Integrand size = 24, antiderivative size = 437 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=-\frac {8 a^3 e \sqrt {e x} (221 A+231 B x) \sqrt {a+c x^2}}{51051 c}-\frac {16 a^4 B e^2 x \sqrt {a+c x^2}}{221 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 a^2 e \sqrt {e x} (221 A+385 B x) \left (a+c x^2\right )^{3/2}}{51051 c}-\frac {2 a e \sqrt {e x} (221 A+495 B x) \left (a+c x^2\right )^{5/2}}{36465 c}+\frac {2 A e \sqrt {e x} \left (a+c x^2\right )^{7/2}}{15 c}+\frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}+\frac {16 a^{17/4} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{221 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a^{15/4} \left (231 \sqrt {a} B+221 A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{51051 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/17*B*(e*x)^(3/2)*(c*x^2+a)^(7/2)/c-4/51051*a^2*e*(385*B*x+221*A)*(c*x^2+ a)^(3/2)*(e*x)^(1/2)/c-2/36465*a*e*(495*B*x+221*A)*(c*x^2+a)^(5/2)*(e*x)^( 1/2)/c+2/15*A*e*(c*x^2+a)^(7/2)*(e*x)^(1/2)/c-16/221*a^4*B*e^2*x*(c*x^2+a) ^(1/2)/c^(3/2)/(a^(1/2)+x*c^(1/2))/(e*x)^(1/2)-8/51051*a^3*e*(231*B*x+221* A)*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c+16/221*a^(17/4)*B*e^2*(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)))*Ellip ticE(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)-8/51051*a^(15/4)*e^2*(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))*(231*B*a^(1/2)+221*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^(7/4)/(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 = 124, normalized size of antiderivative = 0.28 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {2 e \sqrt {e x} \sqrt {a+c x^2} \left ((17 A+15 B x) \left (a+c x^2\right )^3 \sqrt {1+\frac {c x^2}{a}}-17 a^3 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )-15 a^3 B x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{255 c \sqrt {1+\frac {c x^2}{a}}} \]
(2*e*Sqrt[e*x]*Sqrt[a + c*x^2]*((17*A + 15*B*x)*(a + c*x^2)^3*Sqrt[1 + (c* x^2)/a] - 17*a^3*A*Hypergeometric2F1[-5/2, 1/4, 5/4, -((c*x^2)/a)] - 15*a^ 3*B*x*Hypergeometric2F1[-5/2, 3/4, 7/4, -((c*x^2)/a)]))/(255*c*Sqrt[1 + (c *x^2)/a])
Time = 0.57 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.98, 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, 548, 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.
| \(\displaystyle \int (e x)^{3/2} \left (a+c x^2\right )^{5/2} (A+B x) \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {2 e \int \frac {1}{2} \sqrt {e x} (3 a B-17 A c x) \left (c x^2+a\right )^{5/2}dx}{17 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \int \sqrt {e x} (3 a B-17 A c x) \left (c x^2+a\right )^{5/2}dx}{17 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (-\frac {2 e \int -\frac {a c (17 A+45 B x) \left (c x^2+a\right )^{5/2}}{2 \sqrt {e x}}dx}{15 c}-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \int \frac {(17 A+45 B x) \left (c x^2+a\right )^{5/2}}{\sqrt {e x}}dx-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {20}{143} a \int \frac {(221 A+495 B x) \left (c x^2+a\right )^{3/2}}{2 \sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \int \frac {(221 A+495 B x) \left (c x^2+a\right )^{3/2}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {4}{21} a \int \frac {9 (221 A+385 B x) \sqrt {c x^2+a}}{2 \sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \int \frac {(221 A+385 B x) \sqrt {c x^2+a}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 548 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {4}{15} a \int \frac {5 (221 A+231 B x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {2}{3} a \int \frac {221 A+231 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {2 a \sqrt {x} \int \frac {221 A+231 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{3 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \int \frac {221 A+231 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{3 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\left (\frac {231 \sqrt {a} B}{\sqrt {c}}+221 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {231 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\left (\frac {231 \sqrt {a} B}{\sqrt {c}}+221 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {231 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {231 \sqrt {a} B}{\sqrt {c}}+221 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {231 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \left (a+c x^2\right )^{7/2}}{17 c}-\frac {e \left (\frac {1}{15} a e \left (\frac {10}{143} a \left (\frac {6}{7} a \left (\frac {4 a \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {231 \sqrt {a} B}{\sqrt {c}}+221 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {231 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{3 \sqrt {e x}}+\frac {2 \sqrt {e x} \sqrt {a+c x^2} (221 A+231 B x)}{3 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{3/2} (221 A+385 B x)}{7 e}\right )+\frac {2 \sqrt {e x} \left (a+c x^2\right )^{5/2} (221 A+495 B x)}{143 e}\right )-\frac {34}{15} A \sqrt {e x} \left (a+c x^2\right )^{7/2}\right )}{17 c}\) |
(2*B*(e*x)^(3/2)*(a + c*x^2)^(7/2))/(17*c) - (e*((-34*A*Sqrt[e*x]*(a + c*x ^2)^(7/2))/15 + (a*e*((2*Sqrt[e*x]*(221*A + 495*B*x)*(a + c*x^2)^(5/2))/(1 43*e) + (10*a*((2*Sqrt[e*x]*(221*A + 385*B*x)*(a + c*x^2)^(3/2))/(7*e) + ( 6*a*((2*Sqrt[e*x]*(221*A + 231*B*x)*Sqrt[a + c*x^2])/(3*e) + (4*a*Sqrt[x]* ((-231*B*(-((Sqrt[x]*Sqrt[a + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(S qrt[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] + ((221*A + (231*Sqrt[a]*B)/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*a^(1/4)*c^(1/4)*Sqrt[a + c*x^2])))/(3*Sqrt[e*x])))/7))/143))/1 5))/(17*c)
3.5.50.3.1 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 = 1.32 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(-\frac {2 e \sqrt {e x}\, \left (-15015 B \,c^{5} x^{10}-17017 A \,c^{5} x^{9}-56595 B a \,c^{4} x^{8}-66521 A a \,c^{4} x^{7}-75845 B \,a^{2} c^{3} x^{6}+4420 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{4}-95251 A \,a^{2} x^{5} c^{3}+9240 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{5}-4620 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{5}-37345 B \,a^{3} c^{2} x^{4}-54587 A \,a^{3} c^{2} x^{3}-3080 B \,a^{4} c \,x^{2}-8840 A \,a^{4} c x \right )}{255255 x \sqrt {c \,x^{2}+a}\, c^{2}}\) | \(390\) |
| risch | \(\frac {2 \left (15015 B \,c^{3} x^{7}+17017 A \,c^{3} x^{6}+41580 a B \,c^{2} x^{5}+49504 a A \,c^{2} x^{4}+34265 a^{2} B c \,x^{3}+45747 a^{2} A c \,x^{2}+3080 a^{3} B x +8840 A \,a^{3}\right ) x \sqrt {c \,x^{2}+a}\, e^{2}}{255255 c \sqrt {e x}}-\frac {8 a^{4} \left (\frac {221 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {231 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{2} \sqrt {\left (c \,x^{2}+a \right ) e x}}{51051 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(407\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,c^{2} e \,x^{7} \sqrt {c e \,x^{3}+a e x}}{17}+\frac {2 A \,c^{2} e \,x^{6} \sqrt {c e \,x^{3}+a e x}}{15}+\frac {72 B a c e \,x^{5} \sqrt {c e \,x^{3}+a e x}}{221}+\frac {64 a A c e \,x^{4} \sqrt {c e \,x^{3}+a e x}}{165}+\frac {178 B \,a^{2} e \,x^{3} \sqrt {c e \,x^{3}+a e x}}{663}+\frac {138 A \,a^{2} e \,x^{2} \sqrt {c e \,x^{3}+a e x}}{385}+\frac {16 B \,a^{3} e x \sqrt {c e \,x^{3}+a e x}}{663 c}+\frac {16 A \,a^{3} e \sqrt {c e \,x^{3}+a e x}}{231 c}-\frac {8 A \,a^{4} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {8 B \,a^{4} e^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{221 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(502\) |
-2/255255*e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^2*(-15015*B*c^5*x^10-17017*A*c ^5*x^9-56595*B*a*c^4*x^8-66521*A*a*c^4*x^7-75845*B*a^2*c^3*x^6+4420*A*(-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^4-95251*A*a^2*x^5*c^3+9240*B*((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^5-4620*B*((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^5-373 45*B*a^3*c^2*x^4-54587*A*a^3*c^2*x^3-3080*B*a^4*c*x^2-8840*A*a^4*c*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.36 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=-\frac {2 \, {\left (8840 \, \sqrt {c e} A a^{4} e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 9240 \, \sqrt {c e} B a^{4} e {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (15015 \, B c^{4} e x^{7} + 17017 \, A c^{4} e x^{6} + 41580 \, B a c^{3} e x^{5} + 49504 \, A a c^{3} e x^{4} + 34265 \, B a^{2} c^{2} e x^{3} + 45747 \, A a^{2} c^{2} e x^{2} + 3080 \, B a^{3} c e x + 8840 \, A a^{3} c e\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{255255 \, c^{2}} \]
-2/255255*(8840*sqrt(c*e)*A*a^4*e*weierstrassPInverse(-4*a/c, 0, x) - 9240 *sqrt(c*e)*B*a^4*e*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (15015*B*c^4*e*x^7 + 17017*A*c^4*e*x^6 + 41580*B*a*c^3*e*x^5 + 49 504*A*a*c^3*e*x^4 + 34265*B*a^2*c^2*e*x^3 + 45747*A*a^2*c^2*e*x^2 + 3080*B *a^3*c*e*x + 8840*A*a^3*c*e)*sqrt(c*x^2 + a)*sqrt(e*x))/c^2
Result contains complex when optimal does not.
Time = 27.67 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.69 \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\frac {A a^{\frac {5}{2}} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {A a^{\frac {3}{2}} c e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {13}{4}\right )} + \frac {A \sqrt {a} c^{2} e^{\frac {3}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} + \frac {B a^{\frac {5}{2}} e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} + \frac {B a^{\frac {3}{2}} c e^{\frac {3}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {15}{4}\right )} + \frac {B \sqrt {a} c^{2} e^{\frac {3}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {19}{4}\right )} \]
A*a**(5/2)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**2* exp_polar(I*pi)/a)/(2*gamma(9/4)) + A*a**(3/2)*c*e**(3/2)*x**(9/2)*gamma(9 /4)*hyper((-1/2, 9/4), (13/4,), c*x**2*exp_polar(I*pi)/a)/gamma(13/4) + A* sqrt(a)*c**2*e**(3/2)*x**(13/2)*gamma(13/4)*hyper((-1/2, 13/4), (17/4,), c *x**2*exp_polar(I*pi)/a)/(2*gamma(17/4)) + B*a**(5/2)*e**(3/2)*x**(7/2)*ga mma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(11 /4)) + B*a**(3/2)*c*e**(3/2)*x**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15 /4,), c*x**2*exp_polar(I*pi)/a)/gamma(15/4) + B*sqrt(a)*c**2*e**(3/2)*x**( 15/2)*gamma(15/4)*hyper((-1/2, 15/4), (19/4,), c*x**2*exp_polar(I*pi)/a)/( 2*gamma(19/4))
\[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}} \,d x } \]
\[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx=\int {\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right ) \,d x \]