Optimal. Leaf size=111 \[ -\frac {c \sqrt {a+c x^2} (3 A+8 B x)}{8 x^2}-\frac {\left (a+c x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}-\frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}+B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {811, 844, 217, 206, 266, 63, 208} \begin {gather*} -\frac {c \sqrt {a+c x^2} (3 A+8 B x)}{8 x^2}-\frac {\left (a+c x^2\right )^{3/2} (3 A+4 B x)}{12 x^4}-\frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}+B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 206
Rule 208
Rule 217
Rule 266
Rule 811
Rule 844
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{x^5} \, dx &=-\frac {(3 A+4 B x) \left (a+c x^2\right )^{3/2}}{12 x^4}-\frac {\int \frac {(-6 a A c-8 a B c x) \sqrt {a+c x^2}}{x^3} \, dx}{8 a}\\ &=-\frac {c (3 A+8 B x) \sqrt {a+c x^2}}{8 x^2}-\frac {(3 A+4 B x) \left (a+c x^2\right )^{3/2}}{12 x^4}+\frac {\int \frac {12 a^2 A c^2+32 a^2 B c^2 x}{x \sqrt {a+c x^2}} \, dx}{32 a^2}\\ &=-\frac {c (3 A+8 B x) \sqrt {a+c x^2}}{8 x^2}-\frac {(3 A+4 B x) \left (a+c x^2\right )^{3/2}}{12 x^4}+\frac {1}{8} \left (3 A c^2\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx+\left (B c^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx\\ &=-\frac {c (3 A+8 B x) \sqrt {a+c x^2}}{8 x^2}-\frac {(3 A+4 B x) \left (a+c x^2\right )^{3/2}}{12 x^4}+\frac {1}{16} \left (3 A c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )+\left (B c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )\\ &=-\frac {c (3 A+8 B x) \sqrt {a+c x^2}}{8 x^2}-\frac {(3 A+4 B x) \left (a+c x^2\right )^{3/2}}{12 x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )+\frac {1}{8} (3 A c) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )\\ &=-\frac {c (3 A+8 B x) \sqrt {a+c x^2}}{8 x^2}-\frac {(3 A+4 B x) \left (a+c x^2\right )^{3/2}}{12 x^4}+B c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 \sqrt {a}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.08, size = 114, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {a+c x^2} \left (8 a^2 B x \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^2}{a}\right )+9 A c^2 x^4 \tanh ^{-1}\left (\sqrt {\frac {c x^2}{a}+1}\right )+3 a A \left (2 a+5 c x^2\right ) \sqrt {\frac {c x^2}{a}+1}\right )}{24 a x^4 \sqrt {\frac {c x^2}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.63, size = 117, normalized size = 1.05 \begin {gather*} \frac {\sqrt {a+c x^2} \left (-6 a A-8 a B x-15 A c x^2-32 B c x^3\right )}{24 x^4}+\frac {3 A c^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{4 \sqrt {a}}-B c^{3/2} \log \left (\sqrt {a+c x^2}-\sqrt {c} x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 474, normalized size = 4.27 \begin {gather*} \left [\frac {24 \, B a c^{\frac {3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 9 \, A \sqrt {a} c^{2} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (32 \, B a c x^{3} + 15 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a x^{4}}, -\frac {48 \, B a \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 9 \, A \sqrt {a} c^{2} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (32 \, B a c x^{3} + 15 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a x^{4}}, \frac {9 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 12 \, B a c^{\frac {3}{2}} x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - {\left (32 \, B a c x^{3} + 15 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a x^{4}}, -\frac {24 \, B a \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 9 \, A \sqrt {-a} c^{2} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (32 \, B a c x^{3} + 15 \, A a c x^{2} + 8 \, B a^{2} x + 6 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 285, normalized size = 2.57 \begin {gather*} \frac {3 \, A c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} - B c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \frac {15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} A c^{2} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} B a c^{\frac {3}{2}} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} A a c^{2} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} B a^{2} c^{\frac {3}{2}} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} A a^{2} c^{2} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} B a^{3} c^{\frac {3}{2}} + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} A a^{3} c^{2} - 32 \, B a^{4} c^{\frac {3}{2}}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 202, normalized size = 1.82 \begin {gather*} -\frac {3 A \,c^{2} \ln \left (\frac {2 a +2 \sqrt {c \,x^{2}+a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}+B \,c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )+\frac {\sqrt {c \,x^{2}+a}\, B \,c^{2} x}{a}+\frac {3 \sqrt {c \,x^{2}+a}\, A \,c^{2}}{8 a}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}} B \,c^{2} x}{3 a^{2}}+\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}} A \,c^{2}}{8 a^{2}}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} B c}{3 a^{2} x}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A c}{8 a^{2} x^{2}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} B}{3 a \,x^{3}}-\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}} A}{4 a \,x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 164, normalized size = 1.48 \begin {gather*} \frac {\sqrt {c x^{2} + a} B c^{2} x}{a} + B c^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - \frac {3 \, A 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}} A c^{2}}{8 \, a^{2}} + \frac {3 \, \sqrt {c x^{2} + a} A c^{2}}{8 \, a} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c}{3 \, a x} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A c}{8 \, a^{2} x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B}{3 \, a x^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} A}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (A+B\,x\right )}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 9.34, size = 236, normalized size = 2.13 \begin {gather*} - \frac {A a^{2}}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 A a \sqrt {c}}{8 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {A c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} - \frac {A c^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 A c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8 \sqrt {a}} - \frac {B \sqrt {a} c}{x \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {B a \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{3 x^{2}} - \frac {B c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{3} + B c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )} - \frac {B c^{2} x}{\sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________