Integrand size = 22, antiderivative size = 243 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=-\frac {2 A}{3 a^3 x^{3/2}}-\frac {(A b-a B) \sqrt {x}}{4 a^2 \left (a+b x^2\right )^2}-\frac {(15 A b-7 a B) \sqrt {x}}{16 a^3 \left (a+b x^2\right )}+\frac {7 (11 A b-3 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}}-\frac {7 (11 A b-3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{15/4} \sqrt [4]{b}} \] Output:
-2/3*A/a^3/x^(3/2)-1/4*(A*b-B*a)*x^(1/2)/a^2/(b*x^2+a)^2-1/16*(15*A*b-7*B* a)*x^(1/2)/a^3/(b*x^2+a)+7/64*(11*A*b-3*B*a)*arctan(1-2^(1/2)*b^(1/4)*x^(1 /2)/a^(1/4))*2^(1/2)/a^(15/4)/b^(1/4)-7/64*(11*A*b-3*B*a)*arctan(1+2^(1/2) *b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(15/4)/b^(1/4)-7/64*(11*A*b-3*B*a)*arc tanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(15/4) /b^(1/4)
Time = 0.57 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 a^{3/4} \left (77 A b^2 x^4+a^2 \left (32 A-33 B x^2\right )+a b x^2 \left (121 A-21 B x^2\right )\right )}{x^{3/2} \left (a+b x^2\right )^2}+\frac {21 \sqrt {2} (11 A b-3 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {21 \sqrt {2} (-11 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}}{192 a^{15/4}} \] Input:
Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]
Output:
((-4*a^(3/4)*(77*A*b^2*x^4 + a^2*(32*A - 33*B*x^2) + a*b*x^2*(121*A - 21*B *x^2)))/(x^(3/2)*(a + b*x^2)^2) + (21*Sqrt[2]*(11*A*b - 3*a*B)*ArcTan[(Sqr t[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (21*Sqrt[2 ]*(-11*A*b + 3*a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + S qrt[b]*x)])/b^(1/4))/(192*a^(15/4))
Time = 0.49 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.28, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {362, 253, 264, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {(11 A b-3 a B) \int \frac {1}{x^{5/2} \left (b x^2+a\right )^2}dx}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \int \frac {1}{x^{5/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {b \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \int \frac {1}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(11 A b-3 a B) \left (\frac {7 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{4 a}+\frac {1}{2 a x^{3/2} \left (a+b x^2\right )}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2}\) |
Input:
Int[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]
Output:
(A*b - a*B)/(4*a*b*x^(3/2)*(a + b*x^2)^2) + ((11*A*b - 3*a*B)*(1/(2*a*x^(3 /2)*(a + b*x^2)) + (7*(-2/(3*a*x^(3/2)) - (2*b*((-(ArcTan[1 - (Sqrt[2]*b^( 1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^ (1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log [Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b ^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sq rt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a])))/a))/(4*a)))/(8*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.41 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (\frac {15}{32} b^{2} A -\frac {7}{32} a b B \right ) x^{\frac {5}{2}}+\frac {a \left (19 A b -11 B a \right ) \sqrt {x}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {7 \left (11 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a}\right )}{a^{3}}\) | \(173\) |
| default | \(-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 \left (\frac {\left (\frac {15}{32} b^{2} A -\frac {7}{32} a b B \right ) x^{\frac {5}{2}}+\frac {a \left (19 A b -11 B a \right ) \sqrt {x}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {7 \left (11 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a}\right )}{a^{3}}\) | \(173\) |
| risch | \(-\frac {2 A}{3 a^{3} x^{\frac {3}{2}}}-\frac {\frac {2 \left (\frac {15}{32} b^{2} A -\frac {7}{32} a b B \right ) x^{\frac {5}{2}}+\frac {a \left (19 A b -11 B a \right ) \sqrt {x}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {7 \left (11 A b -3 B a \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a}}{a^{3}}\) | \(174\) |
Input:
int((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
-2/3*A/a^3/x^(3/2)-2/a^3*(((15/32*b^2*A-7/32*a*b*B)*x^(5/2)+1/32*a*(19*A*b -11*B*a)*x^(1/2))/(b*x^2+a)^2+7/256*(11*A*b-3*B*a)*(a/b)^(1/4)/a*2^(1/2)*( ln((x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1 /2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2) /(a/b)^(1/4)*x^(1/2)-1)))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 770, normalized size of antiderivative = 3.17 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
-1/192*(21*(a^3*b^2*x^6 + 2*a^4*b*x^4 + a^5*x^2)*(-(81*B^4*a^4 - 1188*A*B^ 3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15* b))^(1/4)*log(7*a^4*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^ 2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4) - 7*(3*B*a - 11*A*b )*sqrt(x)) + 21*(I*a^3*b^2*x^6 + 2*I*a^4*b*x^4 + I*a^5*x^2)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4* b^4)/(a^15*b))^(1/4)*log(7*I*a^4*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A ^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4) - 7*(3 *B*a - 11*A*b)*sqrt(x)) + 21*(-I*a^3*b^2*x^6 - 2*I*a^4*b*x^4 - I*a^5*x^2)* (-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^ 3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*log(-7*I*a^4*(-(81*B^4*a^4 - 1188*A*B^3 *a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b ))^(1/4) - 7*(3*B*a - 11*A*b)*sqrt(x)) - 21*(a^3*b^2*x^6 + 2*a^4*b*x^4 + a ^5*x^2)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^ 3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*log(-7*a^4*(-(81*B^4*a^4 - 1188 *A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/( a^15*b))^(1/4) - 7*(3*B*a - 11*A*b)*sqrt(x)) - 4*(7*(3*B*a*b - 11*A*b^2)*x ^4 - 32*A*a^2 + 11*(3*B*a^2 - 11*A*a*b)*x^2)*sqrt(x))/(a^3*b^2*x^6 + 2*a^4 *b*x^4 + a^5*x^2)
Timed out. \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((B*x**2+A)/x**(5/2)/(b*x**2+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {7 \, {\left (3 \, B a b - 11 \, A b^{2}\right )} x^{4} - 32 \, A a^{2} + 11 \, {\left (3 \, B a^{2} - 11 \, A a b\right )} x^{2}}{48 \, {\left (a^{3} b^{2} x^{\frac {11}{2}} + 2 \, a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {3}{2}}\right )}} + \frac {7 \, {\left (\frac {2 \, \sqrt {2} {\left (3 \, B a - 11 \, A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, B a - 11 \, A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, B a - 11 \, A b\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, B a - 11 \, A b\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, a^{3}} \] Input:
integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/48*(7*(3*B*a*b - 11*A*b^2)*x^4 - 32*A*a^2 + 11*(3*B*a^2 - 11*A*a*b)*x^2) /(a^3*b^2*x^(11/2) + 2*a^4*b*x^(7/2) + a^5*x^(3/2)) + 7/128*(2*sqrt(2)*(3* B*a - 11*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt (x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(3 *B*a - 11*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sq rt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(3 *B*a - 11*A*b)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/ (a^(3/4)*b^(1/4)) - sqrt(2)*(3*B*a - 11*A*b)*log(-sqrt(2)*a^(1/4)*b^(1/4)* sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/a^3
Time = 0.13 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b} + \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b} + \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b} - \frac {7 \, \sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b} - \frac {2 \, A}{3 \, a^{3} x^{\frac {3}{2}}} + \frac {7 \, B a b x^{\frac {5}{2}} - 15 \, A b^{2} x^{\frac {5}{2}} + 11 \, B a^{2} \sqrt {x} - 19 \, A a b \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{3}} \] Input:
integrate((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x, algorithm="giac")
Output:
7/64*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt( 2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) + 7/64*sqrt(2)*( 3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*( a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) + 7/128*sqrt(2)*(3*(a*b^3)^(1 /4)*B*a - 11*(a*b^3)^(1/4)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt (a/b))/(a^4*b) - 7/128*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b )*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b) - 2/3*A/(a^3*x ^(3/2)) + 1/16*(7*B*a*b*x^(5/2) - 15*A*b^2*x^(5/2) + 11*B*a^2*sqrt(x) - 19 *A*a*b*sqrt(x))/((b*x^2 + a)^2*a^3)
Time = 0.73 (sec) , antiderivative size = 888, normalized size of antiderivative = 3.65 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:
int((A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x)
Output:
- ((2*A)/(3*a) + (11*x^2*(11*A*b - 3*B*a))/(48*a^2) + (7*b*x^4*(11*A*b - 3 *B*a))/(48*a^3))/(a^2*x^(3/2) + b^2*x^(11/2) + 2*a*b*x^(7/2)) - (atan((((1 1*A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 529 85856*A*B*a^10*b^4) - (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096* B*a^14*b^3))/(64*(-a)^(15/4)*b^(1/4)))*7i)/(64*(-a)^(15/4)*b^(1/4)) + ((11 *A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 5298 5856*A*B*a^10*b^4) + (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B *a^14*b^3))/(64*(-a)^(15/4)*b^(1/4)))*7i)/(64*(-a)^(15/4)*b^(1/4)))/((7*(1 1*A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 529 85856*A*B*a^10*b^4) - (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096* B*a^14*b^3))/(64*(-a)^(15/4)*b^(1/4))))/(64*(-a)^(15/4)*b^(1/4)) - (7*(11* A*b - 3*B*a)*(x^(1/2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985 856*A*B*a^10*b^4) + (7*(11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B* a^14*b^3))/(64*(-a)^(15/4)*b^(1/4))))/(64*(-a)^(15/4)*b^(1/4))))*(11*A*b - 3*B*a)*7i)/(32*(-a)^(15/4)*b^(1/4)) - (7*atan(((7*(11*A*b - 3*B*a)*(x^(1/ 2)*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985856*A*B*a^10*b^4) - ((11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B*a^14*b^3)*7i)/(64*(-a )^(15/4)*b^(1/4))))/(64*(-a)^(15/4)*b^(1/4)) + (7*(11*A*b - 3*B*a)*(x^(1/2 )*(97140736*A^2*a^9*b^5 + 7225344*B^2*a^11*b^3 - 52985856*A*B*a^10*b^4) + ((11*A*b - 3*B*a)*(80740352*A*a^13*b^4 - 22020096*B*a^14*b^3)*7i)/(64*(...
Time = 0.19 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx=\frac {42 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x +42 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}-42 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x -42 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{3}+21 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x +21 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{3}-21 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x -21 \sqrt {x}\, b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{3}-32 a^{2}-56 a b \,x^{2}}{48 \sqrt {x}\, a^{3} x \left (b \,x^{2}+a \right )} \] Input:
int((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x)
Output:
(42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2* sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*x + 42*sqrt(x)*b**(3/4)*a* *(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1 /4)*a**(1/4)*sqrt(2)))*b*x**3 - 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan( (b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2) ))*a*x - 42*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt (2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b*x**3 + 21*sqrt(x)* b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt( a) + sqrt(b)*x)*a*x + 21*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)* b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**3 - 21*sqrt(x)*b**(3 /4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqr t(b)*x)*a*x - 21*sqrt(x)*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a* *(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**3 - 32*a**2 - 56*a*b*x**2)/(48* sqrt(x)*a**3*x*(a + b*x**2))