Optimal. Leaf size=93 \[ -\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {3 B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {3 B c \sqrt {a+c x^2}}{8 x^2}-\frac {B \left (a+c x^2\right )^{3/2}}{4 x^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {807, 266, 47, 63, 208} \begin {gather*} -\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {3 B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}-\frac {B \left (a+c x^2\right )^{3/2}}{4 x^4}-\frac {3 B c \sqrt {a+c x^2}}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{x^6} \, dx &=-\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}+B \int \frac {\left (a+c x^2\right )^{3/2}}{x^5} \, dx\\ &=-\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}+\frac {1}{2} B \operatorname {Subst}\left (\int \frac {(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {B \left (a+c x^2\right )^{3/2}}{4 x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}+\frac {1}{8} (3 B c) \operatorname {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 B c \sqrt {a+c x^2}}{8 x^2}-\frac {B \left (a+c x^2\right )^{3/2}}{4 x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}+\frac {1}{16} \left (3 B c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )\\ &=-\frac {3 B c \sqrt {a+c x^2}}{8 x^2}-\frac {B \left (a+c x^2\right )^{3/2}}{4 x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}+\frac {1}{8} (3 B c) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )\\ &=-\frac {3 B c \sqrt {a+c x^2}}{8 x^2}-\frac {B \left (a+c x^2\right )^{3/2}}{4 x^4}-\frac {A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac {3 B c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 103, normalized size = 1.11 \begin {gather*} \frac {-\frac {\left (a+c x^2\right ) \left (2 a^2 (4 A+5 B x)+a c x^2 (16 A+25 B x)+8 A c^2 x^4\right )}{a x^5}-15 B c^2 \sqrt {\frac {c x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )}{40 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.63, size = 106, normalized size = 1.14 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-8 a^2 A-10 a^2 B x-16 a A c x^2-25 a B c x^3-8 A c^2 x^4\right )}{40 a x^5}+\frac {3 B c^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 190, normalized size = 2.04 \begin {gather*} \left [\frac {15 \, B \sqrt {a} c^{2} x^{5} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (8 \, A c^{2} x^{4} + 25 \, B a c x^{3} + 16 \, A a c x^{2} + 10 \, B a^{2} x + 8 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{80 \, a x^{5}}, \frac {15 \, B \sqrt {-a} c^{2} x^{5} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (8 \, A c^{2} x^{4} + 25 \, B a c x^{3} + 16 \, A a c x^{2} + 10 \, B a^{2} x + 8 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{40 \, a x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 232, normalized size = 2.49 \begin {gather*} \frac {3 \, B c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} + \frac {25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B c^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A c^{\frac {5}{2}} - 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} B a c^{2} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{2} c^{\frac {5}{2}} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{3} c^{2} - 25 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{4} c^{2} + 8 \, A a^{4} c^{\frac {5}{2}}}{20 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 125, normalized size = 1.34 \begin {gather*} -\frac {3 B \,c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}+\frac {3 \sqrt {c \,x^{2}+a}\, B \,c^{2}}{8 a}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{2}}{8 a^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{4 a \,x^{4}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A}{5 a \,x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 113, normalized size = 1.22 \begin {gather*} -\frac {3 \, B c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{8 \, \sqrt {a}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {c x^{2} + a} B c^{2}}{8 \, a} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c}{8 \, a^{2} x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B}{4 \, a x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.76, size = 73, normalized size = 0.78 \begin {gather*} \frac {3\,B\,a\,\sqrt {c\,x^2+a}}{8\,x^4}-\frac {3\,B\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^2+a}}{\sqrt {a}}\right )}{8\,\sqrt {a}}-\frac {5\,B\,{\left (c\,x^2+a\right )}^{3/2}}{8\,x^4}-\frac {A\,{\left (c\,x^2+a\right )}^{5/2}}{5\,a\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 8.95, size = 199, normalized size = 2.14 \begin {gather*} - \frac {A a \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {2 A c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{2}} - \frac {A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 a} - \frac {B a^{2}}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 B a \sqrt {c}}{8 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {B c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} - \frac {B c^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 B c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________