Integrand size = 22, antiderivative size = 146 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {5}{8} b (4 A b+3 a B) x \sqrt {a+b x^2}+\frac {5 b (4 A b+3 a B) x \left (a+b x^2\right )^{3/2}}{12 a}-\frac {(4 A b+3 a B) \left (a+b x^2\right )^{5/2}}{3 a x}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}+\frac {5}{8} a \sqrt {b} (4 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \]
5/12*b*(4*A*b+3*B*a)*x*(b*x^2+a)^(3/2)/a-1/3*(4*A*b+3*B*a)*(b*x^2+a)^(5/2) /a/x-1/3*A*(b*x^2+a)^(7/2)/a/x^3+5/8*a*(4*A*b+3*B*a)*arctanh(x*b^(1/2)/(b* x^2+a)^(1/2))*b^(1/2)+5/8*b*(4*A*b+3*B*a)*x*(b*x^2+a)^(1/2)
Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 A-56 a A b x^2-24 a^2 B x^2+12 A b^2 x^4+27 a b B x^4+6 b^2 B x^6\right )}{24 x^3}+\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right ) \]
(Sqrt[a + b*x^2]*(-8*a^2*A - 56*a*A*b*x^2 - 24*a^2*B*x^2 + 12*A*b^2*x^4 + 27*a*b*B*x^4 + 6*b^2*B*x^6))/(24*x^3) + (5*a*Sqrt[b]*(4*A*b + 3*a*B)*ArcTa nh[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/4
Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {359, 247, 211, 211, 224, 219}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {(3 a B+4 A b) \int \frac {\left (b x^2+a\right )^{5/2}}{x^2}dx}{3 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {(3 a B+4 A b) \left (5 b \int \left (b x^2+a\right )^{3/2}dx-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )}{3 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(3 a B+4 A b) \left (5 b \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )}{3 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {(3 a B+4 A b) \left (5 b \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )}{3 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {(3 a B+4 A b) \left (5 b \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )}{3 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {(3 a B+4 A b) \left (5 b \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )-\frac {\left (a+b x^2\right )^{5/2}}{x}\right )}{3 a}-\frac {A \left (a+b x^2\right )^{7/2}}{3 a x^3}\) |
-1/3*(A*(a + b*x^2)^(7/2))/(a*x^3) + ((4*A*b + 3*a*B)*(-((a + b*x^2)^(5/2) /x) + 5*b*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcT anh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4)))/(3*a)
3.6.47.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Time = 2.85 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-6 b^{2} B \,x^{6}-12 A \,b^{2} x^{4}-27 B a b \,x^{4}+56 a A b \,x^{2}+24 a^{2} B \,x^{2}+8 a^{2} A \right )}{24 x^{3}}+\frac {5 a \sqrt {b}\, \left (4 A b +3 B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{8}\) | \(97\) |
| pseudoelliptic | \(-\frac {-\frac {15 x^{3} \left (A b +\frac {3 B a}{4}\right ) b a \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{2}+\left (7 x^{2} \left (-\frac {27 x^{2} B}{56}+A \right ) a \,b^{\frac {3}{2}}-\frac {3 x^{4} \left (\frac {x^{2} B}{2}+A \right ) b^{\frac {5}{2}}}{2}+a^{2} \sqrt {b}\, \left (3 x^{2} B +A \right )\right ) \sqrt {b \,x^{2}+a}}{3 \sqrt {b}\, x^{3}}\) | \(101\) |
| default | \(A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{3 a \,x^{3}}+\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\right )}{3 a}\right )+B \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{a}\right )\) | \(212\) |
-1/24*(b*x^2+a)^(1/2)*(-6*B*b^2*x^6-12*A*b^2*x^4-27*B*a*b*x^4+56*A*a*b*x^2 +24*B*a^2*x^2+8*A*a^2)/x^3+5/8*a*b^(1/2)*(4*A*b+3*B*a)*ln(x*b^(1/2)+(b*x^2 +a)^(1/2))
Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\left [\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {b} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, B b^{2} x^{6} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, x^{3}}, -\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (6 \, B b^{2} x^{6} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{24 \, x^{3}}\right ] \]
[1/48*(15*(3*B*a^2 + 4*A*a*b)*sqrt(b)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a) *sqrt(b)*x - a) + 2*(6*B*b^2*x^6 + 3*(9*B*a*b + 4*A*b^2)*x^4 - 8*A*a^2 - 8 *(3*B*a^2 + 7*A*a*b)*x^2)*sqrt(b*x^2 + a))/x^3, -1/24*(15*(3*B*a^2 + 4*A*a *b)*sqrt(-b)*x^3*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (6*B*b^2*x^6 + 3*(9* B*a*b + 4*A*b^2)*x^4 - 8*A*a^2 - 8*(3*B*a^2 + 7*A*a*b)*x^2)*sqrt(b*x^2 + a ))/x^3]
Time = 2.71 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.93 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=- \frac {2 A a^{\frac {3}{2}} b}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {2 A \sqrt {a} b^{2} x}{\sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A a^{2} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {A a b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} + 2 A a b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + A b^{2} \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) - \frac {B a^{\frac {5}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{\frac {3}{2}} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + B a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + 2 B a b \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + B b^{2} \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a x \sqrt {a + b x^{2}}}{8 b} + \frac {x^{3} \sqrt {a + b x^{2}}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \]
-2*A*a**(3/2)*b/(x*sqrt(1 + b*x**2/a)) - 2*A*sqrt(a)*b**2*x/sqrt(1 + b*x** 2/a) - A*a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*x**2) - A*a*b**(3/2)*sqrt(a/ (b*x**2) + 1)/3 + 2*A*a*b**(3/2)*asinh(sqrt(b)*x/sqrt(a)) + A*b**2*Piecewi se((a*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0) ), (x*log(x)/sqrt(b*x**2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sq rt(a)*x, True)) - B*a**(5/2)/(x*sqrt(1 + b*x**2/a)) - B*a**(3/2)*b*x/sqrt( 1 + b*x**2/a) + B*a**2*sqrt(b)*asinh(sqrt(b)*x/sqrt(a)) + 2*B*a*b*Piecewis e((a*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)) , (x*log(x)/sqrt(b*x**2), True))/2 + x*sqrt(a + b*x**2)/2, Ne(b, 0)), (sqr t(a)*x, True)) + B*b**2*Piecewise((-a**2*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt(b*x**2), True))/(8*b) + a*x*sqrt(a + b*x**2)/(8*b) + x**3*sqrt(a + b*x**2)/4, Ne(b, 0)), (sqrt( a)*x**3/3, True))
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {5}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b x + \frac {15}{8} \, \sqrt {b x^{2} + a} B a b x + \frac {5}{2} \, \sqrt {b x^{2} + a} A b^{2} x + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} x}{3 \, a} + \frac {15}{8} \, B a^{2} \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) + \frac {5}{2} \, A a b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{x} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{3 \, a x^{3}} \]
5/4*(b*x^2 + a)^(3/2)*B*b*x + 15/8*sqrt(b*x^2 + a)*B*a*b*x + 5/2*sqrt(b*x^ 2 + a)*A*b^2*x + 5/3*(b*x^2 + a)^(3/2)*A*b^2*x/a + 15/8*B*a^2*sqrt(b)*arcs inh(b*x/sqrt(a*b)) + 5/2*A*a*b^(3/2)*arcsinh(b*x/sqrt(a*b)) - (b*x^2 + a)^ (5/2)*B/x - 4/3*(b*x^2 + a)^(5/2)*A*b/(a*x) - 1/3*(b*x^2 + a)^(7/2)*A/(a*x ^3)
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\frac {1}{8} \, {\left (2 \, B b^{2} x^{2} + \frac {9 \, B a b^{3} + 4 \, A b^{4}}{b^{2}}\right )} \sqrt {b x^{2} + a} x - \frac {5}{16} \, {\left (3 \, B a^{2} \sqrt {b} + 4 \, A a b^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{3} \sqrt {b} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{2} b^{\frac {3}{2}} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{4} \sqrt {b} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{3} b^{\frac {3}{2}} + 3 \, B a^{5} \sqrt {b} + 7 \, A a^{4} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]
1/8*(2*B*b^2*x^2 + (9*B*a*b^3 + 4*A*b^4)/b^2)*sqrt(b*x^2 + a)*x - 5/16*(3* B*a^2*sqrt(b) + 4*A*a*b^(3/2))*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/3* (3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*sqrt(b) + 9*(sqrt(b)*x - sqrt(b*x ^2 + a))^4*A*a^2*b^(3/2) - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*sqrt(b) - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(3/2) + 3*B*a^5*sqrt(b) + 7* A*a^4*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3
Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^4} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{5/2}}{x^4} \,d x \]