Integrand size = 25, antiderivative size = 249 \[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=-\frac {10 a d e^2 \sqrt {e x} \sqrt {a-b x^2}}{21 b^2}-\frac {2 c e (e x)^{3/2} \sqrt {a-b x^2}}{5 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}+\frac {6 a^{7/4} c e^{5/2} \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{5 b^{7/4} \sqrt {a-b x^2}}-\frac {2 a^{7/4} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) e^{5/2} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{105 b^{9/4} \sqrt {a-b x^2}} \] Output:
-10/21*a*d*e^2*(e*x)^(1/2)*(-b*x^2+a)^(1/2)/b^2-2/5*c*e*(e*x)^(3/2)*(-b*x^ 2+a)^(1/2)/b-2/7*d*(e*x)^(5/2)*(-b*x^2+a)^(1/2)/b+6/5*a^(7/4)*c*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^(7/4)/( -b*x^2+a)^(1/2)-2/105*a^(7/4)*(63*b^(1/2)*c-25*a^(1/2)*d)*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^(9/4)/(-b*x^2 +a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.54 \[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=\frac {2 e^2 \sqrt {e x} \left (-\left (\left (a-b x^2\right ) (25 a d+3 b x (7 c+5 d x))\right )+25 a^2 d \sqrt {1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^2}{a}\right )+21 a b c x \sqrt {1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^2}{a}\right )\right )}{105 b^2 \sqrt {a-b x^2}} \] Input:
Integrate[((e*x)^(5/2)*(c + d*x))/Sqrt[a - b*x^2],x]
Output:
(2*e^2*Sqrt[e*x]*(-((a - b*x^2)*(25*a*d + 3*b*x*(7*c + 5*d*x))) + 25*a^2*d *Sqrt[1 - (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (b*x^2)/a] + 21*a*b* c*x*Sqrt[1 - (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (b*x^2)/a]))/(105 *b^2*Sqrt[a - b*x^2])
Time = 0.92 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {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.
| \(\displaystyle \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 e \int \frac {(e x)^{3/2} (5 a d+7 b c x)}{2 \sqrt {a-b x^2}}dx}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {(e x)^{3/2} (5 a d+7 b c x)}{\sqrt {a-b x^2}}dx}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {e \left (\frac {2 e \int \frac {a b \sqrt {e x} (21 c+25 d x)}{2 \sqrt {a-b x^2}}dx}{5 b}-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \int \frac {\sqrt {e x} (21 c+25 d x)}{\sqrt {a-b x^2}}dx-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \int \frac {25 a d+63 b c x}{2 \sqrt {e x} \sqrt {a-b x^2}}dx}{3 b}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {e \int \frac {25 a d+63 b c x}{\sqrt {e x} \sqrt {a-b x^2}}dx}{3 b}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {e \sqrt {x} \int \frac {25 a d+63 b c x}{\sqrt {x} \sqrt {a-b x^2}}dx}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \int \frac {25 a d+63 b c x}{\sqrt {a-b x^2}}d\sqrt {x}}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (63 \sqrt {a} \sqrt {b} c \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^2}}d\sqrt {x}-\sqrt {a} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (63 \sqrt {b} c \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}-\sqrt {a} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (63 \sqrt {b} c \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}-\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {a-b x^2}}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (63 \sqrt {b} c \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}-\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (\frac {63 \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {a-b x^2}}-\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (\frac {63 \sqrt {a} \sqrt {b} c \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {x}}{\sqrt {a-b x^2}}-\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {e \left (\frac {1}{5} a e \left (\frac {2 e \sqrt {x} \left (\frac {63 a^{3/4} \sqrt [4]{b} c \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^2}}-\frac {a^{3/4} \sqrt {1-\frac {b x^2}{a}} \left (63 \sqrt {b} c-25 \sqrt {a} d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}\right )}{3 b \sqrt {e x}}-\frac {50 d \sqrt {e x} \sqrt {a-b x^2}}{3 b}\right )-\frac {14}{5} c (e x)^{3/2} \sqrt {a-b x^2}\right )}{7 b}-\frac {2 d (e x)^{5/2} \sqrt {a-b x^2}}{7 b}\) |
Input:
Int[((e*x)^(5/2)*(c + d*x))/Sqrt[a - b*x^2],x]
Output:
(-2*d*(e*x)^(5/2)*Sqrt[a - b*x^2])/(7*b) + (e*((-14*c*(e*x)^(3/2)*Sqrt[a - b*x^2])/5 + (a*e*((-50*d*Sqrt[e*x]*Sqrt[a - b*x^2])/(3*b) + (2*e*Sqrt[x]* ((63*a^(3/4)*b^(1/4)*c*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[ x])/a^(1/4)], -1])/Sqrt[a - b*x^2] - (a^(3/4)*(63*Sqrt[b]*c - 25*Sqrt[a]*d )*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])))/(3*b*Sqrt[e*x])))/5))/(7*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[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 = 0.95 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(\frac {e^{2} \sqrt {e x}\, \left (25 \sqrt {2}\, \sqrt {a b}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d -126 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b c +63 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b c +30 b^{3} d \,x^{5}+42 b^{3} c \,x^{4}+20 a \,b^{2} d \,x^{3}-42 a \,b^{2} c \,x^{2}-50 a^{2} b d x \right )}{105 x \sqrt {-b \,x^{2}+a}\, b^{3}}\) | \(315\) |
| risch | \(-\frac {2 \left (15 b d \,x^{2}+21 c b x +25 a d \right ) x \sqrt {-b \,x^{2}+a}\, e^{3}}{105 b^{2} \sqrt {e x}}+\frac {a \left (\frac {25 a d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {63 c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {-b e \,x^{3}+a e x}}\right ) e^{3} \sqrt {\left (-b \,x^{2}+a \right ) e x}}{105 b^{2} \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(333\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 d \,e^{2} x^{2} \sqrt {-b e \,x^{3}+a e x}}{7 b}-\frac {2 c \,e^{2} x \sqrt {-b e \,x^{3}+a e x}}{5 b}-\frac {10 d \,e^{2} a \sqrt {-b e \,x^{3}+a e x}}{21 b^{2}}+\frac {5 d \,e^{3} a^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {-b e \,x^{3}+a e x}}+\frac {3 c \,e^{3} a \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b^{2} \sqrt {-b e \,x^{3}+a e x}}\right )}{e x \sqrt {-b \,x^{2}+a}}\) | \(375\) |
Input:
int((e*x)^(5/2)*(d*x+c)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/105*e^2/x*(e*x)^(1/2)/(-b*x^2+a)^(1/2)*(25*2^(1/2)*(a*b)^(1/2)*((b*x+(a* b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/ (a*b)^(1/2))^(1/2)*EllipticF(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^( 1/2))*a^2*d-126*2^(1/2)*((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b) ^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)*EllipticE(((b*x+(a*b)^ (1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b*c+63*2^(1/2)*((b*x+(a*b)^(1/2 ))/(a*b)^(1/2))^(1/2)*((-b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^( 1/2))^(1/2)*EllipticF(((b*x+(a*b)^(1/2))/(a*b)^(1/2))^(1/2),1/2*2^(1/2))*a ^2*b*c+30*b^3*d*x^5+42*b^3*c*x^4+20*a*b^2*d*x^3-42*a*b^2*c*x^2-50*a^2*b*d* x)/b^3
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.43 \[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (25 \, \sqrt {-b e} a^{2} d e^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right ) - 63 \, \sqrt {-b e} a b c e^{2} {\rm weierstrassZeta}\left (\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (15 \, b^{2} d e^{2} x^{2} + 21 \, b^{2} c e^{2} x + 25 \, a b d e^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {e x}\right )}}{105 \, b^{3}} \] Input:
integrate((e*x)^(5/2)*(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/105*(25*sqrt(-b*e)*a^2*d*e^2*weierstrassPInverse(4*a/b, 0, x) - 63*sqrt (-b*e)*a*b*c*e^2*weierstrassZeta(4*a/b, 0, weierstrassPInverse(4*a/b, 0, x )) + (15*b^2*d*e^2*x^2 + 21*b^2*c*e^2*x + 25*a*b*d*e^2)*sqrt(-b*x^2 + a)*s qrt(e*x))/b^3
Time = 11.53 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.39 \[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=\frac {c e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {d e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((e*x)**(5/2)*(d*x+c)/(-b*x**2+a)**(1/2),x)
Output:
c*e**(5/2)*x**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), b*x**2*exp_polar (2*I*pi)/a)/(2*sqrt(a)*gamma(11/4)) + 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)/(2*sqrt(a)*gamma(13/4))
\[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {5}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^(5/2)*(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)*(e*x)^(5/2)/sqrt(-b*x^2 + a), x)
\[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {5}{2}}}{\sqrt {-b x^{2} + a}} \,d x } \] Input:
integrate((e*x)^(5/2)*(d*x+c)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)*(e*x)^(5/2)/sqrt(-b*x^2 + a), x)
Timed out. \[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\left (c+d\,x\right )}{\sqrt {a-b\,x^2}} \,d x \] Input:
int(((e*x)^(5/2)*(c + d*x))/(a - b*x^2)^(1/2),x)
Output:
int(((e*x)^(5/2)*(c + d*x))/(a - b*x^2)^(1/2), x)
\[ \int \frac {(e x)^{5/2} (c+d x)}{\sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, e^{2} \left (-50 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, a d -42 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b c x -30 \sqrt {x}\, \sqrt {-b \,x^{2}+a}\, b d \,x^{2}+25 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{3}+a x}d x \right ) a^{2} d +63 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{2}+a}d x \right ) a b c \right )}{105 b^{2}} \] Input:
int((e*x)^(5/2)*(d*x+c)/(-b*x^2+a)^(1/2),x)
Output:
(sqrt(e)*e**2*( - 50*sqrt(x)*sqrt(a - b*x**2)*a*d - 42*sqrt(x)*sqrt(a - b* x**2)*b*c*x - 30*sqrt(x)*sqrt(a - b*x**2)*b*d*x**2 + 25*int((sqrt(x)*sqrt( a - b*x**2))/(a*x - b*x**3),x)*a**2*d + 63*int((sqrt(x)*sqrt(a - b*x**2))/ (a - b*x**2),x)*a*b*c))/(105*b**2)