Integrand size = 24, antiderivative size = 393 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=-\frac {4 A b \sqrt {a+b x^2}}{45 a d^3 (d x)^{5/2}}-\frac {4 b B \sqrt {a+b x^2}}{21 a d^4 (d x)^{3/2}}+\frac {4 A b^2 \sqrt {a+b x^2}}{15 a^2 d^5 \sqrt {d x}}-\frac {4 A b^{5/2} \sqrt {d x} \sqrt {a+b x^2}}{15 a^2 d^6 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 (7 A+9 B x) \sqrt {a+b x^2}}{63 d (d x)^{9/2}}+\frac {4 A b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{15 a^{7/4} d^{11/2} \sqrt {a+b x^2}}-\frac {2 b^{7/4} \left (7 A \sqrt {b}+5 \sqrt {a} B\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{105 a^{7/4} d^{11/2} \sqrt {a+b x^2}} \] Output:
-4/45*A*b*(b*x^2+a)^(1/2)/a/d^3/(d*x)^(5/2)-4/21*b*B*(b*x^2+a)^(1/2)/a/d^4 /(d*x)^(3/2)+4/15*A*b^2*(b*x^2+a)^(1/2)/a^2/d^5/(d*x)^(1/2)-4/15*A*b^(5/2) *(d*x)^(1/2)*(b*x^2+a)^(1/2)/a^2/d^6/(a^(1/2)+b^(1/2)*x)-2/63*(9*B*x+7*A)* (b*x^2+a)^(1/2)/d/(d*x)^(9/2)+4/15*A*b^(9/4)*(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)/d^(11/2)/(b*x^2+a)^(1/2)-2/105*b^(7 /4)*(7*A*b^(1/2)+5*a^(1/2)*B)*(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)/d^(11/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.22 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=-\frac {2 \sqrt {d x} \sqrt {a+b x^2} \left (7 A \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {1}{2},-\frac {5}{4},-\frac {b x^2}{a}\right )+9 B x \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{63 d^6 x^5 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[((A + B*x)*Sqrt[a + b*x^2])/(d*x)^(11/2),x]
Output:
(-2*Sqrt[d*x]*Sqrt[a + b*x^2]*(7*A*Hypergeometric2F1[-9/4, -1/2, -5/4, -(( b*x^2)/a)] + 9*B*x*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b*x^2)/a)]))/(63 *d^6*x^5*Sqrt[1 + (b*x^2)/a])
Time = 0.99 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {546, 27, 553, 27, 553, 27, 553, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} (A+B x)}{(d x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle -\frac {4 b \int -\frac {7 A+9 B x}{2 (d x)^{7/2} \sqrt {b x^2+a}}dx}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {7 A+9 B x}{(d x)^{7/2} \sqrt {b x^2+a}}dx}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {2 b \left (-\frac {2 \int -\frac {3 (15 a B-7 A b x)}{2 (d x)^{5/2} \sqrt {b x^2+a}}dx}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {3 \int \frac {15 a B-7 A b x}{(d x)^{5/2} \sqrt {b x^2+a}}dx}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {2 \int \frac {3 a b (7 A+5 B x)}{2 (d x)^{3/2} \sqrt {b x^2+a}}dx}{3 a d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \int \frac {7 A+5 B x}{(d x)^{3/2} \sqrt {b x^2+a}}dx}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (-\frac {2 \int -\frac {5 a B+7 A b x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{a d}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {\int \frac {5 a B+7 A b x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{a d}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {\sqrt {x} \int \frac {5 a B+7 A b x}{\sqrt {x} \sqrt {b x^2+a}}dx}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {2 \sqrt {x} \int \frac {5 a B+7 A b x}{\sqrt {b x^2+a}}d\sqrt {x}}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+7 A \sqrt {b}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-7 \sqrt {a} A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (5 \sqrt {a} B+7 A \sqrt {b}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-7 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (5 \sqrt {a} B+7 A \sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}-7 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 b \left (\frac {3 \left (-\frac {b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (5 \sqrt {a} B+7 A \sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}-7 A \sqrt {b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{d}-\frac {10 B \sqrt {a+b x^2}}{d (d x)^{3/2}}\right )}{5 a d}-\frac {14 A \sqrt {a+b x^2}}{5 a d (d x)^{5/2}}\right )}{63 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
Input:
Int[((A + B*x)*Sqrt[a + b*x^2])/(d*x)^(11/2),x]
Output:
(-2*(7*A + 9*B*x)*Sqrt[a + b*x^2])/(63*d*(d*x)^(9/2)) + (2*b*((-14*A*Sqrt[ a + b*x^2])/(5*a*d*(d*x)^(5/2)) + (3*((-10*B*Sqrt[a + b*x^2])/(d*(d*x)^(3/ 2)) - (b*((-14*A*Sqrt[a + b*x^2])/(a*d*Sqrt[d*x]) + (2*Sqrt[x]*(-7*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)*( 7*A*Sqrt[b] + 5*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])))/(a*d*Sqrt[d*x])))/d))/(5*a*d)))/(63*d^2)
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 + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/(e*(m + 1)*(m + 2))), x] - Simp[2*b*(p/(e^2*(m + 1)*(m + 2))) Int[(e*x)^(m + 2)* (a + b*x^2)^(p - 1)*(c*(m + 2) + d*(m + 1)*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[p, 0] && LtQ[m, -2] && !ILtQ[m + 2*p + 3, 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[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 = 1.20 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(-\frac {2 \left (42 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}-21 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}+15 B \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{4}-42 A \,b^{3} x^{6}+30 B a \,b^{2} x^{5}-28 A a \,b^{2} x^{4}+75 a^{2} B b \,x^{3}+49 A \,a^{2} b \,x^{2}+45 B \,a^{3} x +35 a^{3} A \right )}{315 x^{4} \sqrt {b \,x^{2}+a}\, d^{5} \sqrt {d x}\, a^{2}}\) | \(360\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-42 b^{2} A \,x^{4}+30 a b B \,x^{3}+14 a b A \,x^{2}+45 a^{2} B x +35 a^{2} A \right )}{315 x^{4} a^{2} d^{5} \sqrt {d x}}-\frac {2 b^{2} \left (\frac {5 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b d \,x^{3}+a d x}}+\frac {7 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b d \,x^{3}+a d x}}\right ) \sqrt {d \left (b \,x^{2}+a \right ) x}}{105 a^{2} d^{5} \sqrt {d x}\, \sqrt {b \,x^{2}+a}}\) | \(374\) |
| elliptic | \(\frac {\sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (-\frac {2 A \sqrt {b d \,x^{3}+a d x}}{9 d^{6} x^{5}}-\frac {2 B \sqrt {b d \,x^{3}+a d x}}{7 d^{6} x^{4}}-\frac {4 b A \sqrt {b d \,x^{3}+a d x}}{45 a \,d^{6} x^{3}}-\frac {4 B b \sqrt {b d \,x^{3}+a d x}}{21 a \,d^{6} x^{2}}+\frac {4 \left (b d \,x^{2}+a d \right ) A \,b^{2}}{15 a^{2} d^{6} \sqrt {x \left (b d \,x^{2}+a d \right )}}-\frac {2 B b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 a \,d^{5} \sqrt {b d \,x^{3}+a d x}}-\frac {2 A \,b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a^{2} d^{5} \sqrt {b d \,x^{3}+a d x}}\right )}{\sqrt {d x}\, \sqrt {b \,x^{2}+a}}\) | \(443\) |
Input:
int((B*x+A)*(b*x^2+a)^(1/2)/(d*x)^(11/2),x,method=_RETURNVERBOSE)
Output:
-2/315/x^4*(42*A*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)*(-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))*a*b^2*x^4-21*A*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)*(-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))*a*b^2*x^4+15*B*(-a*b)^(1/2)*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)*(-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))*a*b*x^4-42*A*b^3*x^6+30*B*a*b^2*x^5-28*A*a*b^2*x^4+75*a^2*B*b*x^3+ 49*A*a^2*b*x^2+45*B*a^3*x+35*a^3*A)/(b*x^2+a)^(1/2)/d^5/(d*x)^(1/2)/a^2
Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.31 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=-\frac {2 \, {\left (30 \, \sqrt {b d} B a b x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - 42 \, \sqrt {b d} A b^{2} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (42 \, A b^{2} x^{4} - 30 \, B a b x^{3} - 14 \, A a b x^{2} - 45 \, B a^{2} x - 35 \, A a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x}\right )}}{315 \, a^{2} d^{6} x^{5}} \] Input:
integrate((B*x+A)*(b*x^2+a)^(1/2)/(d*x)^(11/2),x, algorithm="fricas")
Output:
-2/315*(30*sqrt(b*d)*B*a*b*x^5*weierstrassPInverse(-4*a/b, 0, x) - 42*sqrt (b*d)*A*b^2*x^5*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) - (42*A*b^2*x^4 - 30*B*a*b*x^3 - 14*A*a*b*x^2 - 45*B*a^2*x - 35*A*a^2) *sqrt(b*x^2 + a)*sqrt(d*x))/(a^2*d^6*x^5)
Result contains complex when optimal does not.
Time = 107.82 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.28 \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 d^{\frac {11}{2}} x^{\frac {9}{2}} \Gamma \left (- \frac {5}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 d^{\frac {11}{2}} x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} \] Input:
integrate((B*x+A)*(b*x**2+a)**(1/2)/(d*x)**(11/2),x)
Output:
A*sqrt(a)*gamma(-9/4)*hyper((-9/4, -1/2), (-5/4,), b*x**2*exp_polar(I*pi)/ a)/(2*d**(11/2)*x**(9/2)*gamma(-5/4)) + B*sqrt(a)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**2*exp_polar(I*pi)/a)/(2*d**(11/2)*x**(7/2)*gamma(-3/ 4))
\[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (B x + A\right )}}{\left (d x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x+A)*(b*x^2+a)^(1/2)/(d*x)^(11/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*(B*x + A)/(d*x)^(11/2), x)
\[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (B x + A\right )}}{\left (d x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x+A)*(b*x^2+a)^(1/2)/(d*x)^(11/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*(B*x + A)/(d*x)^(11/2), x)
Timed out. \[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x\right )}{{\left (d\,x\right )}^{11/2}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(A + B*x))/(d*x)^(11/2),x)
Output:
int(((a + b*x^2)^(1/2)*(A + B*x))/(d*x)^(11/2), x)
\[ \int \frac {(A+B x) \sqrt {a+b x^2}}{(d x)^{11/2}} \, dx=\frac {2 \sqrt {d}\, \left (-5 \sqrt {b \,x^{2}+a}\, a -7 \sqrt {b \,x^{2}+a}\, b x -5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{8}+a \,x^{6}}d x \right ) a^{2} x^{4}-7 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{7}+a \,x^{5}}d x \right ) a b \,x^{4}\right )}{35 \sqrt {x}\, d^{6} x^{4}} \] Input:
int((B*x+A)*(b*x^2+a)^(1/2)/(d*x)^(11/2),x)
Output:
(2*sqrt(d)*( - 5*sqrt(a + b*x**2)*a - 7*sqrt(a + b*x**2)*b*x - 5*sqrt(x)*i nt((sqrt(x)*sqrt(a + b*x**2))/(a*x**6 + b*x**8),x)*a**2*x**4 - 7*sqrt(x)*i nt((sqrt(x)*sqrt(a + b*x**2))/(a*x**5 + b*x**7),x)*a*b*x**4))/(35*sqrt(x)* d**6*x**4)