Integrand size = 24, antiderivative size = 438 \[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=-\frac {8 a^3 A e^3 x \sqrt {a+c x^2}}{65 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {4 a^2 e^2 \sqrt {e x} (65 a B-231 A c x) \sqrt {a+c x^2}}{15015 c^2}+\frac {2 a e^2 \sqrt {e x} (13 a B-77 A c x) \left (a+c x^2\right )^{3/2}}{3003 c^2}-\frac {2 a B e^2 \sqrt {e x} \left (a+c x^2\right )^{5/2}}{33 c^2}+\frac {2 A e (e x)^{3/2} \left (a+c x^2\right )^{5/2}}{13 c}+\frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}+\frac {8 a^{13/4} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {4 a^{13/4} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15015 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/13*A*e*(e*x)^(3/2)*(c*x^2+a)^(5/2)/c+2/15*B*(e*x)^(5/2)*(c*x^2+a)^(5/2)/ c+2/3003*a*e^2*(-77*A*c*x+13*B*a)*(c*x^2+a)^(3/2)*(e*x)^(1/2)/c^2-2/33*a*B *e^2*(c*x^2+a)^(5/2)*(e*x)^(1/2)/c^2-8/65*a^3*A*e^3*x*(c*x^2+a)^(1/2)/c^(3 /2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+4/15015*a^2*e^2*(-231*A*c*x+65*B*a)*(e *x)^(1/2)*(c*x^2+a)^(1/2)/c^2+8/65*a^(13/4)*A*e^3*(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^(7/4)/(e*x)^(1/2)/(c*x^2+a )^(1/2)+4/15015*a^(13/4)*e^3*(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)))*EllipticF(sin(2*arctan(c^(1/4)* x^(1/2)/a^(1/4))),1/2*2^(1/2))*(65*B*a^(1/2)-231*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^(9/4)/(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 = 10.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.31 \[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {2 e^2 \sqrt {e x} \sqrt {a+c x^2} \left (-\left (a+c x^2\right )^2 \sqrt {1+\frac {c x^2}{a}} (65 a B-11 c x (15 A+13 B x))+65 a^3 B \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )-165 a^2 A c x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{2145 c^2 \sqrt {1+\frac {c x^2}{a}}} \]
(2*e^2*Sqrt[e*x]*Sqrt[a + c*x^2]*(-((a + c*x^2)^2*Sqrt[1 + (c*x^2)/a]*(65* a*B - 11*c*x*(15*A + 13*B*x))) + 65*a^3*B*Hypergeometric2F1[-3/2, 1/4, 5/4 , -((c*x^2)/a)] - 165*a^2*A*c*x*Hypergeometric2F1[-3/2, 3/4, 7/4, -((c*x^2 )/a)]))/(2145*c^2*Sqrt[1 + (c*x^2)/a])
Time = 0.58 (sec) , antiderivative size = 427, 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, 548, 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.
| \(\displaystyle \int (e x)^{5/2} \left (a+c x^2\right )^{3/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {2 e \int \frac {5}{2} (e x)^{3/2} (a B-3 A c x) \left (c x^2+a\right )^{3/2}dx}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \int (e x)^{3/2} (a B-3 A c x) \left (c x^2+a\right )^{3/2}dx}{3 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (-\frac {2 e \int -\frac {1}{2} a c \sqrt {e x} (9 A+13 B x) \left (c x^2+a\right )^{3/2}dx}{13 c}-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \int \sqrt {e x} (9 A+13 B x) \left (c x^2+a\right )^{3/2}dx-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {2 e \int \frac {(13 a B-99 A c x) \left (c x^2+a\right )^{3/2}}{2 \sqrt {e x}}dx}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \int \frac {(13 a B-99 A c x) \left (c x^2+a\right )^{3/2}}{\sqrt {e x}}dx}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {4}{21} a \int \frac {9 (13 a B-77 A c x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \int \frac {(13 a B-77 A c x) \sqrt {c x^2+a}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {4}{15} a \int \frac {65 a B-231 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {2}{15} a \int \frac {65 a B-231 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {2 a \sqrt {x} \int \frac {65 a B-231 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \int \frac {65 a B-231 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+231 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\sqrt {a} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+231 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (231 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{5/2} \left (a+c x^2\right )^{5/2}}{15 c}-\frac {e \left (\frac {1}{13} a e \left (\frac {26 B \sqrt {e x} \left (a+c x^2\right )^{5/2}}{11 c}-\frac {e \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (65 \sqrt {a} B-231 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}+231 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{15 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (65 a B-231 A c x)}{15 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (13 a B-77 A c x)}{7 e}\right )}{11 c}\right )-\frac {6}{13} A (e x)^{3/2} \left (a+c x^2\right )^{5/2}\right )}{3 c}\) |
(2*B*(e*x)^(5/2)*(a + c*x^2)^(5/2))/(15*c) - (e*((-6*A*(e*x)^(3/2)*(a + c* x^2)^(5/2))/13 + (a*e*((26*B*Sqrt[e*x]*(a + c*x^2)^(5/2))/(11*c) - (e*((2* Sqrt[e*x]*(13*a*B - 77*A*c*x)*(a + c*x^2)^(3/2))/(7*e) + (6*a*((2*Sqrt[e*x ]*(65*a*B - 231*A*c*x)*Sqrt[a + c*x^2])/(15*e) + (4*a*Sqrt[x]*(231*A*Sqrt[ c]*(-((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])) + (a^(1/4)*( 65*Sqrt[a]*B - 231*A*Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt [a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/( 2*c^(1/4)*Sqrt[a + c*x^2])))/(15*Sqrt[e*x])))/7))/(11*c)))/13))/(3*c)
3.5.42.3.1 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 = 0.89 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {2 e^{2} \sqrt {e x}\, \left (1001 B \,c^{5} x^{9}+1155 A \,c^{5} x^{8}+2548 B a \,c^{4} x^{7}+3080 A a \,c^{4} x^{6}+462 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4} c -924 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4} c +130 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4}+1703 B \,a^{2} c^{3} x^{5}+2233 A \,a^{2} c^{3} x^{4}-104 B \,a^{3} c^{2} x^{3}+308 A \,a^{3} c^{2} x^{2}-260 B \,a^{4} c x \right )}{15015 x \sqrt {c \,x^{2}+a}\, c^{3}}\) | \(384\) |
| risch | \(\frac {2 \left (1001 B \,c^{3} x^{6}+1155 A \,c^{3} x^{5}+1547 a B \,c^{2} x^{4}+1925 a A \,c^{2} x^{3}+156 a^{2} B c \,x^{2}+308 a^{2} A c x -260 B \,a^{3}\right ) x \sqrt {c \,x^{2}+a}\, e^{3}}{15015 c^{2} \sqrt {e x}}-\frac {4 a^{3} \left (-\frac {65 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {231 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) e^{3} \sqrt {\left (c \,x^{2}+a \right ) e x}}{15015 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(396\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B c \,e^{2} x^{6} \sqrt {c e \,x^{3}+a e x}}{15}+\frac {2 A c \,e^{2} x^{5} \sqrt {c e \,x^{3}+a e x}}{13}+\frac {34 B a \,e^{2} x^{4} \sqrt {c e \,x^{3}+a e x}}{165}+\frac {10 A a \,e^{2} x^{3} \sqrt {c e \,x^{3}+a e x}}{39}+\frac {8 B \,a^{2} e^{2} x^{2} \sqrt {c e \,x^{3}+a e x}}{385 c}+\frac {8 a^{2} A \,e^{2} x \sqrt {c e \,x^{3}+a e x}}{195 c}-\frac {8 B \,a^{3} e^{2} \sqrt {c e \,x^{3}+a e x}}{231 c^{2}}+\frac {4 B \,a^{4} e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{3} \sqrt {c e \,x^{3}+a e x}}-\frac {4 a^{3} A \,e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{65 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(490\) |
2/15015*e^2/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)*(1001*B*c^5*x^9+1155*A*c^5*x^8+2 548*B*a*c^4*x^7+3080*A*a*c^4*x^6+462*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^( 1/2)*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))*a^4*c -924*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*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))*a^4*c+130*B*(-a*c)^(1/2)*((c*x+(-a*c) ^(1/2))/(-a*c)^(1/2))^(1/2)*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))*a^4+1703*B*a^2*c^3*x^5+2233*A*a^2*c^3*x^4-104*B*a^3*c^2* x^3+308*A*a^3*c^2*x^2-260*B*a^4*c*x)/c^3
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.38 \[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (260 \, \sqrt {c e} B a^{4} e^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 924 \, \sqrt {c e} A a^{3} c e^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (1001 \, B c^{4} e^{2} x^{6} + 1155 \, A c^{4} e^{2} x^{5} + 1547 \, B a c^{3} e^{2} x^{4} + 1925 \, A a c^{3} e^{2} x^{3} + 156 \, B a^{2} c^{2} e^{2} x^{2} + 308 \, A a^{2} c^{2} e^{2} x - 260 \, B a^{3} c e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{15015 \, c^{3}} \]
2/15015*(260*sqrt(c*e)*B*a^4*e^2*weierstrassPInverse(-4*a/c, 0, x) + 924*s qrt(c*e)*A*a^3*c*e^2*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c , 0, x)) + (1001*B*c^4*e^2*x^6 + 1155*A*c^4*e^2*x^5 + 1547*B*a*c^3*e^2*x^4 + 1925*A*a*c^3*e^2*x^3 + 156*B*a^2*c^2*e^2*x^2 + 308*A*a^2*c^2*e^2*x - 26 0*B*a^3*c*e^2)*sqrt(c*x^2 + a)*sqrt(e*x))/c^3
Result contains complex when optimal does not.
Time = 42.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.45 \[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\frac {A a^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {A \sqrt {a} c e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} + \frac {B a^{\frac {3}{2}} e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B \sqrt {a} c e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} \]
A*a**(3/2)*e**(5/2)*x**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**2 *exp_polar(I*pi)/a)/(2*gamma(11/4)) + A*sqrt(a)*c*e**(5/2)*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)) + B*a**(3/2)*e**(5/2)*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)*c*e**(5/2)*x**(13/2) *gamma(13/4)*hyper((-1/2, 13/4), (17/4,), c*x**2*exp_polar(I*pi)/a)/(2*gam ma(17/4))
\[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}} \,d x } \]
\[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (e x)^{5/2} (A+B x) \left (a+c x^2\right )^{3/2} \, dx=\int {\left (e\,x\right )}^{5/2}\,{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]