Integrand size = 24, antiderivative size = 349 \[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=-\frac {10 a B d^2 \sqrt {d x} \sqrt {a+b x^2}}{21 b^2}+\frac {2 A d (d x)^{3/2} \sqrt {a+b x^2}}{5 b}+\frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {6 a A d^2 \sqrt {d x} \sqrt {a+b x^2}}{5 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {6 a^{5/4} A d^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {a+b x^2}}-\frac {a^{5/4} \left (63 A \sqrt {b}-25 \sqrt {a} B\right ) d^{5/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{105 b^{9/4} \sqrt {a+b x^2}} \] Output:
-10/21*a*B*d^2*(d*x)^(1/2)*(b*x^2+a)^(1/2)/b^2+2/5*A*d*(d*x)^(3/2)*(b*x^2+ a)^(1/2)/b+2/7*B*(d*x)^(5/2)*(b*x^2+a)^(1/2)/b-6/5*a*A*d^2*(d*x)^(1/2)*(b* x^2+a)^(1/2)/b^(3/2)/(a^(1/2)+b^(1/2)*x)+6/5*a^(5/4)*A*d^(5/2)*(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))/b^(7/4)/(b*x^2+a)^(1/2)-1 /105*a^(5/4)*(63*A*b^(1/2)-25*a^(1/2)*B)*d^(5/2)*(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))/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 = 133, normalized size of antiderivative = 0.38 \[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\frac {2 d^2 \sqrt {d x} \left (-\left (\left (a+b x^2\right ) (25 a B-3 b x (7 A+5 B x))\right )+25 a^2 B \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )-21 a A b x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{105 b^2 \sqrt {a+b x^2}} \] Input:
Integrate[((d*x)^(5/2)*(A + B*x))/Sqrt[a + b*x^2],x]
Output:
(2*d^2*Sqrt[d*x]*(-((a + b*x^2)*(25*a*B - 3*b*x*(7*A + 5*B*x))) + 25*a^2*B *Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)] - 21*a *A*b*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 = 0.85 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {552, 27, 552, 27, 552, 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.
| \(\displaystyle \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {2 d \int \frac {(d x)^{3/2} (5 a B-7 A b x)}{2 \sqrt {b x^2+a}}dx}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \int \frac {(d x)^{3/2} (5 a B-7 A b x)}{\sqrt {b x^2+a}}dx}{7 b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (-\frac {2 d \int -\frac {a b \sqrt {d x} (21 A+25 B x)}{2 \sqrt {b x^2+a}}dx}{5 b}-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \int \frac {\sqrt {d x} (21 A+25 B x)}{\sqrt {b x^2+a}}dx-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {2 d \int \frac {25 a B-63 A b x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{3 b}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {d \int \frac {25 a B-63 A b x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{3 b}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {d \sqrt {x} \int \frac {25 a B-63 A b x}{\sqrt {x} \sqrt {b x^2+a}}dx}{3 b \sqrt {d x}}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {2 d \sqrt {x} \int \frac {25 a B-63 A b x}{\sqrt {b x^2+a}}d\sqrt {x}}{3 b \sqrt {d x}}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {2 d \sqrt {x} \left (63 \sqrt {a} A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}-\sqrt {a} \left (63 A \sqrt {b}-25 \sqrt {a} B\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{3 b \sqrt {d x}}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {2 d \sqrt {x} \left (63 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}-\sqrt {a} \left (63 A \sqrt {b}-25 \sqrt {a} B\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{3 b \sqrt {d x}}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {2 d \sqrt {x} \left (63 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (63 A \sqrt {b}-25 \sqrt {a} B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{3 b \sqrt {d x}}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (d x)^{5/2} \sqrt {a+b x^2}}{7 b}-\frac {d \left (\frac {1}{5} a d \left (\frac {50 B \sqrt {d x} \sqrt {a+b x^2}}{3 b}-\frac {2 d \sqrt {x} \left (63 A \sqrt {b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )-\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (63 A \sqrt {b}-25 \sqrt {a} B\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{3 b \sqrt {d x}}\right )-\frac {14}{5} A (d x)^{3/2} \sqrt {a+b x^2}\right )}{7 b}\) |
Input:
Int[((d*x)^(5/2)*(A + B*x))/Sqrt[a + b*x^2],x]
Output:
(2*B*(d*x)^(5/2)*Sqrt[a + b*x^2])/(7*b) - (d*((-14*A*(d*x)^(3/2)*Sqrt[a + b*x^2])/5 + (a*d*((50*B*Sqrt[d*x]*Sqrt[a + b*x^2])/(3*b) - (2*d*Sqrt[x]*(6 3*A*Sqrt[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])) - ( a^(1/4)*(63*A*Sqrt[b] - 25*Sqrt[a]*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*b^(1/4)*Sqrt[a + b*x^2])))/(3*b*Sqrt[d*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[((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] :> 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 = 0.90 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {d^{2} \sqrt {d x}\, \left (126 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b -63 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2} b -25 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-a b}\, a^{2}-30 B \,b^{3} x^{5}-42 A \,b^{3} x^{4}+20 B a \,b^{2} x^{3}-42 A a \,b^{2} x^{2}+50 B \,a^{2} b x \right )}{105 x \sqrt {b \,x^{2}+a}\, b^{3}}\) | \(336\) |
| risch | \(\frac {2 \left (15 B b \,x^{2}+21 A b x -25 B a \right ) x \sqrt {b \,x^{2}+a}\, d^{3}}{105 b^{2} \sqrt {d x}}-\frac {a \left (-\frac {25 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b d \,x^{3}+a d x}}+\frac {63 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b d \,x^{3}+a d x}}\right ) d^{3} \sqrt {d \left (b \,x^{2}+a \right ) x}}{105 b^{2} \sqrt {d x}\, \sqrt {b \,x^{2}+a}}\) | \(348\) |
| elliptic | \(\frac {\sqrt {d \left (b \,x^{2}+a \right ) x}\, \sqrt {d x}\, \left (\frac {2 B \,d^{2} x^{2} \sqrt {b d \,x^{3}+a d x}}{7 b}+\frac {2 A \,d^{2} x \sqrt {b d \,x^{3}+a d x}}{5 b}-\frac {10 B \,d^{2} a \sqrt {b d \,x^{3}+a d x}}{21 b^{2}}+\frac {5 B \,d^{3} a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b d \,x^{3}+a d x}}-\frac {3 A \,d^{3} a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 b^{2} \sqrt {b d \,x^{3}+a d x}}\right )}{\sqrt {b \,x^{2}+a}\, d x}\) | \(388\) |
Input:
int((d*x)^(5/2)*(B*x+A)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/105*d^2/x*(d*x)^(1/2)/(b*x^2+a)^(1/2)*(126*A*((b*x+(-a*b)^(1/2))/(-a*b) ^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b* x)^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^ (1/2)*a^2*b-63*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/ 2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*EllipticF(((b*x+(-a*b) ^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*b-25*B*((b*x+(-a*b)^( 1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a *b)^(1/2)*b*x)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2 *2^(1/2))*2^(1/2)*(-a*b)^(1/2)*a^2-30*B*b^3*x^5-42*A*b^3*x^4+20*B*a*b^2*x^ 3-42*A*a*b^2*x^2+50*B*a^2*b*x)/b^3
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.30 \[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\frac {2 \, {\left (25 \, \sqrt {b d} B a^{2} d^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + 63 \, \sqrt {b d} A a b d^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (15 \, B b^{2} d^{2} x^{2} + 21 \, A b^{2} d^{2} x - 25 \, B a b d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x}\right )}}{105 \, b^{3}} \] Input:
integrate((d*x)^(5/2)*(B*x+A)/(b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
2/105*(25*sqrt(b*d)*B*a^2*d^2*weierstrassPInverse(-4*a/b, 0, x) + 63*sqrt( b*d)*A*a*b*d^2*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x )) + (15*B*b^2*d^2*x^2 + 21*A*b^2*d^2*x - 25*B*a*b*d^2)*sqrt(b*x^2 + a)*sq rt(d*x))/b^3
Result contains complex when optimal does not.
Time = 11.51 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.27 \[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\frac {A d^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {B d^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((d*x)**(5/2)*(B*x+A)/(b*x**2+a)**(1/2),x)
Output:
A*d**(5/2)*x**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), b*x**2*exp_polar (I*pi)/a)/(2*sqrt(a)*gamma(11/4)) + B*d**(5/2)*x**(9/2)*gamma(9/4)*hyper(( 1/2, 9/4), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(13/4))
\[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {5}{2}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x)^(5/2)*(B*x+A)/(b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(d*x)^(5/2)/sqrt(b*x^2 + a), x)
\[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {5}{2}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:
integrate((d*x)^(5/2)*(B*x+A)/(b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(d*x)^(5/2)/sqrt(b*x^2 + a), x)
Timed out. \[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x\right )}^{5/2}\,\left (A+B\,x\right )}{\sqrt {b\,x^2+a}} \,d x \] Input:
int(((d*x)^(5/2)*(A + B*x))/(a + b*x^2)^(1/2),x)
Output:
int(((d*x)^(5/2)*(A + B*x))/(a + b*x^2)^(1/2), x)
\[ \int \frac {(d x)^{5/2} (A+B x)}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {d}\, d^{2} \left (42 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a x -50 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a +30 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b \,x^{2}+25 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{3}+a x}d x \right ) a^{2}-63 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{2}+a}d x \right ) a^{2}\right )}{105 b} \] Input:
int((d*x)^(5/2)*(B*x+A)/(b*x^2+a)^(1/2),x)
Output:
(sqrt(d)*d**2*(42*sqrt(x)*sqrt(a + b*x**2)*a*x - 50*sqrt(x)*sqrt(a + b*x** 2)*a + 30*sqrt(x)*sqrt(a + b*x**2)*b*x**2 + 25*int((sqrt(x)*sqrt(a + b*x** 2))/(a*x + b*x**3),x)*a**2 - 63*int((sqrt(x)*sqrt(a + b*x**2))/(a + b*x**2 ),x)*a**2))/(105*b)