Optimal. Leaf size=146 \[ -\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {5 b (7 A b-4 a B)}{8 a^4 \sqrt {a+b x^2}}+\frac {5 b (7 A b-4 a B)}{24 a^3 \left (a+b x^2\right )^{3/2}}+\frac {7 A b-4 a B}{8 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 150, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \begin {gather*} \frac {5 \sqrt {a+b x^2} (7 A b-4 a B)}{8 a^4 x^2}-\frac {5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt {a+b x^2}}-\frac {7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{9/2}}-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^5 \left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}}+\frac {\left (-\frac {7 A b}{2}+2 a B\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}}-\frac {7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {(5 (7 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )}{24 a^2}\\ &=-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}}-\frac {7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt {a+b x^2}}-\frac {(5 (7 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^2\right )}{8 a^3}\\ &=-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}}-\frac {7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 (7 A b-4 a B) \sqrt {a+b x^2}}{8 a^4 x^2}+\frac {(5 b (7 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{16 a^4}\\ &=-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}}-\frac {7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 (7 A b-4 a B) \sqrt {a+b x^2}}{8 a^4 x^2}+\frac {(5 (7 A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{8 a^4}\\ &=-\frac {A}{4 a x^4 \left (a+b x^2\right )^{3/2}}-\frac {7 A b-4 a B}{12 a^2 x^2 \left (a+b x^2\right )^{3/2}}-\frac {5 (7 A b-4 a B)}{12 a^3 x^2 \sqrt {a+b x^2}}+\frac {5 (7 A b-4 a B) \sqrt {a+b x^2}}{8 a^4 x^2}-\frac {5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 60, normalized size = 0.41 \begin {gather*} \frac {b x^4 (7 A b-4 a B) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\frac {b x^2}{a}+1\right )-3 a^2 A}{12 a^3 x^4 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 126, normalized size = 0.86 \begin {gather*} \frac {5 \left (4 a b B-7 A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 a^{9/2}}+\frac {-6 a^3 A-12 a^3 B x^2+21 a^2 A b x^2-80 a^2 b B x^4+140 a A b^2 x^4-60 a b^2 B x^6+105 A b^3 x^6}{24 a^4 x^4 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 407, normalized size = 2.79 \begin {gather*} \left [-\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4}\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 6 \, A a^{4} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}}, -\frac {15 \, {\left ({\left (4 \, B a b^{3} - 7 \, A b^{4}\right )} x^{8} + 2 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (15 \, {\left (4 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{6} + 6 \, A a^{4} + 20 \, {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{4} + 3 \, {\left (4 \, B a^{4} - 7 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{24 \, {\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 165, normalized size = 1.13 \begin {gather*} -\frac {5 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{4}} - \frac {6 \, {\left (b x^{2} + a\right )} B a b + B a^{2} b - 9 \, {\left (b x^{2} + a\right )} A b^{2} - A a b^{2}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b - 4 \, \sqrt {b x^{2} + a} B a^{2} b - 11 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{2} + 13 \, \sqrt {b x^{2} + a} A a b^{2}}{8 \, a^{4} b^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 187, normalized size = 1.28 \begin {gather*} \frac {35 A \,b^{2}}{24 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}-\frac {5 B b}{6 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}}-\frac {35 A \,b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {9}{2}}}+\frac {5 B b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {7}{2}}}+\frac {35 A \,b^{2}}{8 \sqrt {b \,x^{2}+a}\, a^{4}}-\frac {5 B b}{2 \sqrt {b \,x^{2}+a}\, a^{3}}+\frac {7 A b}{8 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} x^{2}}-\frac {B}{2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,x^{2}}-\frac {A}{4 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 164, normalized size = 1.12 \begin {gather*} \frac {5 \, B b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {7}{2}}} - \frac {35 \, A b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {9}{2}}} - \frac {5 \, B b}{2 \, \sqrt {b x^{2} + a} a^{3}} - \frac {5 \, B b}{6 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {35 \, A b^{2}}{8 \, \sqrt {b x^{2} + a} a^{4}} + \frac {35 \, A b^{2}}{24 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} - \frac {B}{2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{2}} + \frac {7 \, A b}{8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x^{2}} - \frac {A}{4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.02, size = 176, normalized size = 1.21 \begin {gather*} \frac {35\,A\,b^2}{6\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {10\,B\,b}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {35\,A\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{9/2}}-\frac {A}{4\,a\,x^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {B}{2\,a\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {5\,B\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{7/2}}+\frac {7\,A\,b}{8\,a^2\,x^2\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {35\,A\,b^3\,x^2}{8\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {5\,B\,b^2\,x^2}{2\,a^3\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 139.24, size = 1323, normalized size = 9.06
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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