Integrand size = 24, antiderivative size = 427 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}+\frac {28 a^3 B e^4 x \sqrt {a+c x^2}}{195 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {28 a^{13/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 a^{11/4} \left (539 \sqrt {a} B+325 A \sqrt {c}\right ) e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15015 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}} \]
-14/117*a*B*e^2*(e*x)^(3/2)*(c*x^2+a)^(3/2)/c^2+2/11*A*e*(e*x)^(5/2)*(c*x^ 2+a)^(3/2)/c+2/13*B*(e*x)^(7/2)*(c*x^2+a)^(3/2)/c-10/77*a*A*e^3*(c*x^2+a)^ (3/2)*(e*x)^(1/2)/c^2+28/195*a^3*B*e^4*x*(c*x^2+a)^(1/2)/c^(5/2)/(a^(1/2)+ x*c^(1/2))/(e*x)^(1/2)+2/15015*a^2*e^3*(539*B*x+325*A)*(e*x)^(1/2)*(c*x^2+ a)^(1/2)/c^2-28/195*a^(13/4)*B*e^4*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))) ^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(c^ (1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a )/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(11/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)+2/1501 5*a^(11/4)*e^4*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arct an(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4 ))),1/2*2^(1/2))*(539*B*a^(1/2)+325*A*c^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2) *((c*x^2+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(11/4)/(e*x)^(1/2)/(c*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 = 142, normalized size of antiderivative = 0.33 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 e^3 \sqrt {e x} \sqrt {a+c x^2} \left (-\left (\left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \left (-63 c x^2 (13 A+11 B x)+a (585 A+539 B x)\right )\right )+585 a^2 A \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )+539 a^2 B x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{9009 c^2 \sqrt {1+\frac {c x^2}{a}}} \]
(2*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]*(-((a + c*x^2)*Sqrt[1 + (c*x^2)/a]*(-63*c *x^2*(13*A + 11*B*x) + a*(585*A + 539*B*x))) + 585*a^2*A*Hypergeometric2F1 [-1/2, 1/4, 5/4, -((c*x^2)/a)] + 539*a^2*B*x*Hypergeometric2F1[-1/2, 3/4, 7/4, -((c*x^2)/a)]))/(9009*c^2*Sqrt[1 + (c*x^2)/a])
Time = 0.58 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.97, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {552, 27, 552, 27, 552, 27, 552, 27, 548, 27, 556, 555, 1512, 27, 761, 1510}
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.
| \(\displaystyle \int (e x)^{7/2} \sqrt {a+c x^2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {2 e \int \frac {1}{2} (e x)^{5/2} (7 a B-13 A c x) \sqrt {c x^2+a}dx}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \int (e x)^{5/2} (7 a B-13 A c x) \sqrt {c x^2+a}dx}{13 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (-\frac {2 e \int -\frac {1}{2} a c (e x)^{3/2} (65 A+77 B x) \sqrt {c x^2+a}dx}{11 c}-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \int (e x)^{3/2} (65 A+77 B x) \sqrt {c x^2+a}dx-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {2 e \int \frac {3}{2} \sqrt {e x} (77 a B-195 A c x) \sqrt {c x^2+a}dx}{9 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \int \sqrt {e x} (77 a B-195 A c x) \sqrt {c x^2+a}dx}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (-\frac {2 e \int -\frac {a c (195 A+539 B x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx}{7 c}-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \int \frac {(195 A+539 B x) \sqrt {c x^2+a}}{\sqrt {e x}}dx-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4}{15} a \int \frac {3 (325 A+539 B x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {2}{5} a \int \frac {325 A+539 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {2 a \sqrt {x} \int \frac {325 A+539 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \int \frac {325 A+539 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\left (\frac {539 \sqrt {a} B}{\sqrt {c}}+325 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {539 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\left (\frac {539 \sqrt {a} B}{\sqrt {c}}+325 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {539 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {539 \sqrt {a} B}{\sqrt {c}}+325 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {539 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {e \left (\frac {1}{11} a e \left (\frac {154 B (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{9 c}-\frac {e \left (\frac {1}{7} a e \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {539 \sqrt {a} B}{\sqrt {c}}+325 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {539 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{5 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{5 e}\right )-\frac {390}{7} A \sqrt {e x} \left (a+c x^2\right )^{3/2}\right )}{3 c}\right )-\frac {26}{11} A (e x)^{5/2} \left (a+c x^2\right )^{3/2}\right )}{13 c}\) |
(2*B*(e*x)^(7/2)*(a + c*x^2)^(3/2))/(13*c) - (e*((-26*A*(e*x)^(5/2)*(a + c *x^2)^(3/2))/11 + (a*e*((154*B*(e*x)^(3/2)*(a + c*x^2)^(3/2))/(9*c) - (e*( (-390*A*Sqrt[e*x]*(a + c*x^2)^(3/2))/7 + (a*e*((2*Sqrt[e*x]*(325*A + 539*B *x)*Sqrt[a + c*x^2])/(5*e) + (4*a*Sqrt[x]*((-539*B*(-((Sqrt[x]*Sqrt[a + c* x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c* x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4) ], 1/2])/(c^(1/4)*Sqrt[a + c*x^2])))/Sqrt[c] + ((325*A + (539*Sqrt[a]*B)/S qrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*El lipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt [a + c*x^2])))/(5*Sqrt[e*x])))/7))/(3*c)))/11))/(13*c)
3.5.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(e*x)^(m + 1)*(c*(m + 2*p + 2) + d*(m + 2*p + 1)*x)*((a + b*x^ 2)^p/(e*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*a*(p/((m + 2*p + 1)*(m + 2*p + 2))) Int[(e*x)^m*(a + b*x^2)^(p - 1)*(c*(m + 2*p + 2) + d*(m + 2*p + 1)*x), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[ p] || !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e /(b*(m + 2*p + 2)) Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free Q[{a, c, f, g}, x]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb ol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] , x] /; FreeQ[{a, b, c, d, e}, x]
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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Time = 1.04 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(\frac {2 e^{3} \sqrt {e x}\, \left (3465 B \,c^{4} x^{8}+4095 A \,c^{4} x^{7}+4235 a B \,c^{3} x^{6}+975 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{3}+5265 a A \,c^{3} x^{5}+3234 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{4}-1617 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{4}-308 a^{2} B \,c^{2} x^{4}-780 a^{2} A \,c^{2} x^{3}-1078 a^{3} B c \,x^{2}-1950 a^{3} A c x \right )}{45045 x \sqrt {c \,x^{2}+a}\, c^{3}}\) | \(368\) |
| risch | \(-\frac {2 \left (-3465 B \,c^{2} x^{5}-4095 A \,c^{2} x^{4}-770 a B c \,x^{3}-1170 a A c \,x^{2}+1078 a^{2} B x +1950 A \,a^{2}\right ) x \sqrt {c \,x^{2}+a}\, e^{4}}{45045 c^{2} \sqrt {e x}}+\frac {2 a^{3} \left (\frac {325 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {539 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{4} \sqrt {\left (c \,x^{2}+a \right ) e x}}{15015 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(383\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,e^{3} x^{5} \sqrt {c e \,x^{3}+a e x}}{13}+\frac {2 A \,e^{3} x^{4} \sqrt {c e \,x^{3}+a e x}}{11}+\frac {4 B a \,e^{3} x^{3} \sqrt {c e \,x^{3}+a e x}}{117 c}+\frac {4 a A \,e^{3} x^{2} \sqrt {c e \,x^{3}+a e x}}{77 c}-\frac {28 B \,a^{2} e^{3} x \sqrt {c e \,x^{3}+a e x}}{585 c^{2}}-\frac {20 a^{2} A \,e^{3} \sqrt {c e \,x^{3}+a e x}}{231 c^{2}}+\frac {10 a^{3} A \,e^{4} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {14 B \,a^{3} e^{4} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{195 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(466\) |
2/45045*e^3/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^3*(3465*B*c^4*x^8+4095*A*c^4*x ^7+4235*a*B*c^3*x^6+975*A*(-a*c)^(1/2)*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^( 1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*El lipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3+5 265*a*A*c^3*x^5+3234*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*((-c*x+(-a* c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticE(((c*x+(- a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^4-1617*B*((c*x+(-a* c)^(1/2))/(-a*c)^(1/2))^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x /(-a*c)^(1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1 /2*2^(1/2))*2^(1/2)*a^4-308*a^2*B*c^2*x^4-780*a^2*A*c^2*x^3-1078*a^3*B*c*x ^2-1950*a^3*A*c*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.34 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 \, {\left (1950 \, \sqrt {c e} A a^{3} e^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 3234 \, \sqrt {c e} B a^{3} e^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3465 \, B c^{3} e^{3} x^{5} + 4095 \, A c^{3} e^{3} x^{4} + 770 \, B a c^{2} e^{3} x^{3} + 1170 \, A a c^{2} e^{3} x^{2} - 1078 \, B a^{2} c e^{3} x - 1950 \, A a^{2} c e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{45045 \, c^{3}} \]
2/45045*(1950*sqrt(c*e)*A*a^3*e^3*weierstrassPInverse(-4*a/c, 0, x) - 3234 *sqrt(c*e)*B*a^3*e^3*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c , 0, x)) + (3465*B*c^3*e^3*x^5 + 4095*A*c^3*e^3*x^4 + 770*B*a*c^2*e^3*x^3 + 1170*A*a*c^2*e^3*x^2 - 1078*B*a^2*c*e^3*x - 1950*A*a^2*c*e^3)*sqrt(c*x^2 + a)*sqrt(e*x))/c^3
Result contains complex when optimal does not.
Time = 43.64 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.23 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {A \sqrt {a} e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B \sqrt {a} e^{\frac {7}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} \]
A*sqrt(a)*e**(7/2)*x**(9/2)*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**2* exp_polar(I*pi)/a)/(2*gamma(13/4)) + B*sqrt(a)*e**(7/2)*x**(11/2)*gamma(11 /4)*hyper((-1/2, 11/4), (15/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(15/4))
\[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}} \,d x } \]
\[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\int {\left (e\,x\right )}^{7/2}\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \]