) \sqrt {e x} \left (a+b x^2\right )^{3/2}}{77 b e}+\frac {2 B \sqrt {e x} \left (a+b x^2\right )^{5/2}}{11 b e}+\frac {4 a^{7/4} (11 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 )}{77 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 = 10.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.45 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (B \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}}+a (11 A b-a B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{11 b \sqrt {e x} \sqrt {1+\frac {b x^2}{a}}} \]

[In]

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

[Out]

(2*x*Sqrt[a + b*x^2]*(B*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a] + a*(11*A*b - a*B)*Hypergeometric2F1[-3/2, 1/4, 5/4,
-((b*x^2)/a)]))/(11*b*Sqrt[e*x]*Sqrt[1 + (b*x^2)/a])

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {2 \left (7 b^{2} B \,x^{4}+11 A \,b^{2} x^{2}+13 B a b \,x^{2}+33 a b A +4 a^{2} B \right ) x \sqrt {b \,x^{2}+a}}{77 b \sqrt {e x}}+\frac {4 a^{2} \left (11 A b -B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{77 b^{2} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(208\)
default	\(\frac {\frac {2 x^{7} B \,b^{4}}{11}+\frac {4 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2} b}{7}+\frac {2 A \,b^{4} x^{5}}{7}-\frac {4 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{3}}{77}+\frac {40 x^{5} B a \,b^{3}}{77}+\frac {8 A a \,b^{3} x^{3}}{7}+\frac {34 B \,a^{2} b^{2} x^{3}}{77}+\frac {6 A x \,a^{2} b^{2}}{7}+\frac {8 B x \,a^{3} b}{77}}{\sqrt {b \,x^{2}+a}\, b^{2} \sqrt {e x}}\)	\(272\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B b \,x^{4} \sqrt {b e \,x^{3}+a e x}}{11 e}+\frac {2 \left (b^{2} A +\frac {13}{11} a b B \right ) x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b e}+\frac {2 \left (2 a b A +a^{2} B -\frac {5 a \left (b^{2} A +\frac {13}{11} a b B \right )}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}+\frac {\left (a^{2} A -\frac {a \left (2 a b A +a^{2} B -\frac {5 a \left (b^{2} A +\frac {13}{11} a b B \right )}{7 b}\right )}{3 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(285\)

[In]

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

[Out]

2/77/b*(7*B*b^2*x^4+11*A*b^2*x^2+13*B*a*b*x^2+33*A*a*b+4*B*a^2)*x*(b*x^2+a)^(1/2)/(e*x)^(1/2)+4/77*a^2/b^2*(11
*A*b-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)*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 = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.45 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=-\frac {2 \, {\left (4 \, {\left (B a^{3} - 11 \, A a^{2} b\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (7 \, B b^{3} x^{4} + 4 \, B a^{2} b + 33 \, A a b^{2} + {\left (13 \, B a b^{2} + 11 \, A b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{77 \, b^{2} e} \]

[In]

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

[Out]

-2/77*(4*(B*a^3 - 11*A*a^2*b)*sqrt(b*e)*weierstrassPInverse(-4*a/b, 0, x) - (7*B*b^3*x^4 + 4*B*a^2*b + 33*A*a*
b^2 + (13*B*a*b^2 + 11*A*b^3)*x^2)*sqrt(b*x^2 + a)*sqrt(e*x))/(b^2*e)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.42 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{\sqrt {e x}} \, dx=\frac {A a^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {5}{4}\right )} + \frac {A \sqrt {a} b x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {B a^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {e} \Gamma \left (\frac {13}{4}\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[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]

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

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

[In]

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

[Out]

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