Optimal. Leaf size=120 \[ \frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {b \sqrt {a+b x^2} (A b-6 a B)}{16 a x^2}+\frac {\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 47, 63, 208} \[ \frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}+\frac {b \sqrt {a+b x^2} (A b-6 a B)}{16 a x^2}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac {\left (-\frac {A b}{2}+3 a B\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )}{6 a}\\ &=\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {(b (A b-6 a B)) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^2\right )}{16 a}\\ &=\frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {\left (b^2 (A b-6 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac {(b (A b-6 a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{16 a}\\ &=\frac {b (A b-6 a B) \sqrt {a+b x^2}}{16 a x^2}+\frac {(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac {A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 119, normalized size = 0.99 \[ \frac {-\left (a+b x^2\right ) \left (4 a^2 \left (2 A+3 B x^2\right )+2 a b x^2 \left (7 A+15 B x^2\right )+3 A b^2 x^4\right )-3 b^2 x^6 \sqrt {\frac {b x^2}{a}+1} (6 a B-A b) \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{48 a x^6 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 222, normalized size = 1.85 \[ \left [-\frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a^{2} x^{6}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a^{2} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 159, normalized size = 1.32 \[ \frac {\frac {3 \, {\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {30 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a b^{3} - 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 18 \, \sqrt {b x^{2} + a} B a^{3} b^{3} + 3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{4} + 8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A a b^{4} - 3 \, \sqrt {b x^{2} + a} A a^{2} b^{4}}{a b^{3} x^{6}}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 233, normalized size = 1.94 \[ \frac {A \,b^{3} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {3}{2}}}-\frac {3 B \,b^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}-\frac {\sqrt {b \,x^{2}+a}\, A \,b^{3}}{16 a^{2}}+\frac {3 \sqrt {b \,x^{2}+a}\, B \,b^{2}}{8 a}-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A \,b^{3}}{48 a^{3}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B \,b^{2}}{8 a^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A \,b^{2}}{48 a^{3} x^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B b}{8 a^{2} x^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A b}{24 a^{2} x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} B}{4 a \,x^{4}}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} A}{6 a \,x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.09, size = 210, normalized size = 1.75 \[ -\frac {3 \, B b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {A b^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {b x^{2} + a} B b^{2}}{8 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{3}}{48 \, a^{3}} - \frac {\sqrt {b x^{2} + a} A b^{3}}{16 \, a^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B b}{8 \, a^{2} x^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b^{2}}{48 \, a^{3} x^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B}{4 \, a x^{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A b}{24 \, a^{2} x^{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{6 \, a x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.63, size = 130, normalized size = 1.08 \[ \frac {A\,a\,\sqrt {b\,x^2+a}}{16\,x^6}-\frac {5\,B\,{\left (b\,x^2+a\right )}^{3/2}}{8\,x^4}-\frac {3\,B\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}}{6\,x^6}+\frac {3\,B\,a\,\sqrt {b\,x^2+a}}{8\,x^4}-\frac {A\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a\,x^6}-\frac {A\,b^3\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{16\,a^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________