Integrand size = 26, antiderivative size = 212 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {4 a (11 A b-5 a B) e \sqrt {e x} \sqrt {a+b x^2}}{231 b^2}+\frac {2 (11 A b-5 a B) (e x)^{5/2} \sqrt {a+b x^2}}{77 b e}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}-\frac {2 a^{7/4} (11 A b-5 a B) e^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 b^{9/4} \sqrt {a+b x^2}} \] Output:
4/231*a*(11*A*b-5*B*a)*e*(e*x)^(1/2)*(b*x^2+a)^(1/2)/b^2+2/77*(11*A*b-5*B* a)*(e*x)^(5/2)*(b*x^2+a)^(1/2)/b/e+2/11*B*(e*x)^(5/2)*(b*x^2+a)^(3/2)/b/e- 2/231*a^(7/4)*(11*A*b-5*B*a)*e^(3/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/ 2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4 )/e^(1/2)),1/2*2^(1/2))/b^(9/4)/(b*x^2+a)^(1/2)
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 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 e \sqrt {e x} \sqrt {a+b x^2} \left (-\left (\left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (-11 A b+5 a B-7 b B x^2\right )\right )+a (-11 A b+5 a B) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{77 b^2 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[(e*x)^(3/2)*Sqrt[a + b*x^2]*(A + B*x^2),x]
Output:
(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])
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {363, 248, 262, 266, 761}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {(11 A b-5 a B) \int (e x)^{3/2} \sqrt {b x^2+a}dx}{11 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {(11 A b-5 a B) \left (\frac {2}{7} a \int \frac {(e x)^{3/2}}{\sqrt {b x^2+a}}dx+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )}{11 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {(11 A b-5 a B) \left (\frac {2}{7} a \left (\frac {2 e \sqrt {e x} \sqrt {a+b x^2}}{3 b}-\frac {a e^2 \int \frac {1}{\sqrt {e x} \sqrt {b x^2+a}}dx}{3 b}\right )+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )}{11 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(11 A b-5 a B) \left (\frac {2}{7} a \left (\frac {2 e \sqrt {e x} \sqrt {a+b x^2}}{3 b}-\frac {2 a e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{3 b}\right )+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )}{11 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(11 A b-5 a B) \left (\frac {2}{7} a \left (\frac {2 e \sqrt {e x} \sqrt {a+b x^2}}{3 b}-\frac {a^{3/4} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 b^{5/4} \sqrt {a+b x^2}}\right )+\frac {2 (e x)^{5/2} \sqrt {a+b x^2}}{7 e}\right )}{11 b}+\frac {2 B (e x)^{5/2} \left (a+b x^2\right )^{3/2}}{11 b e}\) |
Input:
Int[(e*x)^(3/2)*Sqrt[a + b*x^2]*(A + B*x^2),x]
Output:
(2*B*(e*x)^(5/2)*(a + b*x^2)^(3/2))/(11*b*e) + ((11*A*b - 5*a*B)*((2*(e*x) ^(5/2)*Sqrt[a + b*x^2])/(7*e) + (2*a*((2*e*Sqrt[e*x]*Sqrt[a + b*x^2])/(3*b ) - (a^(3/4)*Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(S qrt[a]*e + Sqrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4) *Sqrt[e])], 1/2])/(3*b^(5/4)*Sqrt[a + b*x^2])))/7))/(11*b)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
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]
Time = 0.71 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.01
| method | result | size |
| risch | \(\frac {2 \left (21 b^{2} B \,x^{4}+33 A \,b^{2} x^{2}+6 B a b \,x^{2}+22 a b A -10 a^{2} B \right ) x \sqrt {b \,x^{2}+a}\, e^{2}}{231 b^{2} \sqrt {e x}}-\frac {2 a^{2} \left (11 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {\left (b \,x^{2}+a \right ) e x}}{231 b^{3} \sqrt {b e \,x^{3}+a e x}\, \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(214\) |
| default | \(-\frac {2 e \sqrt {e x}\, \left (-21 B \,b^{4} x^{7}+11 A \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -33 A \,b^{4} x^{5}-5 B \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-27 B a \,b^{3} x^{5}-55 A a \,b^{3} x^{3}+4 B \,a^{2} b^{2} x^{3}-22 A x \,a^{2} b^{2}+10 B x \,a^{3} b \right )}{231 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(276\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {2 B e \,x^{4} \sqrt {b e \,x^{3}+a e x}}{11}+\frac {2 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) x^{2} \sqrt {b e \,x^{3}+a e x}}{7 b e}+\frac {2 \left (A a \,e^{2}-\frac {5 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) a}{7 b}\right ) \sqrt {b e \,x^{3}+a e x}}{3 b e}-\frac {\left (A a \,e^{2}-\frac {5 \left (\left (A b +B a \right ) e^{2}-\frac {9 B a \,e^{2}}{11}\right ) a}{7 b}\right ) a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(294\) |
Input:
int((e*x)^(3/2)*(b*x^2+a)^(1/2)*(B*x^2+A),x,method=_RETURNVERBOSE)
Output:
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+ 1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a*b)^ (1/2))^(1/2)*(-b/(-a*b)^(1/2)*x)^(1/2)/(b*e*x^3+a*e*x)^(1/2)*EllipticF(((x +1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(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)
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.48 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 \, {\left (2 \, {\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} \sqrt {b e} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (21 \, B b^{3} e x^{4} + 3 \, {\left (2 \, B a b^{2} + 11 \, A b^{3}\right )} e x^{2} - 2 \, {\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{231 \, b^{3}} \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^(1/2)*(B*x^2+A),x, algorithm="fricas")
Output:
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
Result contains complex when optimal does not.
Time = 5.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.46 \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {A \sqrt {a} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {B \sqrt {a} e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((e*x)**(3/2)*(b*x**2+a)**(1/2)*(B*x**2+A),x)
Output:
A*sqrt(a)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), b*x**2*e xp_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))
\[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \left (e x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^(1/2)*(B*x^2+A),x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*sqrt(b*x^2 + a)*(e*x)^(3/2), x)
\[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int { {\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \left (e x\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x)^(3/2)*(b*x^2+a)^(1/2)*(B*x^2+A),x, algorithm="giac")
Output:
integrate((B*x^2 + A)*sqrt(b*x^2 + a)*(e*x)^(3/2), x)
Timed out. \[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^{3/2}\,\sqrt {b\,x^2+a} \,d x \] Input:
int((A + B*x^2)*(e*x)^(3/2)*(a + b*x^2)^(1/2),x)
Output:
int((A + B*x^2)*(e*x)^(3/2)*(a + b*x^2)^(1/2), x)
\[ \int (e x)^{3/2} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx=\frac {2 \sqrt {e}\, e \left (4 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2}+13 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a b \,x^{2}+7 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b^{2} x^{4}-2 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{3}+a x}d x \right ) a^{3}\right )}{77 b} \] Input:
int((e*x)^(3/2)*(b*x^2+a)^(1/2)*(B*x^2+A),x)
Output:
(2*sqrt(e)*e*(4*sqrt(x)*sqrt(a + b*x**2)*a**2 + 13*sqrt(x)*sqrt(a + b*x**2 )*a*b*x**2 + 7*sqrt(x)*sqrt(a + b*x**2)*b**2*x**4 - 2*int((sqrt(x)*sqrt(a + b*x**2))/(a*x + b*x**3),x)*a**3))/(77*b)