Optimal. Leaf size=113 \[ -\frac {3 d}{2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {1}{2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {444, 51, 63, 208} \[ -\frac {3 d}{2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {1}{2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}+\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 208
Rule 444
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {(3 d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 (b c-a d)}\\ &=-\frac {3 d}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {(3 b d) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 (b c-a d)^2}\\ &=-\frac {3 d}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 (b c-a d)^2}\\ &=-\frac {3 d}{2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {3 \sqrt {b} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 52, normalized size = 0.46 \[ -\frac {d \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b \left (d x^2+c\right )}{a d-b c}\right )}{\sqrt {c+d x^2} (a d-b c)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.62, size = 537, normalized size = 4.75 \[ \left [\frac {3 \, {\left (b d^{2} x^{4} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (3 \, b d x^{2} + b c + 2 \, a d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (b d^{2} x^{4} + a c d + {\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, {\left (3 \, b d x^{2} + b c + 2 \, a d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.39, size = 153, normalized size = 1.35 \[ -\frac {3 \, b d \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {3 \, {\left (d x^{2} + c\right )} b d - 2 \, b c d + 2 \, a d^{2}}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{2} + c} b c + \sqrt {d x^{2} + c} a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 989, normalized size = 8.75 \[ \frac {3 d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{4 \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {3 d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{4 \left (a d -b c \right )^{2} \sqrt {-\frac {a d -b c}{b}}}+\frac {3 \sqrt {-a b}\, d^{2} x}{4 \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, b c}-\frac {3 \sqrt {-a b}\, d^{2} x}{4 \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, b c}-\frac {3 d}{4 \left (a d -b c \right )^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}-\frac {3 d}{4 \left (a d -b c \right )^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}-\frac {\sqrt {-a b}\, d x}{2 \left (a d -b c \right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a b c}+\frac {\sqrt {-a b}\, d x}{2 \left (a d -b c \right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a b c}-\frac {\sqrt {-a b}}{4 \left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right ) \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a b}+\frac {\sqrt {-a b}}{4 \left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right ) \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.99, size = 130, normalized size = 1.15 \[ -\frac {\frac {d}{a\,d-b\,c}+\frac {3\,b\,d\,\left (d\,x^2+c\right )}{2\,{\left (a\,d-b\,c\right )}^2}}{b\,{\left (d\,x^2+c\right )}^{3/2}+\sqrt {d\,x^2+c}\,\left (a\,d-b\,c\right )}-\frac {3\,\sqrt {b}\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^{5/2}}\right )}{2\,{\left (a\,d-b\,c\right )}^{5/2}} \]
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]
________________________________________________________________________________________