Integrand size = 27, antiderivative size = 317 \[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=-\frac {20 a c d e^2 \sqrt {e x} \sqrt {a-b x^2}}{21 b^2}-\frac {2 \left (9 b c^2+7 a d^2\right ) e (e x)^{3/2} \sqrt {a-b x^2}}{45 b^2}-\frac {4 c d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}+\frac {2 a^{7/4} \left (9 b c^2+7 a d^2\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{15 b^{11/4} \sqrt {a-b x^2}}-\frac {2 a^{7/4} \left (63 b c^2-50 \sqrt {a} \sqrt {b} c d+49 a d^2\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{105 b^{11/4} \sqrt {a-b x^2}} \] Output:
-20/21*a*c*d*e^2*(e*x)^(1/2)*(-b*x^2+a)^(1/2)/b^2-2/45*(7*a*d^2+9*b*c^2)*e *(e*x)^(3/2)*(-b*x^2+a)^(1/2)/b^2-4/7*c*d*(e*x)^(5/2)*(-b*x^2+a)^(1/2)/b-2 /9*d^2*(e*x)^(7/2)*(-b*x^2+a)^(1/2)/b/e+2/15*a^(7/4)*(7*a*d^2+9*b*c^2)*e^( 5/2)*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),I)/b^ (11/4)/(-b*x^2+a)^(1/2)-2/105*a^(7/4)*(63*b*c^2-50*a^(1/2)*b^(1/2)*c*d+49* a*d^2)*e^(5/2)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^( 1/2),I)/b^(11/4)/(-b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.20 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.52 \[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=\frac {2 e^2 \sqrt {e x} \left (-\left (\left (a-b x^2\right ) \left (a d (150 c+49 d x)+b x \left (63 c^2+90 c d x+35 d^2 x^2\right )\right )\right )+150 a^2 c d \sqrt {1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^2}{a}\right )+7 a \left (9 b c^2+7 a d^2\right ) x \sqrt {1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^2}{a}\right )\right )}{315 b^2 \sqrt {a-b x^2}} \] Input:
Integrate[((e*x)^(5/2)*(c + d*x)^2)/Sqrt[a - b*x^2],x]
Output:
(2*e^2*Sqrt[e*x]*(-((a - b*x^2)*(a*d*(150*c + 49*d*x) + b*x*(63*c^2 + 90*c *d*x + 35*d^2*x^2))) + 150*a^2*c*d*Sqrt[1 - (b*x^2)/a]*Hypergeometric2F1[1 /4, 1/2, 5/4, (b*x^2)/a] + 7*a*(9*b*c^2 + 7*a*d^2)*x*Sqrt[1 - (b*x^2)/a]*H ypergeometric2F1[1/2, 3/4, 7/4, (b*x^2)/a]))/(315*b^2*Sqrt[a - b*x^2])
Time = 1.17 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {559, 27, 552, 27, 552, 27, 552, 27, 556, 555, 1513, 27, 765, 762, 1390, 1389, 327}
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.
| \(\displaystyle \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle -\frac {2 \int -\frac {(e x)^{5/2} \left (9 b c^2+18 b d x c+7 a d^2\right )}{2 \sqrt {a-b x^2}}dx}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(e x)^{5/2} \left (9 b c^2+18 b d x c+7 a d^2\right )}{\sqrt {a-b x^2}}dx}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {\frac {2 e \int \frac {b (e x)^{3/2} \left (90 a c d+7 \left (9 b c^2+7 a d^2\right ) x\right )}{2 \sqrt {a-b x^2}}dx}{7 b}-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} e \int \frac {(e x)^{3/2} \left (90 a c d+7 \left (9 b c^2+7 a d^2\right ) x\right )}{\sqrt {a-b x^2}}dx-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {2 e \int \frac {3 a \sqrt {e x} \left (7 \left (9 b c^2+7 a d^2\right )+150 b c d x\right )}{2 \sqrt {a-b x^2}}dx}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \int \frac {\sqrt {e x} \left (7 \left (9 b c^2+7 a d^2\right )+150 b c d x\right )}{\sqrt {a-b x^2}}dx}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \int \frac {3 b \left (50 a c d+7 \left (9 b c^2+7 a d^2\right ) x\right )}{2 \sqrt {e x} \sqrt {a-b x^2}}dx}{3 b}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (e \int \frac {50 a c d+7 \left (9 b c^2+7 a d^2\right ) x}{\sqrt {e x} \sqrt {a-b x^2}}dx-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {e \sqrt {x} \int \frac {50 a c d+7 \left (9 b c^2+7 a d^2\right ) x}{\sqrt {x} \sqrt {a-b x^2}}dx}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \int \frac {50 a c d+7 \left (9 b c^2+7 a d^2\right ) x}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\sqrt {a} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}+\frac {7 \sqrt {a} \left (7 a d^2+9 b c^2\right ) \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\sqrt {a} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}+\frac {7 \left (7 a d^2+9 b c^2\right ) \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\frac {7 \left (7 a d^2+9 b c^2\right ) \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}+\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {a-b x^2}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\frac {7 \left (7 a d^2+9 b c^2\right ) \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}}{\sqrt {b}}+\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\frac {7 \sqrt {1-\frac {b x^2}{a}} \left (7 a d^2+9 b c^2\right ) \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {b} \sqrt {a-b x^2}}+\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\frac {7 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (7 a d^2+9 b c^2\right ) \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {x}}{\sqrt {b} \sqrt {a-b x^2}}+\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{7} e \left (\frac {3 a e \left (\frac {2 e \sqrt {x} \left (\frac {7 a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (7 a d^2+9 b c^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}+\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (50 \sqrt {a} c d-\frac {7 \left (7 a d^2+9 b c^2\right )}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{\sqrt {e x}}-100 c d \sqrt {e x} \sqrt {a-b x^2}\right )}{5 b}-\frac {14 (e x)^{3/2} \sqrt {a-b x^2} \left (7 a d^2+9 b c^2\right )}{5 b}\right )-\frac {36}{7} c d (e x)^{5/2} \sqrt {a-b x^2}}{9 b}-\frac {2 d^2 (e x)^{7/2} \sqrt {a-b x^2}}{9 b e}\) |
Input:
Int[((e*x)^(5/2)*(c + d*x)^2)/Sqrt[a - b*x^2],x]
Output:
(-2*d^2*(e*x)^(7/2)*Sqrt[a - b*x^2])/(9*b*e) + ((-36*c*d*(e*x)^(5/2)*Sqrt[ a - b*x^2])/7 + (e*((-14*(9*b*c^2 + 7*a*d^2)*(e*x)^(3/2)*Sqrt[a - b*x^2])/ (5*b) + (3*a*e*(-100*c*d*Sqrt[e*x]*Sqrt[a - b*x^2] + (2*e*Sqrt[x]*((7*a^(3 /4)*(9*b*c^2 + 7*a*d^2)*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt [x])/a^(1/4)], -1])/(b^(3/4)*Sqrt[a - b*x^2]) + (a^(3/4)*(50*Sqrt[a]*c*d - (7*(9*b*c^2 + 7*a*d^2))/Sqrt[b])*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^ (1/4)*Sqrt[x])/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^2])))/Sqrt[e*x]))/(5*b )))/7)/(9*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/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]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Time = 1.36 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(-\frac {2 \left (35 b \,d^{2} x^{3}+90 b c d \,x^{2}+49 a \,d^{2} x +63 c^{2} b x +150 a c d \right ) x \sqrt {-b \,x^{2}+a}\, e^{3}}{315 b^{2} \sqrt {e x}}+\frac {a \left (\frac {\left (49 a \,d^{2}+63 b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {50 a c d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}\right ) e^{3} \sqrt {\left (-b \,x^{2}+a \right ) e x}}{105 b^{2} \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(368\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 d^{2} e^{2} x^{3} \sqrt {-b e \,x^{3}+a e x}}{9 b}-\frac {4 c d \,e^{2} x^{2} \sqrt {-b e \,x^{3}+a e x}}{7 b}-\frac {2 \left (c^{2} e^{3}+\frac {7 d^{2} e^{3} a}{9 b}\right ) x \sqrt {-b e \,x^{3}+a e x}}{5 b e}-\frac {20 c d \,e^{2} a \sqrt {-b e \,x^{3}+a e x}}{21 b^{2}}+\frac {10 c d \,e^{3} a^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {-b e \,x^{3}+a e x}}+\frac {3 \left (c^{2} e^{3}+\frac {7 d^{2} e^{3} a}{9 b}\right ) a \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b^{2} \sqrt {-b e \,x^{3}+a e x}}\right )}{e x \sqrt {-b \,x^{2}+a}}\) | \(441\) |
| default | \(\frac {e^{2} \sqrt {e x}\, \left (70 b^{3} d^{2} x^{6}+150 \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c d -294 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3} d^{2}-378 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,c^{2}+147 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3} d^{2}+189 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,c^{2}+180 b^{3} c d \,x^{5}+28 a \,b^{2} d^{2} x^{4}+126 b^{3} c^{2} x^{4}+120 a \,b^{2} c d \,x^{3}-98 x^{2} a^{2} b \,d^{2}-126 a \,b^{2} c^{2} x^{2}-300 a^{2} x b c d \right )}{315 x \sqrt {-b \,x^{2}+a}\, b^{3}}\) | \(522\) |
Input:
int((e*x)^(5/2)*(d*x+c)^2/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/315*(35*b*d^2*x^3+90*b*c*d*x^2+49*a*d^2*x+63*b*c^2*x+150*a*c*d)/b^2*x*( -b*x^2+a)^(1/2)*e^3/(e*x)^(1/2)+1/105*a/b^2*((49*a*d^2+63*b*c^2)/b*(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*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)*(-2/b*(a *b)^(1/2)*EllipticE(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2^(1/2)) +1/b*(a*b)^(1/2)*EllipticF(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),1/2*2 ^(1/2)))+50*a*c*d/b*(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*x/(a*b)^(1/2))^(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^3*((-b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(-b*x^2+a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.47 \[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (150 \, \sqrt {-b e} a^{2} c d e^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right ) - 21 \, {\left (9 \, a b c^{2} + 7 \, a^{2} d^{2}\right )} \sqrt {-b e} e^{2} {\rm weierstrassZeta}\left (\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (35 \, b^{2} d^{2} e^{2} x^{3} + 90 \, b^{2} c d e^{2} x^{2} + 150 \, a b c d e^{2} + 7 \, {\left (9 \, b^{2} c^{2} + 7 \, a b d^{2}\right )} e^{2} x\right )} \sqrt {-b x^{2} + a} \sqrt {e x}\right )}}{315 \, b^{3}} \] Input:
integrate((e*x)^(5/2)*(d*x+c)^2/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/315*(150*sqrt(-b*e)*a^2*c*d*e^2*weierstrassPInverse(4*a/b, 0, x) - 21*( 9*a*b*c^2 + 7*a^2*d^2)*sqrt(-b*e)*e^2*weierstrassZeta(4*a/b, 0, weierstras sPInverse(4*a/b, 0, x)) + (35*b^2*d^2*e^2*x^3 + 90*b^2*c*d*e^2*x^2 + 150*a *b*c*d*e^2 + 7*(9*b^2*c^2 + 7*a*b*d^2)*e^2*x)*sqrt(-b*x^2 + a)*sqrt(e*x))/ b^3
Time = 18.77 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.47 \[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=\frac {c^{2} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {c d e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {13}{4}\right )} + \frac {d^{2} e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate((e*x)**(5/2)*(d*x+c)**2/(-b*x**2+a)**(1/2),x)
Output:
c**2*e**(5/2)*x**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), b*x**2*exp_po lar(2*I*pi)/a)/(2*sqrt(a)*gamma(11/4)) + c*d*e**(5/2)*x**(9/2)*gamma(9/4)* hyper((1/2, 9/4), (13/4,), b*x**2*exp_polar(2*I*pi)/a)/(sqrt(a)*gamma(13/4 )) + d**2*e**(5/2)*x**(11/2)*gamma(11/4)*hyper((1/2, 11/4), (15/4,), b*x** 2*exp_polar(2*I*pi)/a)/(2*sqrt(a)*gamma(15/4))
\[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^(5/2)*(d*x+c)^2/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^2*(e*x)^(5/2)/sqrt(-b*x^2 + a), x)
\[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^(5/2)*(d*x+c)^2/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^2*(e*x)^(5/2)/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^2}{\sqrt {a-b\,x^2}} \,d x \] Input:
int(((e*x)^(5/2)*(c + d*x)^2)/(a - b*x^2)^(1/2),x)
Output:
int(((e*x)^(5/2)*(c + d*x)^2)/(a - b*x^2)^(1/2), x)
\[ \int \frac {(e x)^{5/2} (c+d x)^2}{\sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, e^{2} \left (-300 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, a c d -98 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, a \,d^{2} x -126 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b \,c^{2} x -180 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b c d \,x^{2}-70 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b \,d^{2} x^{3}+150 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{3}+a x}d x \right ) a^{2} c d +147 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{2}+a}d x \right ) a^{2} d^{2}+189 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{2}+a}d x \right ) a b \,c^{2}\right )}{315 b^{2}} \] Input:
int((e*x)^(5/2)*(d*x+c)^2/(-b*x^2+a)^(1/2),x)
Output:
(sqrt(e)*e**2*( - 300*sqrt(x)*sqrt(a - b*x**2)*a*c*d - 98*sqrt(x)*sqrt(a - b*x**2)*a*d**2*x - 126*sqrt(x)*sqrt(a - b*x**2)*b*c**2*x - 180*sqrt(x)*sq rt(a - b*x**2)*b*c*d*x**2 - 70*sqrt(x)*sqrt(a - b*x**2)*b*d**2*x**3 + 150* int((sqrt(x)*sqrt(a - b*x**2))/(a*x - b*x**3),x)*a**2*c*d + 147*int((sqrt( x)*sqrt(a - b*x**2))/(a - b*x**2),x)*a**2*d**2 + 189*int((sqrt(x)*sqrt(a - b*x**2))/(a - b*x**2),x)*a*b*c**2))/(315*b**2)