\(\int \frac {(e x)^{7/2} (A+B x^2)}{(a+b x^2)^{3/2}} \, dx\) [807]

Optimal result

Integrand size = 26, antiderivative size = 211 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {5 a^{3/4} (7 A b-9 a B) e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 b^{13/4} \sqrt {a+b x^2}} \]

[Out]

-1/7*(7*A*b-9*B*a)*e*(e*x)^(5/2)/b^2/(b*x^2+a)^(1/2)+2/7*B*(e*x)^(9/2)/b/e/(b*x^2+a)^(1/2)+5/21*(7*A*b-9*B*a)*
e^3*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^3-5/42*a^(3/4)*(7*A*b-9*B*a)*e^(7/2)*(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^(13/4)/(b*x
^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 211, 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, 294, 327, 335, 226} \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {5 a^{3/4} e^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (7 A b-9 a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 b^{13/4} \sqrt {a+b x^2}}+\frac {5 e^3 \sqrt {e x} \sqrt {a+b x^2} (7 A b-9 a B)}{21 b^3}-\frac {e (e x)^{5/2} (7 A b-9 a B)}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}} \]

[In]

Int[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

-1/7*((7*A*b - 9*a*B)*e*(e*x)^(5/2))/(b^2*Sqrt[a + b*x^2]) + (2*B*(e*x)^(9/2))/(7*b*e*Sqrt[a + b*x^2]) + (5*(7
*A*b - 9*a*B)*e^3*Sqrt[e*x]*Sqrt[a + b*x^2])/(21*b^3) - (5*a^(3/4)*(7*A*b - 9*a*B)*e^(7/2)*(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])/
(42*b^(13/4)*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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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)^{9/2}}{7 b e \sqrt {a+b x^2}}-\frac {\left (2 \left (-\frac {7 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {(e x)^{7/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{7 b} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {\left (5 (7 A b-9 a B) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^2}} \, dx}{14 b^2} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {\left (5 a (7 A b-9 a B) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {a+b x^2}} \, dx}{42 b^3} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {\left (5 a (7 A b-9 a B) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^3} \\ & = -\frac {(7 A b-9 a B) e (e x)^{5/2}}{7 b^2 \sqrt {a+b x^2}}+\frac {2 B (e x)^{9/2}}{7 b e \sqrt {a+b x^2}}+\frac {5 (7 A b-9 a B) e^3 \sqrt {e x} \sqrt {a+b x^2}}{21 b^3}-\frac {5 a^{3/4} (7 A b-9 a B) e^{7/2} \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 )}{42 b^{13/4} \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 = 111, normalized size of antiderivative = 0.53 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {e^3 \sqrt {e x} \left (-45 a^2 B+a b \left (35 A-18 B x^2\right )+2 b^2 x^2 \left (7 A+3 B x^2\right )+5 a (-7 A b+9 a B) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{21 b^3 \sqrt {a+b x^2}} \]

[In]

Integrate[((e*x)^(7/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]

[Out]

(e^3*Sqrt[e*x]*(-45*a^2*B + a*b*(35*A - 18*B*x^2) + 2*b^2*x^2*(7*A + 3*B*x^2) + 5*a*(-7*A*b + 9*a*B)*Sqrt[1 +
(b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)]))/(21*b^3*Sqrt[a + b*x^2])

Maple [A] (verified)

Time = 3.65 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {e^{3} \sqrt {e x}\, \left (35 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b -45 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a^{2}-12 b^{3} B \,x^{5}-28 A \,b^{3} x^{3}+36 B a \,b^{2} x^{3}-70 a \,b^{2} A x +90 a^{2} b B x \right )}{42 x \sqrt {b \,x^{2}+a}\, b^{4}}\)	\(252\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {e^{4} x a \left (A b -B a \right )}{b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {2 B \,e^{3} x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b^{2}}+\frac {2 \left (\frac {\left (A b -B a \right ) e^{4}}{b^{2}}-\frac {5 B \,e^{4} a}{7 b^{2}}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}+\frac {\left (-\frac {a \left (A b -B a \right ) e^{4}}{2 b^{3}}-\frac {\left (\frac {\left (A b -B a \right ) e^{4}}{b^{2}}-\frac {5 B \,e^{4} a}{7 b^{2}}\right ) a}{3 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\)	\(298\)
risch	\(\frac {2 \left (3 b B \,x^{2}+7 A b -12 B a \right ) x \sqrt {b \,x^{2}+a}\, e^{4}}{21 b^{3} \sqrt {e x}}-\frac {a \left (\frac {28 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}-\frac {33 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}}-21 a \left (A b -B a \right ) \left (\frac {x}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) e^{4} \sqrt {\left (b \,x^{2}+a \right ) e x}}{21 b^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(442\)

[In]

int((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/42*e^3/x*(e*x)^(1/2)*(35*A*2^(1/2)*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2
))^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)
*a*b-45*B*2^(1/2)*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x/(-a*b)^
(1/2)*b)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a^2-12*b^3*B*x^5-28
*A*b^3*x^3+36*B*a*b^2*x^3-70*a*b^2*A*x+90*a^2*b*B*x)/(b*x^2+a)^(1/2)/b^4

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {5 \, {\left ({\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} e^{3} x^{2} + {\left (9 \, B a^{3} - 7 \, A a^{2} b\right )} e^{3}\right )} \sqrt {b e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (6 \, B b^{3} e^{3} x^{4} - 2 \, {\left (9 \, B a b^{2} - 7 \, A b^{3}\right )} e^{3} x^{2} - 5 \, {\left (9 \, B a^{2} b - 7 \, A a b^{2}\right )} e^{3}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{21 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

1/21*(5*((9*B*a^2*b - 7*A*a*b^2)*e^3*x^2 + (9*B*a^3 - 7*A*a^2*b)*e^3)*sqrt(b*e)*weierstrassPInverse(-4*a/b, 0,
x) + (6*B*b^3*e^3*x^4 - 2*(9*B*a*b^2 - 7*A*b^3)*e^3*x^2 - 5*(9*B*a^2*b - 7*A*a*b^2)*e^3)*sqrt(b*x^2 + a)*sqrt
(e*x))/(b^5*x^2 + a*b^4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 163.87 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} + \frac {B e^{\frac {7}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {17}{4}\right )} \]

[In]

integrate((e*x)**(7/2)*(B*x**2+A)/(b*x**2+a)**(3/2),x)

[Out]

A*e**(7/2)*x**(9/2)*gamma(9/4)*hyper((3/2, 9/4), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*gamma(13/4)) +
B*e**(7/2)*x**(13/2)*gamma(13/4)*hyper((3/2, 13/4), (17/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*gamma(17/4
))

Maxima [F]

\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(3/2), x)

Giac [F]

\[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(e*x)^(7/2)/(b*x^2 + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^{7/2}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]

[In]

int(((A + B*x^2)*(e*x)^(7/2))/(a + b*x^2)^(3/2),x)

[Out]

int(((A + B*x^2)*(e*x)^(7/2))/(a + b*x^2)^(3/2), x)