Integrand size = 24, antiderivative size = 367 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=-\frac {8 A b^2 \sqrt {a+b x^2}}{15 a d^5 \sqrt {d x}}+\frac {8 A b^{5/2} \sqrt {d x} \sqrt {a+b x^2}}{15 a d^6 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {4 b (7 A+15 B x) \sqrt {a+b x^2}}{105 d^3 (d x)^{5/2}}-\frac {2 (7 A+9 B x) \left (a+b x^2\right )^{3/2}}{63 d (d x)^{9/2}}-\frac {8 A b^{9/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{15 a^{3/4} d^{11/2} \sqrt {a+b x^2}}+\frac {4 b^{7/4} \left (7 A \sqrt {b}+15 \sqrt {a} B\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{105 a^{3/4} d^{11/2} \sqrt {a+b x^2}} \] Output:
-8/15*A*b^2*(b*x^2+a)^(1/2)/a/d^5/(d*x)^(1/2)+8/15*A*b^(5/2)*(d*x)^(1/2)*( b*x^2+a)^(1/2)/a/d^6/(a^(1/2)+b^(1/2)*x)-4/105*b*(15*B*x+7*A)*(b*x^2+a)^(1 /2)/d^3/(d*x)^(5/2)-2/63*(9*B*x+7*A)*(b*x^2+a)^(3/2)/d/(d*x)^(9/2)-8/15*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)*Ellipt icE(sin(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))),1/2*2^(1/2))/a^(3/4 )/d^(11/2)/(b*x^2+a)^(1/2)+4/105*b^(7/4)*(7*A*b^(1/2)+15*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*ar ctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2)),1/2*2^(1/2))/a^(3/4)/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 = 89, normalized size of antiderivative = 0.24 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=-\frac {2 a \sqrt {d x} \sqrt {a+b x^2} \left (7 A \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {b x^2}{a}\right )+9 B x \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{63 d^6 x^5 \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[((A + B*x)*(a + b*x^2)^(3/2))/(d*x)^(11/2),x]
Output:
(-2*a*Sqrt[d*x]*Sqrt[a + b*x^2]*(7*A*Hypergeometric2F1[-9/4, -3/2, -5/4, - ((b*x^2)/a)] + 9*B*x*Hypergeometric2F1[-7/4, -3/2, -3/4, -((b*x^2)/a)]))/( 63*d^6*x^5*Sqrt[1 + (b*x^2)/a])
Time = 0.90 (sec) , antiderivative size = 365, 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 = {546, 27, 546, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} (A+B x)}{(d x)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle -\frac {4 b \int -\frac {(7 A+9 B x) \sqrt {b x^2+a}}{2 (d x)^{7/2}}dx}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {(7 A+9 B x) \sqrt {b x^2+a}}{(d x)^{7/2}}dx}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 546 |
| \(\displaystyle \frac {2 b \left (-\frac {4 b \int -\frac {3 (7 A+15 B x)}{2 (d x)^{3/2} \sqrt {b x^2+a}}dx}{15 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \int \frac {7 A+15 B x}{(d x)^{3/2} \sqrt {b x^2+a}}dx}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (-\frac {2 \int -\frac {15 a B+7 A b x}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{a d}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {\int \frac {15 a B+7 A b x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{a d}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {\sqrt {x} \int \frac {15 a B+7 A b x}{\sqrt {x} \sqrt {b x^2+a}}dx}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 \sqrt {x} \int \frac {15 a B+7 A b x}{\sqrt {b x^2+a}}d\sqrt {x}}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (15 \sqrt {a} B+7 A \sqrt {b}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-7 \sqrt {a} A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 \sqrt {x} \left (\sqrt {a} \left (15 \sqrt {a} B+7 A \sqrt {b}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-7 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (15 \sqrt {a} B+7 A \sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}-7 A \sqrt {b} \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 b \left (\frac {2 b \left (\frac {2 \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (15 \sqrt {a} B+7 A \sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}-7 A \sqrt {b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{a d \sqrt {d x}}-\frac {14 A \sqrt {a+b x^2}}{a d \sqrt {d x}}\right )}{5 d^2}-\frac {2 \sqrt {a+b x^2} (7 A+15 B x)}{5 d (d x)^{5/2}}\right )}{21 d^2}-\frac {2 \left (a+b x^2\right )^{3/2} (7 A+9 B x)}{63 d (d x)^{9/2}}\) |
Input:
Int[((A + B*x)*(a + b*x^2)^(3/2))/(d*x)^(11/2),x]
Output:
(-2*(7*A + 9*B*x)*(a + b*x^2)^(3/2))/(63*d*(d*x)^(9/2)) + (2*b*((-2*(7*A + 15*B*x)*Sqrt[a + b*x^2])/(5*d*(d*x)^(5/2)) + (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] + 15*Sqrt[a]*B)*(Sqr t[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*Ar cTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*b^(1/4)*Sqrt[a + b*x^2])))/(a*d* Sqrt[d*x])))/(5*d^2)))/(21*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 = 1.63 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {\frac {8 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}}{15}-\frac {4 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}}{15}+\frac {4 B \sqrt {-a b}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{4}}{7}-\frac {8 A \,b^{3} x^{6}}{15}-\frac {6 B a \,b^{2} x^{5}}{7}-\frac {46 A a \,b^{2} x^{4}}{45}-\frac {8 a^{2} B b \,x^{3}}{7}-\frac {32 A \,a^{2} b \,x^{2}}{45}-\frac {2 B \,a^{3} x}{7}-\frac {2 a^{3} A}{9}}{x^{4} \sqrt {b \,x^{2}+a}\, a \,d^{5} \sqrt {d x}}\) | \(360\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (84 b^{2} A \,x^{4}+135 a b B \,x^{3}+77 a b A \,x^{2}+45 a^{2} B x +35 a^{2} A \right )}{315 x^{4} a \,d^{5} \sqrt {d x}}+\frac {4 b^{2} \left (\frac {15 B a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b d \,x^{3}+a d x}}+\frac {7 A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b d \,x^{3}+a d x}}\right ) \sqrt {d \left (b \,x^{2}+a \right ) x}}{105 a \,d^{5} \sqrt {d x}\, \sqrt {b \,x^{2}+a}}\) | \(374\) |
| elliptic | \(\frac {\sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (-\frac {2 a A \sqrt {b d \,x^{3}+a d x}}{9 d^{6} x^{5}}-\frac {2 B a \sqrt {b d \,x^{3}+a d x}}{7 d^{6} x^{4}}-\frac {22 b A \sqrt {b d \,x^{3}+a d x}}{45 d^{6} x^{3}}-\frac {6 B b \sqrt {b d \,x^{3}+a d x}}{7 d^{6} x^{2}}-\frac {8 \left (b d \,x^{2}+a d \right ) A \,b^{2}}{15 a \,d^{6} \sqrt {x \left (b d \,x^{2}+a d \right )}}+\frac {4 B b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{7 d^{5} \sqrt {b d \,x^{3}+a d x}}+\frac {4 A \,b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 a \,d^{5} \sqrt {b d \,x^{3}+a d x}}\right )}{\sqrt {d x}\, \sqrt {b \,x^{2}+a}}\) | \(436\) |
Input:
int((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(11/2),x,method=_RETURNVERBOSE)
Output:
2/315/x^4*(84*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^4-42*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)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2) )^(1/2),1/2*2^(1/2))*a*b^2*x^4+90*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^4-84*A*b^3*x^6-135*B*a*b^2*x^5-161*A*a*b^2*x^4-180*a^2*B*b*x^ 3-112*A*a^2*b*x^2-45*B*a^3*x-35*a^3*A)/(b*x^2+a)^(1/2)/a/d^5/(d*x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=\frac {2 \, {\left (180 \, \sqrt {b d} B a b x^{5} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - 84 \, \sqrt {b d} A b^{2} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (84 \, A b^{2} x^{4} + 135 \, B a b x^{3} + 77 \, A a b x^{2} + 45 \, B a^{2} x + 35 \, A a^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x}\right )}}{315 \, a d^{6} x^{5}} \] Input:
integrate((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(11/2),x, algorithm="fricas")
Output:
2/315*(180*sqrt(b*d)*B*a*b*x^5*weierstrassPInverse(-4*a/b, 0, x) - 84*sqrt (b*d)*A*b^2*x^5*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) - (84*A*b^2*x^4 + 135*B*a*b*x^3 + 77*A*a*b*x^2 + 45*B*a^2*x + 35*A*a^2 )*sqrt(b*x^2 + a)*sqrt(d*x))/(a*d^6*x^5)
Timed out. \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)*(b*x**2+a)**(3/2)/(d*x)**(11/2),x)
Output:
Timed out
\[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (d x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(11/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/2)*(B*x + A)/(d*x)^(11/2), x)
\[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{\left (d x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(11/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*(B*x + A)/(d*x)^(11/2), x)
Timed out. \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d\,x\right )}^{11/2}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(A + B*x))/(d*x)^(11/2),x)
Output:
int(((a + b*x^2)^(3/2)*(A + B*x))/(d*x)^(11/2), x)
\[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{(d x)^{11/2}} \, dx=\frac {2 \sqrt {d}\, \left (-5 \sqrt {b \,x^{2}+a}\, a^{2}-35 \sqrt {b \,x^{2}+a}\, a b \,x^{2}+21 \sqrt {b \,x^{2}+a}\, a b x -105 \sqrt {b \,x^{2}+a}\, b^{2} x^{3}+30 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{8}+a \,x^{6}}d x \right ) a^{3} x^{4}+126 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{7}+a \,x^{5}}d x \right ) a^{2} b \,x^{4}\right )}{105 \sqrt {x}\, d^{6} x^{4}} \] Input:
int((B*x+A)*(b*x^2+a)^(3/2)/(d*x)^(11/2),x)
Output:
(2*sqrt(d)*( - 5*sqrt(a + b*x**2)*a**2 - 35*sqrt(a + b*x**2)*a*b*x**2 + 21 *sqrt(a + b*x**2)*a*b*x - 105*sqrt(a + b*x**2)*b**2*x**3 + 30*sqrt(x)*int( (sqrt(x)*sqrt(a + b*x**2))/(a*x**6 + b*x**8),x)*a**3*x**4 + 126*sqrt(x)*in t((sqrt(x)*sqrt(a + b*x**2))/(a*x**5 + b*x**7),x)*a**2*b*x**4))/(105*sqrt( x)*d**6*x**4)