Optimal. Leaf size=144 \[ -\frac {b (3 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^2}-\frac {(3 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{48 c}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2}+\frac {b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2059, 808, 678,
626, 634, 212} \begin {gather*} \frac {b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{5/2}}-\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (3 b B-8 A c)}{128 c^2}-\frac {\left (b x^2+c x^4\right )^{3/2} (3 b B-8 A c)}{48 c}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 626
Rule 634
Rule 678
Rule 808
Rule 2059
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2}+\frac {\left (b B-A c+\frac {5}{2} (-b B+2 A c)\right ) \text {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )}{8 c}\\ &=-\frac {(3 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{48 c}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2}-\frac {(b (3 b B-8 A c)) \text {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{32 c}\\ &=-\frac {b (3 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^2}-\frac {(3 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{48 c}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2}+\frac {\left (b^3 (3 b B-8 A c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{256 c^2}\\ &=-\frac {b (3 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^2}-\frac {(3 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{48 c}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2}+\frac {\left (b^3 (3 b B-8 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^2}\\ &=-\frac {b (3 b B-8 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{128 c^2}-\frac {(3 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{48 c}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{8 c x^2}+\frac {b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{128 c^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 149, normalized size = 1.03 \begin {gather*} \frac {x \sqrt {b+c x^2} \left (\sqrt {c} x \sqrt {b+c x^2} \left (-9 b^3 B+6 b^2 c \left (4 A+B x^2\right )+16 c^3 x^4 \left (4 A+3 B x^2\right )+8 b c^2 x^2 \left (14 A+9 B x^2\right )\right )+3 b^3 (-3 b B+8 A c) \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{384 c^{5/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.37, size = 202, normalized size = 1.40
method | result | size |
risch | \(\frac {\left (48 B \,c^{3} x^{6}+64 A \,c^{3} x^{4}+72 B b \,c^{2} x^{4}+112 A b \,c^{2} x^{2}+6 B \,b^{2} c \,x^{2}+24 A \,b^{2} c -9 B \,b^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{384 c^{2}}+\frac {\left (-\frac {b^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) A}{16 c^{\frac {3}{2}}}+\frac {3 b^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) B}{128 c^{\frac {5}{2}}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\) | \(159\) |
default | \(\frac {\left (x^{4} c +b \,x^{2}\right )^{\frac {3}{2}} \left (48 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {3}{2}} x^{3}+64 A \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{\frac {3}{2}} x -24 B \left (c \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {c}\, b x -16 A \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} b x +6 B \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\, b^{2} x -24 A \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} b^{2} x +9 B \sqrt {c \,x^{2}+b}\, \sqrt {c}\, b^{3} x -24 A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{3} c +9 B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{4}\right )}{384 x^{3} \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {5}{2}}}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 216, normalized size = 1.50 \begin {gather*} \frac {1}{96} \, {\left (12 \, \sqrt {c x^{4} + b x^{2}} b x^{2} - \frac {3 \, b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {3}{2}}} + 16 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} b^{2}}{c}\right )} A + \frac {1}{256} \, {\left (32 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{2} - \frac {12 \, \sqrt {c x^{4} + b x^{2}} b^{2} x^{2}}{c} + \frac {3 \, b^{4} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {5}{2}}} - \frac {6 \, \sqrt {c x^{4} + b x^{2}} b^{3}}{c^{2}} + \frac {16 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b}{c}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.78, size = 275, normalized size = 1.91 \begin {gather*} \left [-\frac {3 \, {\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (48 \, B c^{4} x^{6} - 9 \, B b^{3} c + 24 \, A b^{2} c^{2} + 8 \, {\left (9 \, B b c^{3} + 8 \, A c^{4}\right )} x^{4} + 2 \, {\left (3 \, B b^{2} c^{2} + 56 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{768 \, c^{3}}, -\frac {3 \, {\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - {\left (48 \, B c^{4} x^{6} - 9 \, B b^{3} c + 24 \, A b^{2} c^{2} + 8 \, {\left (9 \, B b c^{3} + 8 \, A c^{4}\right )} x^{4} + 2 \, {\left (3 \, B b^{2} c^{2} + 56 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{384 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.57, size = 178, normalized size = 1.24 \begin {gather*} \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, B c x^{2} \mathrm {sgn}\left (x\right ) + \frac {9 \, B b c^{6} \mathrm {sgn}\left (x\right ) + 8 \, A c^{7} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} x^{2} + \frac {3 \, B b^{2} c^{5} \mathrm {sgn}\left (x\right ) + 56 \, A b c^{6} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} x^{2} - \frac {3 \, {\left (3 \, B b^{3} c^{4} \mathrm {sgn}\left (x\right ) - 8 \, A b^{2} c^{5} \mathrm {sgn}\left (x\right )\right )}}{c^{6}}\right )} \sqrt {c x^{2} + b} x - \frac {{\left (3 \, B b^{4} \mathrm {sgn}\left (x\right ) - 8 \, A b^{3} c \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{128 \, c^{\frac {5}{2}}} + \frac {{\left (3 \, B b^{4} \log \left ({\left | b \right |}\right ) - 8 \, A b^{3} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{256 \, c^{\frac {5}{2}}} \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 (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________