Integrand size = 24, antiderivative size = 331 \[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}+\frac {d \sqrt {d x} (A+3 B x)}{6 a b \sqrt {a+b x^2}}-\frac {B d \sqrt {d x} \sqrt {a+b x^2}}{2 a b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {B d^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{2 a^{3/4} b^{7/4} \sqrt {a+b x^2}}+\frac {\left (A \sqrt {b}-3 \sqrt {a} B\right ) d^{3/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{12 a^{5/4} b^{7/4} \sqrt {a+b x^2}} \] Output:
-1/3*d*(d*x)^(1/2)*(B*x+A)/b/(b*x^2+a)^(3/2)+1/6*d*(d*x)^(1/2)*(3*B*x+A)/a /b/(b*x^2+a)^(1/2)-1/2*B*d*(d*x)^(1/2)*(b*x^2+a)^(1/2)/a/b^(3/2)/(a^(1/2)+ b^(1/2)*x)+1/2*B*d^(3/2)*(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^(3/4)/b^(7/4)/(b*x^2+a)^(1/2)+1/12*(A*b^(1/2)-3*a^(1/2)*B)*d^ (3/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJ acobiAM(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2)),1/2*2^(1/2))/a^(5/4) /b^(7/4)/(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 = 138, normalized size of antiderivative = 0.42 \[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \sqrt {d x} \left (a A-a B x-A b x^2-3 b B x^3-A \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )+B x \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{6 a b \left (a+b x^2\right )^{3/2}} \] Input:
Integrate[((d*x)^(3/2)*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
-1/6*(d*Sqrt[d*x]*(a*A - a*B*x - A*b*x^2 - 3*b*B*x^3 - A*(a + b*x^2)*Sqrt[ 1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)] + B*x*(a + b *x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)])) /(a*b*(a + b*x^2)^(3/2))
Time = 0.47 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {549, 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.
| \(\displaystyle \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {d^2 \int \frac {A+3 B x}{2 \sqrt {d x} \left (b x^2+a\right )^{3/2}}dx}{3 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {A+3 B x}{\sqrt {d x} \left (b x^2+a\right )^{3/2}}dx}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}-\frac {\int -\frac {A-3 B x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{a}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {\int \frac {A-3 B x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{2 a}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {x} \int \frac {A-3 B x}{\sqrt {x} \sqrt {b x^2+a}}dx}{2 a \sqrt {d x}}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {x} \int \frac {A-3 B x}{\sqrt {b x^2+a}}d\sqrt {x}}{a \sqrt {d x}}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {x} \left (\left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}+\frac {3 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {x} \left (\left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}+\frac {3 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {x} \left (\frac {3 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}+\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {d^2 \left (\frac {\sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}+\frac {3 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {b}}\right )}{a \sqrt {d x}}+\frac {\sqrt {d x} (A+3 B x)}{a d \sqrt {a+b x^2}}\right )}{6 b}-\frac {d \sqrt {d x} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((d*x)^(3/2)*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
-1/3*(d*Sqrt[d*x]*(A + B*x))/(b*(a + b*x^2)^(3/2)) + (d^2*((Sqrt[d*x]*(A + 3*B*x))/(a*d*Sqrt[a + b*x^2]) + (Sqrt[x]*((3*B*(-((Sqrt[x]*Sqrt[a + b*x^2 ])/(Sqrt[a] + Sqrt[b]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(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])))/Sqrt[b] + ((A - (3*Sqrt[a]*B)/Sqrt[b])*( Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2 *ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[a + b*x^ 2])))/(a*Sqrt[d*x])))/(6*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*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b *x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[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 = 1.23 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.21
| method | result | size |
| elliptic | \(\frac {\sqrt {d x}\, \sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (\frac {\left (-\frac {d B x}{3 b^{3}}-\frac {d A}{3 b^{3}}\right ) \sqrt {b d \,x^{3}+a d x}}{\left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {2 b d x \left (-\frac {d B x}{4 a \,b^{2}}-\frac {A d}{12 a \,b^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b d x}}+\frac {d^{2} A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 b^{2} a \sqrt {b d \,x^{3}+a d x}}-\frac {d^{2} B \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 b^{2} a \sqrt {b d \,x^{3}+a d x}}\right )}{d x \sqrt {b \,x^{2}+a}}\) | \(402\) |
| default | \(\frac {\left (A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b \,x^{2}-6 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}+3 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}+A \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a -6 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}+3 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}+6 x^{4} B \,b^{2}+2 A \,x^{3} b^{2}+2 B a \,x^{2} b -2 a b A x \right ) d \sqrt {d x}}{12 x \,b^{2} a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(583\) |
Input:
int((d*x)^(3/2)*(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*d/b^3*B*x-1 /3*d/b^3*A)*(b*d*x^3+a*d*x)^(1/2)/(x^2+a/b)^2-2*b*d*x*(-1/4*d/a*B/b^2*x-1/ 12*A/a*d/b^2)/((x^2+a/b)*b*d*x)^(1/2)+1/12/b^2*d^2/a*A*(-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/b^2*d^2/a*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)*(-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 = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.51 \[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {{\left (A b^{2} d x^{4} + 2 \, A a b d x^{2} + A a^{2} d\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + 3 \, {\left (B b^{2} d x^{4} + 2 \, B a b d x^{2} + B a^{2} d\right )} \sqrt {b d} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, B b^{2} d x^{3} + A b^{2} d x^{2} + B a b d x - A a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x}}{6 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \] Input:
integrate((d*x)^(3/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/6*((A*b^2*d*x^4 + 2*A*a*b*d*x^2 + A*a^2*d)*sqrt(b*d)*weierstrassPInverse (-4*a/b, 0, x) + 3*(B*b^2*d*x^4 + 2*B*a*b*d*x^2 + B*a^2*d)*sqrt(b*d)*weier strassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + (3*B*b^2*d*x^3 + A*b^2*d*x^2 + B*a*b*d*x - A*a*b*d)*sqrt(b*x^2 + a)*sqrt(d*x))/(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)
Result contains complex when optimal does not.
Time = 70.49 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.28 \[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {A d^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B d^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {5}{2} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((d*x)**(3/2)*(B*x+A)/(b*x**2+a)**(5/2),x)
Output:
A*d**(3/2)*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)) + B*d**(3/2)*x**(7/2)*gamma(7/4)*hyper((7 /4, 5/2), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*gamma(11/4))
\[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(3/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(d*x)^(3/2)/(b*x^2 + a)^(5/2), x)
\[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(3/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(d*x)^(3/2)/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((d*x)^(3/2)*(A + B*x))/(a + b*x^2)^(5/2),x)
Output:
int(((d*x)^(3/2)*(A + B*x))/(a + b*x^2)^(5/2), x)
\[ \int \frac {(d x)^{3/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {d}\, d \left (-6 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a -10 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b x +15 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{3} b +30 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{2} b^{2} x^{2}+15 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a \,b^{3} x^{4}+3 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{4}+6 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{3} b \,x^{2}+3 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{2} b^{2} x^{4}\right )}{15 b \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((d*x)^(3/2)*(B*x+A)/(b*x^2+a)^(5/2),x)
Output:
(sqrt(d)*d*( - 6*sqrt(x)*sqrt(a + b*x**2)*a - 10*sqrt(x)*sqrt(a + b*x**2)* b*x + 15*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*b + 30*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**2*x**2 + 15*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**3*x**4 + 3*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**4 + 6*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*b*x**2 + 3*int((sq rt(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**2*x**4))/(15*b*(a**2 + 2*a*b*x**2 + b**2*x**4))