Integrand size = 27, antiderivative size = 232 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 c^2 \sqrt {a-b x^2}}{3 a e (e x)^{3/2}}-\frac {4 c d \sqrt {a-b x^2}}{a e^2 \sqrt {e x}}-\frac {4 \sqrt [4]{b} c d \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{\sqrt [4]{a} e^{5/2} \sqrt {a-b x^2}}+\frac {2 \left (b c^2+6 \sqrt {a} \sqrt {b} c d+3 a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{3 a^{3/4} \sqrt [4]{b} e^{5/2} \sqrt {a-b x^2}} \] Output:
-2/3*c^2*(-b*x^2+a)^(1/2)/a/e/(e*x)^(3/2)-4*c*d*(-b*x^2+a)^(1/2)/a/e^2/(e* x)^(1/2)-4*b^(1/4)*c*d*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/a^( 1/4)/e^(1/2),I)/a^(1/4)/e^(5/2)/(-b*x^2+a)^(1/2)+2/3*(b*c^2+6*a^(1/2)*b^(1 /2)*c*d+3*a*d^2)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e ^(1/2),I)/a^(3/4)/b^(1/4)/e^(5/2)/(-b*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 = 110, normalized size of antiderivative = 0.47 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {2 x \sqrt {1-\frac {b x^2}{a}} \left (-c^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},\frac {b x^2}{a}\right )+3 d x \left (-2 c \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\frac {b x^2}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^2}{a}\right )\right )\right )}{3 (e x)^{5/2} \sqrt {a-b x^2}} \] Input:
Integrate[(c + d*x)^2/((e*x)^(5/2)*Sqrt[a - b*x^2]),x]
Output:
(2*x*Sqrt[1 - (b*x^2)/a]*(-(c^2*Hypergeometric2F1[-3/4, 1/2, 1/4, (b*x^2)/ a]) + 3*d*x*(-2*c*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^2)/a] + d*x*Hyper geometric2F1[1/4, 1/2, 5/4, (b*x^2)/a])))/(3*(e*x)^(5/2)*Sqrt[a - b*x^2])
Time = 1.05 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.19, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {559, 27, 553, 27, 553, 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.
| \(\displaystyle \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle \frac {2 \int \frac {b c^2+2 b d x c+3 a d^2}{2 (e x)^{5/2} \sqrt {a-b x^2}}dx}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b c^2+2 b d x c+3 a d^2}{(e x)^{5/2} \sqrt {a-b x^2}}dx}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b \left (6 a c d+\left (b c^2+3 a d^2\right ) x\right )}{2 (e x)^{3/2} \sqrt {a-b x^2}}dx}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \int \frac {6 a c d+\left (b c^2+3 a d^2\right ) x}{(e x)^{3/2} \sqrt {a-b x^2}}dx}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {\frac {b \left (-\frac {2 \int -\frac {a \left (b c^2-6 b d x c+3 a d^2\right )}{2 \sqrt {e x} \sqrt {a-b x^2}}dx}{a e}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \left (\frac {\int \frac {b c^2-6 b d x c+3 a d^2}{\sqrt {e x} \sqrt {a-b x^2}}dx}{e}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {\frac {b \left (\frac {\sqrt {x} \int \frac {b c^2-6 b d x c+3 a d^2}{\sqrt {x} \sqrt {a-b x^2}}dx}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \int \frac {b c^2-6 b d x c+3 a d^2}{\sqrt {a-b x^2}}d\sqrt {x}}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}-6 \sqrt {a} \sqrt {b} c d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^2}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {x}-6 \sqrt {b} c d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {a-b x^2}}-6 \sqrt {b} c d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-6 \sqrt {b} c d \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {a-b x^2}}d\sqrt {x}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {6 \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {b} x+\sqrt {a}}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {x}}{\sqrt {a-b x^2}}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {6 \sqrt {a} \sqrt {b} c d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {x}}{\sqrt {a-b x^2}}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^2}{a}} \left (6 \sqrt {a} \sqrt {b} c d+3 a d^2+b c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {6 a^{3/4} \sqrt [4]{b} c d \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^2}}\right )}{e \sqrt {e x}}-\frac {12 c d \sqrt {a-b x^2}}{e \sqrt {e x}}\right )}{3 a e}-\frac {2 \sqrt {a-b x^2} \left (3 a d^2+b c^2\right )}{3 a e (e x)^{3/2}}}{b}+\frac {2 d^2 \sqrt {a-b x^2}}{b e (e x)^{3/2}}\) |
Input:
Int[(c + d*x)^2/((e*x)^(5/2)*Sqrt[a - b*x^2]),x]
Output:
(2*d^2*Sqrt[a - b*x^2])/(b*e*(e*x)^(3/2)) + ((-2*(b*c^2 + 3*a*d^2)*Sqrt[a - b*x^2])/(3*a*e*(e*x)^(3/2)) + (b*((-12*c*d*Sqrt[a - b*x^2])/(e*Sqrt[e*x] ) + (2*Sqrt[x]*((-6*a^(3/4)*b^(1/4)*c*d*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcS in[(b^(1/4)*Sqrt[x])/a^(1/4)], -1])/Sqrt[a - b*x^2] + (a^(1/4)*(b*c^2 + 6* Sqrt[a]*Sqrt[b]*c*d + 3*a*d^2)*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])))/(e*Sqrt[e*x])))/(3* a*e))/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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(180)=360\).
Time = 1.38 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.60
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 c^{2} \sqrt {-b e \,x^{3}+a e x}}{3 e^{3} a \,x^{2}}-\frac {4 \left (-b e \,x^{2}+a e \right ) c d}{e^{3} a \sqrt {x \left (-b e \,x^{2}+a e \right )}}+\frac {\left (\frac {d^{2}}{e^{2}}+\frac {b \,c^{2}}{3 a \,e^{2}}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}-\frac {2 c d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{a \,e^{2} \sqrt {-b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(371\) |
| default | \(-\frac {6 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c d x -3 \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {a b}\, a \,d^{2} x -\sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {a b}\, b \,c^{2} x -12 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) a b c d x \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}-12 b^{2} c d \,x^{3}-2 b^{2} c^{2} x^{2}+12 a b c d x +2 a b \,c^{2}}{3 x \sqrt {-b \,x^{2}+a}\, b a \,e^{2} \sqrt {e x}}\) | \(392\) |
| risch | \(-\frac {2 \sqrt {-b \,x^{2}+a}\, c \left (6 d x +c \right )}{3 a x \,e^{2} \sqrt {e x}}+\frac {\left (\frac {3 a \,d^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-b e \,x^{3}+a e x}}+\frac {c^{2} \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-b e \,x^{3}+a e x}}-\frac {6 d c \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {-b e \,x^{3}+a e x}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) e x}}{3 a \,e^{2} \sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(429\) |
Input:
int((d*x+c)^2/(e*x)^(5/2)/(-b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((-b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(-b*x^2+a)^(1/2)*(-2/3/e^3/a*c^2*(-b*e* x^3+a*e*x)^(1/2)/x^2-4*(-b*e*x^2+a*e)/e^3/a*c*d/(x*(-b*e*x^2+a*e))^(1/2)+( d^2/e^2+1/3/a*b/e^2*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)*EllipticF(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1 /2),1/2*2^(1/2))-2/a/e^2*c*d*(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))))
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.45 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left (6 \, \sqrt {-b e} b c d x^{2} {\rm weierstrassZeta}\left (\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (b c^{2} + 3 \, a d^{2}\right )} \sqrt {-b e} x^{2} {\rm weierstrassPInverse}\left (\frac {4 \, a}{b}, 0, x\right ) + {\left (6 \, b c d x + b c^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {e x}\right )}}{3 \, a b e^{3} x^{2}} \] Input:
integrate((d*x+c)^2/(e*x)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
-2/3*(6*sqrt(-b*e)*b*c*d*x^2*weierstrassZeta(4*a/b, 0, weierstrassPInverse (4*a/b, 0, x)) + (b*c^2 + 3*a*d^2)*sqrt(-b*e)*x^2*weierstrassPInverse(4*a/ b, 0, x) + (6*b*c*d*x + b*c^2)*sqrt(-b*x^2 + a)*sqrt(e*x))/(a*b*e^3*x^2)
Time = 8.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {c^{2} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {c d \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt {a} e^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {d^{2} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((d*x+c)**2/(e*x)**(5/2)/(-b*x**2+a)**(1/2),x)
Output:
c**2*gamma(-3/4)*hyper((-3/4, 1/2), (1/4,), b*x**2*exp_polar(2*I*pi)/a)/(2 *sqrt(a)*e**(5/2)*x**(3/2)*gamma(1/4)) + c*d*gamma(-1/4)*hyper((-1/4, 1/2) , (3/4,), b*x**2*exp_polar(2*I*pi)/a)/(sqrt(a)*e**(5/2)*sqrt(x)*gamma(3/4) ) + d**2*sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**2*exp_polar(2*I *pi)/a)/(2*sqrt(a)*e**(5/2)*gamma(5/4))
\[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\sqrt {-b x^{2} + a} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^2/(e*x)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^2/(sqrt(-b*x^2 + a)*(e*x)^(5/2)), x)
\[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\sqrt {-b x^{2} + a} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x+c)^2/(e*x)^(5/2)/(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^2/(sqrt(-b*x^2 + a)*(e*x)^(5/2)), x)
Timed out. \[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (e\,x\right )}^{5/2}\,\sqrt {a-b\,x^2}} \,d x \] Input:
int((c + d*x)^2/((e*x)^(5/2)*(a - b*x^2)^(1/2)),x)
Output:
int((c + d*x)^2/((e*x)^(5/2)*(a - b*x^2)^(1/2)), x)
\[ \int \frac {(c+d x)^2}{(e x)^{5/2} \sqrt {a-b x^2}} \, dx=\frac {\sqrt {e}\, \left (\left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{5}+a \,x^{3}}d x \right ) c^{2}+2 \left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{4}+a \,x^{2}}d x \right ) c d +\left (\int \frac {\sqrt {x}\, \sqrt {-b \,x^{2}+a}}{-b \,x^{3}+a x}d x \right ) d^{2}\right )}{e^{3}} \] Input:
int((d*x+c)^2/(e*x)^(5/2)/(-b*x^2+a)^(1/2),x)
Output:
(sqrt(e)*(int((sqrt(x)*sqrt(a - b*x**2))/(a*x**3 - b*x**5),x)*c**2 + 2*int ((sqrt(x)*sqrt(a - b*x**2))/(a*x**2 - b*x**4),x)*c*d + int((sqrt(x)*sqrt(a - b*x**2))/(a*x - b*x**3),x)*d**2))/e**3