Integrand size = 24, antiderivative size = 339 \[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\sqrt {d x} (a B+3 A b x)}{6 a^2 b \sqrt {a+b x^2}}-\frac {A \sqrt {d x} \sqrt {a+b x^2}}{2 a^2 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {A \sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} \sqrt {a+b x^2}}-\frac {\left (3 A \sqrt {b}-\sqrt {a} B\right ) \sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{12 a^{7/4} b^{5/4} \sqrt {a+b x^2}} \] Output:
-1/3*(d*x)^(1/2)*(-A*b*x+B*a)/a/b/(b*x^2+a)^(3/2)+1/6*(d*x)^(1/2)*(3*A*b*x +B*a)/a^2/b/(b*x^2+a)^(1/2)-1/2*A*(d*x)^(1/2)*(b*x^2+a)^(1/2)/a^2/b^(1/2)/ (a^(1/2)+b^(1/2)*x)+1/2*A*d^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+ b^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^( 1/2))),1/2*2^(1/2))/a^(7/4)/b^(3/4)/(b*x^2+a)^(1/2)-1/12*(3*A*b^(1/2)-a^(1 /2)*B)*d^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2) *InverseJacobiAM(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2)),1/2*2^(1/2) )/a^(7/4)/b^(5/4)/(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 = 145, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {d x} \left (-a^2 B+5 a A b x+a b B x^2+3 A b^2 x^3+a B \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )-A b x \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{6 a^2 b \left (a+b x^2\right )^{3/2}} \] Input:
Integrate[(Sqrt[d*x]*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
(Sqrt[d*x]*(-(a^2*B) + 5*a*A*b*x + a*b*B*x^2 + 3*A*b^2*x^3 + a*B*(a + b*x^ 2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)] - A* b*x*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b* x^2)/a)]))/(6*a^2*b*(a + b*x^2)^(3/2))
Time = 0.48 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {550, 27, 551, 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.
| \(\displaystyle \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 550 |
| \(\displaystyle \frac {d \int \frac {a B+3 A b x}{2 \sqrt {d x} \left (b x^2+a\right )^{3/2}}dx}{3 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \frac {a B+3 A b x}{\sqrt {d x} \left (b x^2+a\right )^{3/2}}dx}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}-\frac {\int -\frac {a B-3 A b x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{a}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {\int \frac {a B-3 A b x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{2 a}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {x} \int \frac {a B-3 A b x}{\sqrt {x} \sqrt {b x^2+a}}dx}{2 a \sqrt {d x}}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {x} \int \frac {a B-3 A b x}{\sqrt {b x^2+a}}d\sqrt {x}}{a \sqrt {d x}}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {x} \left (3 \sqrt {a} A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}-\sqrt {a} \left (3 A \sqrt {b}-\sqrt {a} B\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {x} \left (3 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}-\sqrt {a} \left (3 A \sqrt {b}-\sqrt {a} B\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {x} \left (3 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (3 A \sqrt {b}-\sqrt {a} B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {x} \left (3 A \sqrt {b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (3 A \sqrt {b}-\sqrt {a} B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (a B+3 A b x)}{a d \sqrt {a+b x^2}}\right )}{6 a b}-\frac {\sqrt {d x} (a B-A b x)}{3 a b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[(Sqrt[d*x]*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
-1/3*(Sqrt[d*x]*(a*B - A*b*x))/(a*b*(a + b*x^2)^(3/2)) + (d*((Sqrt[d*x]*(a *B + 3*A*b*x))/(a*d*Sqrt[a + b*x^2]) + (Sqrt[x]*(3*A*Sqrt[b]*(-((Sqrt[x]*S qrt[a + b*x^2])/(Sqrt[a] + Sqrt[b]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sq rt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x ])/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])) - (a^(1/4)*(3*A*Sqrt[b] - Sq rt[a]*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*E llipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*b^(1/4)*Sqrt[a + b* x^2])))/(a*Sqrt[d*x])))/(6*a*b)
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*(a*d - b*c*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x ] - Simp[e/(2*a*b*(p + 1)) Int[(e*x)^(m - 1)*(a*d*m - b*c*(m + 2*p + 3)*x )*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && LtQ[0, m, 1]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
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 = 0.46 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.17
| method | result | size |
| elliptic | \(\frac {\sqrt {d x}\, \sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (\frac {\left (\frac {A x}{3 a \,b^{2}}-\frac {B}{3 b^{3}}\right ) \sqrt {b d \,x^{3}+a d x}}{\left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {2 b d x \left (-\frac {A x}{4 b \,a^{2}}-\frac {B}{12 a \,b^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b d x}}+\frac {d B \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 a \,b^{2} \sqrt {b d \,x^{3}+a d x}}-\frac {d A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 a^{2} b \sqrt {b d \,x^{3}+a d x}}\right )}{d x \sqrt {b \,x^{2}+a}}\) | \(397\) |
| default | \(\frac {\left (3 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-6 A \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, a \,b^{2} x^{2}+B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a b \,x^{2}+3 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b -6 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b +B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a b}\, a^{2}+6 A \,b^{3} x^{4}+2 B a \,b^{2} x^{3}+10 A a \,b^{2} x^{2}-2 B \,a^{2} b x \right ) \sqrt {d x}}{12 a^{2} x \,b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(596\) |
Input:
int((d*x)^(1/2)*(B*x+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/d/x*(d*x)^(1/2)/(b*x^2+a)^(1/2)*(d*(b*x^2+a)*x)^(1/2)*((1/3/a/b^2*A*x-1/ 3*B/b^3)*(b*d*x^3+a*d*x)^(1/2)/(x^2+a/b)^2-2*b*d*x*(-1/4*A/b/a^2*x-1/12*B/ a/b^2)/((x^2+a/b)*b*d*x)^(1/2)+1/12*d/a/b^2*B*(-a*b)^(1/2)*((x+(-a*b)^(1/2 )/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(- 1/(-a*b)^(1/2)*b*x)^(1/2)/(b*d*x^3+a*d*x)^(1/2)*EllipticF(((x+(-a*b)^(1/2) /b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))-1/4*d/a^2*A*(-a*b)^(1/2)/b*((x+(-a* b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^( 1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*d*x^3+a*d*x)^(1/2)*(-2*(-a*b)^(1/2)/b* EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^(1 /2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))))
Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (B a b^{2} x^{4} + 2 \, B a^{2} b x^{2} + B a^{3}\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + 3 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \sqrt {b d} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, A b^{3} x^{3} + B a b^{2} x^{2} + 5 \, A a b^{2} x - B a^{2} b\right )} \sqrt {b x^{2} + a} \sqrt {d x}}{6 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \] Input:
integrate((d*x)^(1/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/6*((B*a*b^2*x^4 + 2*B*a^2*b*x^2 + B*a^3)*sqrt(b*d)*weierstrassPInverse(- 4*a/b, 0, x) + 3*(A*b^3*x^4 + 2*A*a*b^2*x^2 + A*a^2*b)*sqrt(b*d)*weierstra ssZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + (3*A*b^3*x^3 + B*a* b^2*x^2 + 5*A*a*b^2*x - B*a^2*b)*sqrt(b*x^2 + a)*sqrt(d*x))/(a^2*b^4*x^4 + 2*a^3*b^3*x^2 + a^4*b^2)
Result contains complex when optimal does not.
Time = 35.85 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {A \sqrt {d} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {d} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((d*x)**(1/2)*(B*x+A)/(b*x**2+a)**(5/2),x)
Output:
A*sqrt(d)*x**(3/2)*gamma(3/4)*hyper((3/4, 5/2), (7/4,), b*x**2*exp_polar(I *pi)/a)/(2*a**(5/2)*gamma(7/4)) + B*sqrt(d)*x**(5/2)*gamma(5/4)*hyper((5/4 , 5/2), (9/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*gamma(9/4))
\[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {d x}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(1/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*sqrt(d*x)/(b*x^2 + a)^(5/2), x)
\[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {d x}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(1/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*sqrt(d*x)/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d\,x}\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((d*x)^(1/2)*(A + B*x))/(a + b*x^2)^(5/2),x)
Output:
int(((d*x)^(1/2)*(A + B*x))/(a + b*x^2)^(5/2), x)
\[ \int \frac {\sqrt {d x} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {d}\, \left (-2 \sqrt {x}\, \sqrt {b \,x^{2}+a}+5 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{3}+10 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{2} b \,x^{2}+5 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a \,b^{2} x^{4}+\left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{3}+2 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{2} b \,x^{2}+\left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a \,b^{2} x^{4}\right )}{5 b^{2} x^{4}+10 a b \,x^{2}+5 a^{2}} \] Input:
int((d*x)^(1/2)*(B*x+A)/(b*x^2+a)^(5/2),x)
Output:
(sqrt(d)*( - 2*sqrt(x)*sqrt(a + b*x**2) + 5*int((sqrt(x)*sqrt(a + b*x**2)) /(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6),x)*a**3 + 10*int((sqrt (x)*sqrt(a + b*x**2))/(a**3 + 3*a**2*b*x**2 + 3*a*b**2*x**4 + b**3*x**6),x )*a**2*b*x**2 + 5*int((sqrt(x)*sqrt(a + b*x**2))/(a**3 + 3*a**2*b*x**2 + 3 *a*b**2*x**4 + b**3*x**6),x)*a*b**2*x**4 + int((sqrt(x)*sqrt(a + b*x**2))/ (a**3*x + 3*a**2*b*x**3 + 3*a*b**2*x**5 + b**3*x**7),x)*a**3 + 2*int((sqrt (x)*sqrt(a + b*x**2))/(a**3*x + 3*a**2*b*x**3 + 3*a*b**2*x**5 + b**3*x**7) ,x)*a**2*b*x**2 + int((sqrt(x)*sqrt(a + b*x**2))/(a**3*x + 3*a**2*b*x**3 + 3*a*b**2*x**5 + b**3*x**7),x)*a*b**2*x**4))/(5*(a**2 + 2*a*b*x**2 + b**2* x**4))