Integrand size = 24, antiderivative size = 391 \[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}-\frac {d^3 (d x)^{5/2} (9 A+11 B x)}{6 b^2 \sqrt {a+b x^2}}+\frac {5 A d^5 \sqrt {d x} \sqrt {a+b x^2}}{2 b^3}+\frac {77 B d^4 (d x)^{3/2} \sqrt {a+b x^2}}{30 b^3}-\frac {77 a B d^5 \sqrt {d x} \sqrt {a+b x^2}}{10 b^{7/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {77 a^{5/4} B d^{11/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{10 b^{15/4} \sqrt {a+b x^2}}-\frac {a^{3/4} \left (25 A \sqrt {b}+77 \sqrt {a} B\right ) d^{11/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{20 b^{15/4} \sqrt {a+b x^2}} \] Output:
-1/3*d*(d*x)^(9/2)*(B*x+A)/b/(b*x^2+a)^(3/2)-1/6*d^3*(d*x)^(5/2)*(11*B*x+9 *A)/b^2/(b*x^2+a)^(1/2)+5/2*A*d^5*(d*x)^(1/2)*(b*x^2+a)^(1/2)/b^3+77/30*B* d^4*(d*x)^(3/2)*(b*x^2+a)^(1/2)/b^3-77/10*a*B*d^5*(d*x)^(1/2)*(b*x^2+a)^(1 /2)/b^(7/2)/(a^(1/2)+b^(1/2)*x)+77/10*a^(5/4)*B*d^(11/2)*(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^(15/4)/(b*x^2+a)^(1/2)-1/20*a ^(3/4)*(25*A*b^(1/2)+77*a^(1/2)*B)*d^(11/2)*(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^(15/4)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.42 \[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {d^5 \sqrt {d x} \left (75 a^2 A+77 a^2 B x+105 a A b x^2+99 a b B x^3+20 A b^2 x^4+12 b^2 B x^5-75 a A \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )-77 a B x \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{30 b^3 \left (a+b x^2\right )^{3/2}} \] Input:
Integrate[((d*x)^(11/2)*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
(d^5*Sqrt[d*x]*(75*a^2*A + 77*a^2*B*x + 105*a*A*b*x^2 + 99*a*b*B*x^3 + 20* A*b^2*x^4 + 12*b^2*B*x^5 - 75*a*A*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeo metric2F1[1/4, 1/2, 5/4, -((b*x^2)/a)] - 77*a*B*x*(a + b*x^2)*Sqrt[1 + (b* x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^2)/a)]))/(30*b^3*(a + b*x^ 2)^(3/2))
Time = 0.57 (sec) , antiderivative size = 391, 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 = {549, 27, 549, 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.
| \(\displaystyle \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {d^2 \int \frac {(d x)^{7/2} (9 A+11 B x)}{2 \left (b x^2+a\right )^{3/2}}dx}{3 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \int \frac {(d x)^{7/2} (9 A+11 B x)}{\left (b x^2+a\right )^{3/2}}dx}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {(d x)^{3/2} (45 A+77 B x)}{2 \sqrt {b x^2+a}}dx}{b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \int \frac {(d x)^{3/2} (45 A+77 B x)}{\sqrt {b x^2+a}}dx}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {2 d \int \frac {3 \sqrt {d x} (77 a B-75 A b x)}{2 \sqrt {b x^2+a}}dx}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \int \frac {\sqrt {d x} (77 a B-75 A b x)}{\sqrt {b x^2+a}}dx}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (-\frac {2 d \int -\frac {3 a b (25 A+77 B x)}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{3 b}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (a d \int \frac {25 A+77 B x}{\sqrt {d x} \sqrt {b x^2+a}}dx-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (\frac {a d \sqrt {x} \int \frac {25 A+77 B x}{\sqrt {x} \sqrt {b x^2+a}}dx}{\sqrt {d x}}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (\frac {2 a d \sqrt {x} \int \frac {25 A+77 B x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {d x}}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (\frac {2 a d \sqrt {x} \left (\left (\frac {77 \sqrt {a} B}{\sqrt {b}}+25 A\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-\frac {77 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {d x}}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (\frac {2 a d \sqrt {x} \left (\left (\frac {77 \sqrt {a} B}{\sqrt {b}}+25 A\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}-\frac {77 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {d x}}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (\frac {2 a d \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (\frac {77 \sqrt {a} B}{\sqrt {b}}+25 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}-\frac {77 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {d x}}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {d^2 \left (\frac {d^2 \left (\frac {154 B (d x)^{3/2} \sqrt {a+b x^2}}{5 b}-\frac {3 d \left (\frac {2 a d \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (\frac {77 \sqrt {a} B}{\sqrt {b}}+25 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}-\frac {77 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {b}}\right )}{\sqrt {d x}}-50 A \sqrt {d x} \sqrt {a+b x^2}\right )}{5 b}\right )}{2 b}-\frac {d (d x)^{5/2} (9 A+11 B x)}{b \sqrt {a+b x^2}}\right )}{6 b}-\frac {d (d x)^{9/2} (A+B x)}{3 b \left (a+b x^2\right )^{3/2}}\) |
Input:
Int[((d*x)^(11/2)*(A + B*x))/(a + b*x^2)^(5/2),x]
Output:
-1/3*(d*(d*x)^(9/2)*(A + B*x))/(b*(a + b*x^2)^(3/2)) + (d^2*(-((d*(d*x)^(5 /2)*(9*A + 11*B*x))/(b*Sqrt[a + b*x^2])) + (d^2*((154*B*(d*x)^(3/2)*Sqrt[a + b*x^2])/(5*b) - (3*d*(-50*A*Sqrt[d*x]*Sqrt[a + b*x^2] + (2*a*d*Sqrt[x]* ((-77*B*(-((Sqrt[x]*Sqrt[a + b*x^2])/(Sqrt[a] + Sqrt[b]*x)) + (a^(1/4)*(Sq rt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*A rcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])))/Sqrt[b ] + ((25*A + (77*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])))/Sqrt[d*x]))/(5*b)))/(2*b)))/(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[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 = 2.98 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\sqrt {d x}\, \sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (\frac {\left (-\frac {a^{2} d^{5} B x}{3 b^{5}}-\frac {a^{2} d^{5} A}{3 b^{5}}\right ) \sqrt {b d \,x^{3}+a d x}}{\left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {2 b d x \left (-\frac {5 a \,d^{5} B x}{4 b^{4}}-\frac {13 a \,d^{5} A}{12 b^{4}}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b d x}}+\frac {2 B \,d^{5} x \sqrt {b d \,x^{3}+a d x}}{5 b^{3}}+\frac {2 A \,d^{5} \sqrt {b d \,x^{3}+a d x}}{3 b^{3}}-\frac {5 A a \,d^{6} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{4 b^{4} \sqrt {b d \,x^{3}+a d x}}-\frac {77 B a \,d^{6} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{20 b^{4} \sqrt {b d \,x^{3}+a d x}}\right )}{d x \sqrt {b \,x^{2}+a}}\) | \(453\) |
| default | \(-\frac {\left (75 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}+462 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}-231 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}-24 B \,b^{3} x^{6}+75 A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-40 A \,b^{3} x^{5}+462 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-231 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-198 B a \,b^{2} x^{4}-210 A a \,x^{3} b^{2}-154 B \,a^{2} b \,x^{2}-150 A \,a^{2} b x \right ) d^{5} \sqrt {d x}}{60 x \,b^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(615\) |
| risch | \(\text {Expression too large to display}\) | \(1050\) |
Input:
int((d*x)^(11/2)*(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^2*d^5/b^5 *B*x-1/3*a^2*d^5/b^5*A)*(b*d*x^3+a*d*x)^(1/2)/(x^2+a/b)^2-2*b*d*x*(-5/4*a* d^5*B/b^4*x-13/12*a*d^5*A/b^4)/((x^2+a/b)*b*d*x)^(1/2)+2/5*B/b^3*d^5*x*(b* d*x^3+a*d*x)^(1/2)+2/3*A/b^3*d^5*(b*d*x^3+a*d*x)^(1/2)-5/4*A*a*d^6/b^4*(-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)*E llipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))-77/20*B*a* d^6/b^4*(-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 = 225, normalized size of antiderivative = 0.58 \[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {75 \, {\left (A a b^{2} d^{5} x^{4} + 2 \, A a^{2} b d^{5} x^{2} + A a^{3} d^{5}\right )} \sqrt {b d} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - 231 \, {\left (B a b^{2} d^{5} x^{4} + 2 \, B a^{2} b d^{5} x^{2} + B a^{3} d^{5}\right )} \sqrt {b d} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (12 \, B b^{3} d^{5} x^{5} + 20 \, A b^{3} d^{5} x^{4} + 99 \, B a b^{2} d^{5} x^{3} + 105 \, A a b^{2} d^{5} x^{2} + 77 \, B a^{2} b d^{5} x + 75 \, A a^{2} b d^{5}\right )} \sqrt {b x^{2} + a} \sqrt {d x}}{30 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \] Input:
integrate((d*x)^(11/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/30*(75*(A*a*b^2*d^5*x^4 + 2*A*a^2*b*d^5*x^2 + A*a^3*d^5)*sqrt(b*d)*weie rstrassPInverse(-4*a/b, 0, x) - 231*(B*a*b^2*d^5*x^4 + 2*B*a^2*b*d^5*x^2 + B*a^3*d^5)*sqrt(b*d)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/ b, 0, x)) - (12*B*b^3*d^5*x^5 + 20*A*b^3*d^5*x^4 + 99*B*a*b^2*d^5*x^3 + 10 5*A*a*b^2*d^5*x^2 + 77*B*a^2*b*d^5*x + 75*A*a^2*b*d^5)*sqrt(b*x^2 + a)*sqr t(d*x))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)
Timed out. \[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x)**(11/2)*(B*x+A)/(b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {11}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(11/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)*(d*x)^(11/2)/(b*x^2 + a)^(5/2), x)
\[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (d x\right )^{\frac {11}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((d*x)^(11/2)*(B*x+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)*(d*x)^(11/2)/(b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{11/2}\,\left (A+B\,x\right )}{{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int(((d*x)^(11/2)*(A + B*x))/(a + b*x^2)^(5/2),x)
Output:
int(((d*x)^(11/2)*(A + B*x))/(a + b*x^2)^(5/2), x)
\[ \int \frac {(d x)^{11/2} (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {d}\, d^{5} \left (90 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{3}+90 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2}-154 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a^{2} b x +10 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4}-66 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{3}+6 \sqrt {x}\, \sqrt {b \,x^{2}+a}\, b^{3} x^{5}+231 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{5} b +462 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{4} b^{2} x^{2}+231 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}}d x \right ) a^{3} b^{3} x^{4}-45 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{6}-90 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{5} b \,x^{2}-45 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{3} x^{7}+3 a \,b^{2} x^{5}+3 a^{2} b \,x^{3}+a^{3} x}d x \right ) a^{4} b^{2} x^{4}\right )}{15 b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )} \] Input:
int((d*x)^(11/2)*(B*x+A)/(b*x^2+a)^(5/2),x)
Output:
(sqrt(d)*d**5*(90*sqrt(x)*sqrt(a + b*x**2)*a**3 + 90*sqrt(x)*sqrt(a + b*x* *2)*a**2*b*x**2 - 154*sqrt(x)*sqrt(a + b*x**2)*a**2*b*x + 10*sqrt(x)*sqrt( a + b*x**2)*a*b**2*x**4 - 66*sqrt(x)*sqrt(a + b*x**2)*a*b**2*x**3 + 6*sqrt (x)*sqrt(a + b*x**2)*b**3*x**5 + 231*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**5*b + 462*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* *4*b**2*x**2 + 231*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**3*x**4 - 45*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**6 - 90*i nt((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**5*b*x**2 - 45*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*b**2*x**4))/(15*b**3*(a**2 + 2*a*b*x**2 + b**2*x**4))