Optimal. Leaf size=130 \[ \frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}+\frac {b \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.14, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 103, 156, 63, 208} \begin {gather*} \frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}+\frac {b \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 156
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2 \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {b c-a d+\frac {b d x}{2}}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a (b c-a d)}\\ &=\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 a^2}-\frac {(b (2 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^2 (b c-a d)}\\ &=\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{a^2 d}-\frac {(b (2 b c-3 a d)) \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 a^2 d (b c-a d)}\\ &=\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) \left (a+b x^2\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}+\frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 (b c-a d)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 123, normalized size = 0.95 \begin {gather*} \frac {\frac {a b \sqrt {c+d x^2}}{\left (a+b x^2\right ) (b c-a d)}+\frac {\sqrt {b} (2 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 144, normalized size = 1.11 \begin {gather*} \frac {\left (3 a \sqrt {b} d-2 b^{3/2} c\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2} \sqrt {a d-b c}}{b c-a d}\right )}{2 a^2 (a d-b c)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2 \sqrt {c}}-\frac {b \sqrt {c+d x^2}}{2 a \left (a+b x^2\right ) (a d-b c)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.86, size = 1037, normalized size = 7.98 \begin {gather*} \left [\frac {4 \, \sqrt {d x^{2} + c} a b c + {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\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 (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right )}{8 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}, \frac {4 \, \sqrt {d x^{2} + c} a b c + 8 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\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 )}{8 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}, \frac {2 \, \sqrt {d x^{2} + c} a b c - {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\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 (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right )}{4 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}, \frac {2 \, \sqrt {d x^{2} + c} a b c - {\left (2 \, a b c^{2} - 3 \, a^{2} c d + {\left (2 \, b^{2} c^{2} - 3 \, a b c d\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 ) + 4 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right )}{4 \, {\left (a^{3} b c^{2} - a^{4} c d + {\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 138, normalized size = 1.06 \begin {gather*} \frac {\sqrt {d x^{2} + c} b d}{2 \, {\left (a b c - a^{2} d\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} - \frac {{\left (2 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{2} b c - a^{3} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 838, normalized size = 6.45 \begin {gather*} \frac {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 ) \sqrt {-\frac {a d -b c}{b}}\, a}+\frac {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 ) \sqrt {-\frac {a d -b c}{b}}\, a}+\frac {\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}{4 \sqrt {-a b}\, \left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right ) a}-\frac {\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}{4 \sqrt {-a b}\, \left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right ) a}+\frac {\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 )}{2 \sqrt {-\frac {a d -b c}{b}}\, a^{2}}+\frac {\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 )}{2 \sqrt {-\frac {a d -b c}{b}}\, a^{2}}-\frac {\ln \left (\frac {2 c +2 \sqrt {d \,x^{2}+c}\, \sqrt {c}}{x}\right )}{a^{2} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.94, size = 3023, normalized size = 23.25
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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